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- How to Write a Research Proposal | Examples & Templates

## How to Write a Research Proposal | Examples & Templates

Published on October 12, 2022 by Shona McCombes and Tegan George. Revised on June 13, 2023.

A research proposal describes what you will investigate, why it’s important, and how you will conduct your research.

The format of a research proposal varies between fields, but most proposals will contain at least these elements:

## Introduction

Literature review.

- Research design

## Reference list

While the sections may vary, the overall objective is always the same. A research proposal serves as a blueprint and guide for your research plan, helping you get organized and feel confident in the path forward you choose to take.

## Table of contents

Research proposal purpose, research proposal examples, research design and methods, contribution to knowledge, research schedule, other interesting articles, frequently asked questions about research proposals.

Academics often have to write research proposals to get funding for their projects. As a student, you might have to write a research proposal as part of a grad school application , or prior to starting your thesis or dissertation .

In addition to helping you figure out what your research can look like, a proposal can also serve to demonstrate why your project is worth pursuing to a funder, educational institution, or supervisor.

## Research proposal length

The length of a research proposal can vary quite a bit. A bachelor’s or master’s thesis proposal can be just a few pages, while proposals for PhD dissertations or research funding are usually much longer and more detailed. Your supervisor can help you determine the best length for your work.

One trick to get started is to think of your proposal’s structure as a shorter version of your thesis or dissertation , only without the results , conclusion and discussion sections.

Download our research proposal template

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Writing a research proposal can be quite challenging, but a good starting point could be to look at some examples. We’ve included a few for you below.

- Example research proposal #1: “A Conceptual Framework for Scheduling Constraint Management”
- Example research proposal #2: “Medical Students as Mediators of Change in Tobacco Use”

Like your dissertation or thesis, the proposal will usually have a title page that includes:

- The proposed title of your project
- Your supervisor’s name
- Your institution and department

The first part of your proposal is the initial pitch for your project. Make sure it succinctly explains what you want to do and why.

Your introduction should:

- Introduce your topic
- Give necessary background and context
- Outline your problem statement and research questions

To guide your introduction , include information about:

- Who could have an interest in the topic (e.g., scientists, policymakers)
- How much is already known about the topic
- What is missing from this current knowledge
- What new insights your research will contribute
- Why you believe this research is worth doing

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As you get started, it’s important to demonstrate that you’re familiar with the most important research on your topic. A strong literature review shows your reader that your project has a solid foundation in existing knowledge or theory. It also shows that you’re not simply repeating what other people have already done or said, but rather using existing research as a jumping-off point for your own.

In this section, share exactly how your project will contribute to ongoing conversations in the field by:

- Comparing and contrasting the main theories, methods, and debates
- Examining the strengths and weaknesses of different approaches
- Explaining how will you build on, challenge, or synthesize prior scholarship

Following the literature review, restate your main objectives . This brings the focus back to your own project. Next, your research design or methodology section will describe your overall approach, and the practical steps you will take to answer your research questions.

To finish your proposal on a strong note, explore the potential implications of your research for your field. Emphasize again what you aim to contribute and why it matters.

For example, your results might have implications for:

- Improving best practices
- Informing policymaking decisions
- Strengthening a theory or model
- Challenging popular or scientific beliefs
- Creating a basis for future research

Last but not least, your research proposal must include correct citations for every source you have used, compiled in a reference list . To create citations quickly and easily, you can use our free APA citation generator .

Some institutions or funders require a detailed timeline of the project, asking you to forecast what you will do at each stage and how long it may take. While not always required, be sure to check the requirements of your project.

Here’s an example schedule to help you get started. You can also download a template at the button below.

Download our research schedule template

If you are applying for research funding, chances are you will have to include a detailed budget. This shows your estimates of how much each part of your project will cost.

Make sure to check what type of costs the funding body will agree to cover. For each item, include:

- Cost : exactly how much money do you need?
- Justification : why is this cost necessary to complete the research?
- Source : how did you calculate the amount?

To determine your budget, think about:

- Travel costs : do you need to go somewhere to collect your data? How will you get there, and how much time will you need? What will you do there (e.g., interviews, archival research)?
- Materials : do you need access to any tools or technologies?
- Help : do you need to hire any research assistants for the project? What will they do, and how much will you pay them?

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

Methodology

- Sampling methods
- Simple random sampling
- Stratified sampling
- Cluster sampling
- Likert scales
- Reproducibility

Statistics

- Null hypothesis
- Statistical power
- Probability distribution
- Effect size
- Poisson distribution

Research bias

- Optimism bias
- Cognitive bias
- Implicit bias
- Hawthorne effect
- Anchoring bias
- Explicit bias

Once you’ve decided on your research objectives , you need to explain them in your paper, at the end of your problem statement .

Keep your research objectives clear and concise, and use appropriate verbs to accurately convey the work that you will carry out for each one.

I will compare …

A research aim is a broad statement indicating the general purpose of your research project. It should appear in your introduction at the end of your problem statement , before your research objectives.

Research objectives are more specific than your research aim. They indicate the specific ways you’ll address the overarching aim.

A PhD, which is short for philosophiae doctor (doctor of philosophy in Latin), is the highest university degree that can be obtained. In a PhD, students spend 3–5 years writing a dissertation , which aims to make a significant, original contribution to current knowledge.

A PhD is intended to prepare students for a career as a researcher, whether that be in academia, the public sector, or the private sector.

A master’s is a 1- or 2-year graduate degree that can prepare you for a variety of careers.

All master’s involve graduate-level coursework. Some are research-intensive and intend to prepare students for further study in a PhD; these usually require their students to write a master’s thesis . Others focus on professional training for a specific career.

Critical thinking refers to the ability to evaluate information and to be aware of biases or assumptions, including your own.

Like information literacy , it involves evaluating arguments, identifying and solving problems in an objective and systematic way, and clearly communicating your ideas.

The best way to remember the difference between a research plan and a research proposal is that they have fundamentally different audiences. A research plan helps you, the researcher, organize your thoughts. On the other hand, a dissertation proposal or research proposal aims to convince others (e.g., a supervisor, a funding body, or a dissertation committee) that your research topic is relevant and worthy of being conducted.

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## Six Approaches to Justify Sample Sizes

Six ways to evaluate which effect sizes are interesting, the value of information, what is your inferential goal, additional considerations when designing an informative study, competing interests, data availability, sample size justification.

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Daniël Lakens; Sample Size Justification. Collabra: Psychology 5 January 2022; 8 (1): 33267. doi: https://doi.org/10.1525/collabra.33267

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An important step when designing an empirical study is to justify the sample size that will be collected. The key aim of a sample size justification for such studies is to explain how the collected data is expected to provide valuable information given the inferential goals of the researcher. In this overview article six approaches are discussed to justify the sample size in a quantitative empirical study: 1) collecting data from (almost) the entire population, 2) choosing a sample size based on resource constraints, 3) performing an a-priori power analysis, 4) planning for a desired accuracy, 5) using heuristics, or 6) explicitly acknowledging the absence of a justification. An important question to consider when justifying sample sizes is which effect sizes are deemed interesting, and the extent to which the data that is collected informs inferences about these effect sizes. Depending on the sample size justification chosen, researchers could consider 1) what the smallest effect size of interest is, 2) which minimal effect size will be statistically significant, 3) which effect sizes they expect (and what they base these expectations on), 4) which effect sizes would be rejected based on a confidence interval around the effect size, 5) which ranges of effects a study has sufficient power to detect based on a sensitivity power analysis, and 6) which effect sizes are expected in a specific research area. Researchers can use the guidelines presented in this article, for example by using the interactive form in the accompanying online Shiny app, to improve their sample size justification, and hopefully, align the informational value of a study with their inferential goals.

Scientists perform empirical studies to collect data that helps to answer a research question. The more data that is collected, the more informative the study will be with respect to its inferential goals. A sample size justification should consider how informative the data will be given an inferential goal, such as estimating an effect size, or testing a hypothesis. Even though a sample size justification is sometimes requested in manuscript submission guidelines, when submitting a grant to a funder, or submitting a proposal to an ethical review board, the number of observations is often simply stated , but not justified . This makes it difficult to evaluate how informative a study will be. To prevent such concerns from emerging when it is too late (e.g., after a non-significant hypothesis test has been observed), researchers should carefully justify their sample size before data is collected.

Researchers often find it difficult to justify their sample size (i.e., a number of participants, observations, or any combination thereof). In this review article six possible approaches are discussed that can be used to justify the sample size in a quantitative study (see Table 1 ). This is not an exhaustive overview, but it includes the most common and applicable approaches for single studies. 1 The first justification is that data from (almost) the entire population has been collected. The second justification centers on resource constraints, which are almost always present, but rarely explicitly evaluated. The third and fourth justifications are based on a desired statistical power or a desired accuracy. The fifth justification relies on heuristics, and finally, researchers can choose a sample size without any justification. Each of these justifications can be stronger or weaker depending on which conclusions researchers want to draw from the data they plan to collect.

All of these approaches to the justification of sample sizes, even the ‘no justification’ approach, give others insight into the reasons that led to the decision for a sample size in a study. It should not be surprising that the ‘heuristics’ and ‘no justification’ approaches are often unlikely to impress peers. However, it is important to note that the value of the information that is collected depends on the extent to which the final sample size allows a researcher to achieve their inferential goals, and not on the sample size justification that is chosen.

The extent to which these approaches make other researchers judge the data that is collected as informative depends on the details of the question a researcher aimed to answer and the parameters they chose when determining the sample size for their study. For example, a badly performed a-priori power analysis can quickly lead to a study with very low informational value. These six justifications are not mutually exclusive, and multiple approaches can be considered when designing a study.

The informativeness of the data that is collected depends on the inferential goals a researcher has, or in some cases, the inferential goals scientific peers will have. A shared feature of the different inferential goals considered in this review article is the question which effect sizes a researcher considers meaningful to distinguish. This implies that researchers need to evaluate which effect sizes they consider interesting. These evaluations rely on a combination of statistical properties and domain knowledge. In Table 2 six possibly useful considerations are provided. This is not intended to be an exhaustive overview, but it presents common and useful approaches that can be applied in practice. Not all evaluations are equally relevant for all types of sample size justifications. The online Shiny app accompanying this manuscript provides researchers with an interactive form that guides researchers through the considerations for a sample size justification. These considerations often rely on the same information (e.g., effect sizes, the number of observations, the standard deviation, etc.) so these six considerations should be seen as a set of complementary approaches that can be used to evaluate which effect sizes are of interest.

To start, researchers should consider what their smallest effect size of interest is. Second, although only relevant when performing a hypothesis test, researchers should consider which effect sizes could be statistically significant given a choice of an alpha level and sample size. Third, it is important to consider the (range of) effect sizes that are expected. This requires a careful consideration of the source of this expectation and the presence of possible biases in these expectations. Fourth, it is useful to consider the width of the confidence interval around possible values of the effect size in the population, and whether we can expect this confidence interval to reject effects we considered a-priori plausible. Fifth, it is worth evaluating the power of the test across a wide range of possible effect sizes in a sensitivity power analysis. Sixth, a researcher can consider the effect size distribution of related studies in the literature.

Since all scientists are faced with resource limitations, they need to balance the cost of collecting each additional datapoint against the increase in information that datapoint provides. This is referred to as the value of information (Eckermann et al., 2010) . Calculating the value of information is notoriously difficult (Detsky, 1990) . Researchers need to specify the cost of collecting data, and weigh the costs of data collection against the increase in utility that having access to the data provides. From a value of information perspective not every data point that can be collected is equally valuable (J. Halpern et al., 2001; Wilson, 2015) . Whenever additional observations do not change inferences in a meaningful way, the costs of data collection can outweigh the benefits.

The value of additional information will in most cases be a non-monotonic function, especially when it depends on multiple inferential goals. A researcher might be interested in comparing an effect against a previously observed large effect in the literature, a theoretically predicted medium effect, and the smallest effect that would be practically relevant. In such a situation the expected value of sampling information will lead to different optimal sample sizes for each inferential goal. It could be valuable to collect informative data about a large effect, with additional data having less (or even a negative) marginal utility, up to a point where the data becomes increasingly informative about a medium effect size, with the value of sampling additional information decreasing once more until the study becomes increasingly informative about the presence or absence of a smallest effect of interest.

Because of the difficulty of quantifying the value of information, scientists typically use less formal approaches to justify the amount of data they set out to collect in a study. Even though the cost-benefit analysis is not always made explicit in reported sample size justifications, the value of information perspective is almost always implicitly the underlying framework that sample size justifications are based on. Throughout the subsequent discussion of sample size justifications, the importance of considering the value of information given inferential goals will repeatedly be highlighted.

## Measuring (Almost) the Entire Population

In some instances it might be possible to collect data from (almost) the entire population under investigation. For example, researchers might use census data, are able to collect data from all employees at a firm or study a small population of top athletes. Whenever it is possible to measure the entire population, the sample size justification becomes straightforward: the researcher used all the data that is available.

## Resource Constraints

A common reason for the number of observations in a study is that resource constraints limit the amount of data that can be collected at a reasonable cost (Lenth, 2001) . In practice, sample sizes are always limited by the resources that are available. Researchers practically always have resource limitations, and therefore even when resource constraints are not the primary justification for the sample size in a study, it is always a secondary justification.

Despite the omnipresence of resource limitations, the topic often receives little attention in texts on experimental design (for an example of an exception, see Bulus and Dong (2021) ). This might make it feel like acknowledging resource constraints is not appropriate, but the opposite is true: Because resource limitations always play a role, a responsible scientist carefully evaluates resource constraints when designing a study. Resource constraint justifications are based on a trade-off between the costs of data collection, and the value of having access to the information the data provides. Even if researchers do not explicitly quantify this trade-off, it is revealed in their actions. For example, researchers rarely spend all the resources they have on a single study. Given resource constraints, researchers are confronted with an optimization problem of how to spend resources across multiple research questions.

Time and money are two resource limitations all scientists face. A PhD student has a certain time to complete a PhD thesis, and is typically expected to complete multiple research lines in this time. In addition to time limitations, researchers have limited financial resources that often directly influence how much data can be collected. A third limitation in some research lines is that there might simply be a very small number of individuals from whom data can be collected, such as when studying patients with a rare disease. A resource constraint justification puts limited resources at the center of the justification for the sample size that will be collected, and starts with the resources a scientist has available. These resources are translated into an expected number of observations ( N ) that a researcher expects they will be able to collect with an amount of money in a given time. The challenge is to evaluate whether collecting N observations is worthwhile. How do we decide if a study will be informative, and when should we conclude that data collection is not worthwhile?

When evaluating whether resource constraints make data collection uninformative, researchers need to explicitly consider which inferential goals they have when collecting data (Parker & Berman, 2003) . Having data always provides more knowledge about the research question than not having data, so in an absolute sense, all data that is collected has value. However, it is possible that the benefits of collecting the data are outweighed by the costs of data collection.

It is most straightforward to evaluate whether data collection has value when we know for certain that someone will make a decision, with or without data. In such situations any additional data will reduce the error rates of a well-calibrated decision process, even if only ever so slightly. For example, without data we will not perform better than a coin flip if we guess which of two conditions has a higher true mean score on a measure. With some data, we can perform better than a coin flip by picking the condition that has the highest mean. With a small amount of data we would still very likely make a mistake, but the error rate is smaller than without any data. In these cases, the value of information might be positive, as long as the reduction in error rates is more beneficial than the cost of data collection.

Another way in which a small dataset can be valuable is if its existence eventually makes it possible to perform a meta-analysis (Maxwell & Kelley, 2011) . This argument in favor of collecting a small dataset requires 1) that researchers share the data in a way that a future meta-analyst can find it, and 2) that there is a decent probability that someone will perform a high-quality meta-analysis that will include this data in the future (S. D. Halpern et al., 2002) . The uncertainty about whether there will ever be such a meta-analysis should be weighed against the costs of data collection.

One way to increase the probability of a future meta-analysis is if researchers commit to performing this meta-analysis themselves, by combining several studies they have performed into a small-scale meta-analysis (Cumming, 2014) . For example, a researcher might plan to repeat a study for the next 12 years in a class they teach, with the expectation that after 12 years a meta-analysis of 12 studies would be sufficient to draw informative inferences (but see ter Schure and Grünwald (2019) ). If it is not plausible that a researcher will collect all the required data by themselves, they can attempt to set up a collaboration where fellow researchers in their field commit to collecting similar data with identical measures. If it is not likely that sufficient data will emerge over time to reach the inferential goals, there might be no value in collecting the data.

Even if a researcher believes it is worth collecting data because a future meta-analysis will be performed, they will most likely perform a statistical test on the data. To make sure their expectations about the results of such a test are well-calibrated, it is important to consider which effect sizes are of interest, and to perform a sensitivity power analysis to evaluate the probability of a Type II error for effects of interest. From the six ways to evaluate which effect sizes are interesting that will be discussed in the second part of this review, it is useful to consider the smallest effect size that can be statistically significant, the expected width of the confidence interval around the effect size, and effects that can be expected in a specific research area, and to evaluate the power for these effect sizes in a sensitivity power analysis. If a decision or claim is made, a compromise power analysis is worthwhile to consider when deciding upon the error rates while planning the study. When reporting a resource constraints sample size justification it is recommended to address the five considerations in Table 3 . Addressing these points explicitly facilitates evaluating if the data is worthwhile to collect. To make it easier to address all relevant points explicitly, an interactive form to implement the recommendations in this manuscript can be found at https://shiny.ieis.tue.nl/sample_size_justification/ .

## A-priori Power Analysis

When designing a study where the goal is to test whether a statistically significant effect is present, researchers often want to make sure their sample size is large enough to prevent erroneous conclusions for a range of effect sizes they care about. In this approach to justifying a sample size, the value of information is to collect observations up to the point that the probability of an erroneous inference is, in the long run, not larger than a desired value. If a researcher performs a hypothesis test, there are four possible outcomes:

A false positive (or Type I error), determined by the α level. A test yields a significant result, even though the null hypothesis is true.

A false negative (or Type II error), determined by β , or 1 - power. A test yields a non-significant result, even though the alternative hypothesis is true.

A true negative, determined by 1- α . A test yields a non-significant result when the null hypothesis is true.

A true positive, determined by 1- β . A test yields a significant result when the alternative hypothesis is true.

Given a specified effect size, alpha level, and power, an a-priori power analysis can be used to calculate the number of observations required to achieve the desired error rates, given the effect size. 3 Figure 1 illustrates how the statistical power increases as the number of observations (per group) increases in an independent t test with a two-sided alpha level of 0.05. If we are interested in detecting an effect of d = 0.5, a sample size of 90 per condition would give us more than 90% power. Statistical power can be computed to determine the number of participants, or the number of items (Westfall et al., 2014) but can also be performed for single case studies (Ferron & Onghena, 1996; McIntosh & Rittmo, 2020)

Although it is common to set the Type I error rate to 5% and aim for 80% power, error rates should be justified (Lakens, Adolfi, et al., 2018) . As explained in the section on compromise power analysis, the default recommendation to aim for 80% power lacks a solid justification. In general, the lower the error rates (and thus the higher the power), the more informative a study will be, but the more resources are required. Researchers should carefully weigh the costs of increasing the sample size against the benefits of lower error rates, which would probably make studies designed to achieve a power of 90% or 95% more common for articles reporting a single study. An additional consideration is whether the researcher plans to publish an article consisting of a set of replication and extension studies, in which case the probability of observing multiple Type I errors will be very low, but the probability of observing mixed results even when there is a true effect increases (Lakens & Etz, 2017) , which would also be a reason to aim for studies with low Type II error rates, perhaps even by slightly increasing the alpha level for each individual study.

Figure 2 visualizes two distributions. The left distribution (dashed line) is centered at 0. This is a model for the null hypothesis. If the null hypothesis is true a statistically significant result will be observed if the effect size is extreme enough (in a two-sided test either in the positive or negative direction), but any significant result would be a Type I error (the dark grey areas under the curve). If there is no true effect, formally statistical power for a null hypothesis significance test is undefined. Any significant effects observed if the null hypothesis is true are Type I errors, or false positives, which occur at the chosen alpha level. The right distribution (solid line) is centered on an effect of d = 0.5. This is the specified model for the alternative hypothesis in this study, illustrating the expectation of an effect of d = 0.5 if the alternative hypothesis is true. Even though there is a true effect, studies will not always find a statistically significant result. This happens when, due to random variation, the observed effect size is too close to 0 to be statistically significant. Such results are false negatives (the light grey area under the curve on the right). To increase power, we can collect a larger sample size. As the sample size increases, the distributions become more narrow, reducing the probability of a Type II error. 4

It is important to highlight that the goal of an a-priori power analysis is not to achieve sufficient power for the true effect size. The true effect size is unknown. The goal of an a-priori power analysis is to achieve sufficient power, given a specific assumption of the effect size a researcher wants to detect. Just like a Type I error rate is the maximum probability of making a Type I error conditional on the assumption that the null hypothesis is true, an a-priori power analysis is computed under the assumption of a specific effect size. It is unknown if this assumption is correct. All a researcher can do is to make sure their assumptions are well justified. Statistical inferences based on a test where the Type II error rate is controlled are conditional on the assumption of a specific effect size. They allow the inference that, assuming the true effect size is at least as large as that used in the a-priori power analysis, the maximum Type II error rate in a study is not larger than a desired value.

This point is perhaps best illustrated if we consider a study where an a-priori power analysis is performed both for a test of the presence of an effect, as for a test of the absence of an effect. When designing a study, it essential to consider the possibility that there is no effect (e.g., a mean difference of zero). An a-priori power analysis can be performed both for a null hypothesis significance test, as for a test of the absence of a meaningful effect, such as an equivalence test that can statistically provide support for the null hypothesis by rejecting the presence of effects that are large enough to matter (Lakens, 2017; Meyners, 2012; Rogers et al., 1993) . When multiple primary tests will be performed based on the same sample, each analysis requires a dedicated sample size justification. If possible, a sample size is collected that guarantees that all tests are informative, which means that the collected sample size is based on the largest sample size returned by any of the a-priori power analyses.

For example, if the goal of a study is to detect or reject an effect size of d = 0.4 with 90% power, and the alpha level is set to 0.05 for a two-sided independent t test, a researcher would need to collect 133 participants in each condition for an informative null hypothesis test, and 136 participants in each condition for an informative equivalence test. Therefore, the researcher should aim to collect 272 participants in total for an informative result for both tests that are planned. This does not guarantee a study has sufficient power for the true effect size (which can never be known), but it guarantees the study has sufficient power given an assumption of the effect a researcher is interested in detecting or rejecting. Therefore, an a-priori power analysis is useful, as long as a researcher can justify the effect sizes they are interested in.

If researchers correct the alpha level when testing multiple hypotheses, the a-priori power analysis should be based on this corrected alpha level. For example, if four tests are performed, an overall Type I error rate of 5% is desired, and a Bonferroni correction is used, the a-priori power analysis should be based on a corrected alpha level of .0125.

An a-priori power analysis can be performed analytically, or by performing computer simulations. Analytic solutions are faster but less flexible. A common challenge researchers face when attempting to perform power analyses for more complex or uncommon tests is that available software does not offer analytic solutions. In these cases simulations can provide a flexible solution to perform power analyses for any test (Morris et al., 2019) . The following code is an example of a power analysis in R based on 10000 simulations for a one-sample t test against zero for a sample size of 20, assuming a true effect of d = 0.5. All simulations consist of first randomly generating data based on assumptions of the data generating mechanism (e.g., a normal distribution with a mean of 0.5 and a standard deviation of 1), followed by a test performed on the data. By computing the percentage of significant results, power can be computed for any design.

p <- numeric(10000) # to store p-values for (i in 1:10000) { #simulate 10k tests x <- rnorm(n = 20, mean = 0.5, sd = 1) p[i] <- t.test(x)$p.value # store p-value } sum(p < 0.05) / 10000 # Compute power

There is a wide range of tools available to perform power analyses. Whichever tool a researcher decides to use, it will take time to learn how to use the software correctly to perform a meaningful a-priori power analysis. Resources to educate psychologists about power analysis consist of book-length treatments (Aberson, 2019; Cohen, 1988; Julious, 2004; Murphy et al., 2014) , general introductions (Baguley, 2004; Brysbaert, 2019; Faul et al., 2007; Maxwell et al., 2008; Perugini et al., 2018) , and an increasing number of applied tutorials for specific tests (Brysbaert & Stevens, 2018; DeBruine & Barr, 2019; P. Green & MacLeod, 2016; Kruschke, 2013; Lakens & Caldwell, 2021; Schoemann et al., 2017; Westfall et al., 2014) . It is important to be trained in the basics of power analysis, and it can be extremely beneficial to learn how to perform simulation-based power analyses. At the same time, it is often recommended to enlist the help of an expert, especially when a researcher lacks experience with a power analysis for a specific test.

When reporting an a-priori power analysis, make sure that the power analysis is completely reproducible. If power analyses are performed in R it is possible to share the analysis script and information about the version of the package. In many software packages it is possible to export the power analysis that is performed as a PDF file. For example, in G*Power analyses can be exported under the ‘protocol of power analysis’ tab. If the software package provides no way to export the analysis, add a screenshot of the power analysis to the supplementary files.

The reproducible report needs to be accompanied by justifications for the choices that were made with respect to the values used in the power analysis. If the effect size used in the power analysis is based on previous research the factors presented in Table 5 (if the effect size is based on a meta-analysis) or Table 6 (if the effect size is based on a single study) should be discussed. If an effect size estimate is based on the existing literature, provide a full citation, and preferably a direct quote from the article where the effect size estimate is reported. If the effect size is based on a smallest effect size of interest, this value should not just be stated, but justified (e.g., based on theoretical predictions or practical implications, see Lakens, Scheel, and Isager (2018) ). For an overview of all aspects that should be reported when describing an a-priori power analysis, see Table 4 .

## Planning for Precision

Some researchers have suggested to justify sample sizes based on a desired level of precision of the estimate (Cumming & Calin-Jageman, 2016; Kruschke, 2018; Maxwell et al., 2008) . The goal when justifying a sample size based on precision is to collect data to achieve a desired width of the confidence interval around a parameter estimate. The width of the confidence interval around the parameter estimate depends on the standard deviation and the number of observations. The only aspect a researcher needs to justify for a sample size justification based on accuracy is the desired width of the confidence interval with respect to their inferential goal, and their assumption about the population standard deviation of the measure.

If a researcher has determined the desired accuracy, and has a good estimate of the true standard deviation of the measure, it is straightforward to calculate the sample size needed for a desired level of accuracy. For example, when measuring the IQ of a group of individuals a researcher might desire to estimate the IQ score within an error range of 2 IQ points for 95% of the observed means, in the long run. The required sample size to achieve this desired level of accuracy (assuming normally distributed data) can be computed by:

where N is the number of observations, z is the critical value related to the desired confidence interval, sd is the standard deviation of IQ scores in the population, and error is the width of the confidence interval within which the mean should fall, with the desired error rate. In this example, (1.96 × 15 / 2)^2 = 216.1 observations. If a researcher desires 95% of the means to fall within a 2 IQ point range around the true population mean, 217 observations should be collected. If a desired accuracy for a non-zero mean difference is computed, accuracy is based on a non-central t -distribution. For these calculations an expected effect size estimate needs to be provided, but it has relatively little influence on the required sample size (Maxwell et al., 2008) . It is also possible to incorporate uncertainty about the observed effect size in the sample size calculation, known as assurance (Kelley & Rausch, 2006) . The MBESS package in R provides functions to compute sample sizes for a wide range of tests (Kelley, 2007) .

What is less straightforward is to justify how a desired level of accuracy is related to inferential goals. There is no literature that helps researchers to choose a desired width of the confidence interval. Morey (2020) convincingly argues that most practical use-cases of planning for precision involve an inferential goal of distinguishing an observed effect from other effect sizes (for a Bayesian perspective, see Kruschke (2018) ). For example, a researcher might expect an effect size of r = 0.4 and would treat observed correlations that differ more than 0.2 (i.e., 0.2 < r < 0.6) differently, in that effects of r = 0.6 or larger are considered too large to be caused by the assumed underlying mechanism (Hilgard, 2021) , while effects smaller than r = 0.2 are considered too small to support the theoretical prediction. If the goal is indeed to get an effect size estimate that is precise enough so that two effects can be differentiated with high probability, the inferential goal is actually a hypothesis test, which requires designing a study with sufficient power to reject effects (e.g., testing a range prediction of correlations between 0.2 and 0.6).

If researchers do not want to test a hypothesis, for example because they prefer an estimation approach over a testing approach, then in the absence of clear guidelines that help researchers to justify a desired level of precision, one solution might be to rely on a generally accepted norm of precision to aim for. This norm could be based on ideas about a certain resolution below which measurements in a research area no longer lead to noticeably different inferences. Just as researchers normatively use an alpha level of 0.05, they could plan studies to achieve a desired confidence interval width around the observed effect that is determined by a norm. Future work is needed to help researchers choose a confidence interval width when planning for accuracy.

When a researcher uses a heuristic, they are not able to justify their sample size themselves, but they trust in a sample size recommended by some authority. When I started as a PhD student in 2005 it was common to collect 15 participants in each between subject condition. When asked why this was a common practice, no one was really sure, but people trusted there was a justification somewhere in the literature. Now, I realize there was no justification for the heuristics we used. As Berkeley (1735) already observed: “Men learn the elements of science from others: And every learner hath a deference more or less to authority, especially the young learners, few of that kind caring to dwell long upon principles, but inclining rather to take them upon trust: And things early admitted by repetition become familiar: And this familiarity at length passeth for evidence.”

Some papers provide researchers with simple rules of thumb about the sample size that should be collected. Such papers clearly fill a need, and are cited a lot, even when the advice in these articles is flawed. For example, Wilson VanVoorhis and Morgan (2007) translate an absolute minimum of 50+8 observations for regression analyses suggested by a rule of thumb examined in S. B. Green (1991) into the recommendation to collect ~50 observations. Green actually concludes in his article that “In summary, no specific minimum number of subjects or minimum ratio of subjects-to-predictors was supported”. He does discuss how a general rule of thumb of N = 50 + 8 provided an accurate minimum number of observations for the ‘typical’ study in the social sciences because these have a ‘medium’ effect size, as Green claims by citing Cohen (1988) . Cohen actually didn’t claim that the typical study in the social sciences has a ‘medium’ effect size, and instead said (1988, p. 13) : “Many effects sought in personality, social, and clinical-psychological research are likely to be small effects as here defined”. We see how a string of mis-citations eventually leads to a misleading rule of thumb.

Rules of thumb seem to primarily emerge due to mis-citations and/or overly simplistic recommendations. Simonsohn, Nelson, and Simmons (2011) recommended that “Authors must collect at least 20 observations per cell”. A later recommendation by the same authors presented at a conference suggested to use n > 50, unless you study large effects (Simmons et al., 2013) . Regrettably, this advice is now often mis-cited as a justification to collect no more than 50 observations per condition without considering the expected effect size. If authors justify a specific sample size (e.g., n = 50) based on a general recommendation in another paper, either they are mis-citing the paper, or the paper they are citing is flawed.

Another common heuristic is to collect the same number of observations as were collected in a previous study. This strategy is not recommended in scientific disciplines with widespread publication bias, and/or where novel and surprising findings from largely exploratory single studies are published. Using the same sample size as a previous study is only a valid approach if the sample size justification in the previous study also applies to the current study. Instead of stating that you intend to collect the same sample size as an earlier study, repeat the sample size justification, and update it in light of any new information (such as the effect size in the earlier study, see Table 6 ).

Peer reviewers and editors should carefully scrutinize rules of thumb sample size justifications, because they can make it seem like a study has high informational value for an inferential goal even when the study will yield uninformative results. Whenever one encounters a sample size justification based on a heuristic, ask yourself: ‘Why is this heuristic used?’ It is important to know what the logic behind a heuristic is to determine whether the heuristic is valid for a specific situation. In most cases, heuristics are based on weak logic, and not widely applicable. It might be possible that fields develop valid heuristics for sample size justifications. For example, it is possible that a research area reaches widespread agreement that effects smaller than d = 0.3 are too small to be of interest, and all studies in a field use sequential designs (see below) that have 90% power to detect a d = 0.3. Alternatively, it is possible that a field agrees that data should be collected with a desired level of accuracy, irrespective of the true effect size. In these cases, valid heuristics would exist based on generally agreed goals of data collection. For example, Simonsohn (2015) suggests to design replication studies that have 2.5 times as large sample sizes as the original study, as this provides 80% power for an equivalence test against an equivalence bound set to the effect the original study had 33% power to detect, assuming the true effect size is 0. As original authors typically do not specify which effect size would falsify their hypothesis, the heuristic underlying this ‘small telescopes’ approach is a good starting point for a replication study with the inferential goal to reject the presence of an effect as large as was described in an earlier publication. It is the responsibility of researchers to gain the knowledge to distinguish valid heuristics from mindless heuristics, and to be able to evaluate whether a heuristic will yield an informative result given the inferential goal of the researchers in a specific study, or not.

## No Justification

It might sound like a contradictio in terminis , but it is useful to distinguish a final category where researchers explicitly state they do not have a justification for their sample size. Perhaps the resources were available to collect more data, but they were not used. A researcher could have performed a power analysis, or planned for precision, but they did not. In those cases, instead of pretending there was a justification for the sample size, honesty requires you to state there is no sample size justification. This is not necessarily bad. It is still possible to discuss the smallest effect size of interest, the minimal statistically detectable effect, the width of the confidence interval around the effect size, and to plot a sensitivity power analysis, in relation to the sample size that was collected. If a researcher truly had no specific inferential goals when collecting the data, such an evaluation can perhaps be performed based on reasonable inferential goals peers would have when they learn about the existence of the collected data.

Do not try to spin a story where it looks like a study was highly informative when it was not. Instead, transparently evaluate how informative the study was given effect sizes that were of interest, and make sure that the conclusions follow from the data. The lack of a sample size justification might not be problematic, but it might mean that a study was not informative for most effect sizes of interest, which makes it especially difficult to interpret non-significant effects, or estimates with large uncertainty.

The inferential goal of data collection is often in some way related to the size of an effect. Therefore, to design an informative study, researchers will want to think about which effect sizes are interesting. First, it is useful to consider three effect sizes when determining the sample size. The first is the smallest effect size a researcher is interested in, the second is the smallest effect size that can be statistically significant (only in studies where a significance test will be performed), and the third is the effect size that is expected. Beyond considering these three effect sizes, it can be useful to evaluate ranges of effect sizes. This can be done by computing the width of the expected confidence interval around an effect size of interest (for example, an effect size of zero), and examine which effects could be rejected. Similarly, it can be useful to plot a sensitivity curve and evaluate the range of effect sizes the design has decent power to detect, as well as to consider the range of effects for which the design has low power. Finally, there are situations where it is useful to consider a range of effect that is likely to be observed in a specific research area.

## What is the Smallest Effect Size of Interest?

The strongest possible sample size justification is based on an explicit statement of the smallest effect size that is considered interesting. A smallest effect size of interest can be based on theoretical predictions or practical considerations. For a review of approaches that can be used to determine a smallest effect size of interest in randomized controlled trials, see Cook et al. (2014) and Keefe et al. (2013) , for reviews of different methods to determine a smallest effect size of interest, see King (2011) and Copay, Subach, Glassman, Polly, and Schuler (2007) , and for a discussion focused on psychological research, see Lakens, Scheel, et al. (2018) .

It can be challenging to determine the smallest effect size of interest whenever theories are not very developed, or when the research question is far removed from practical applications, but it is still worth thinking about which effects would be too small to matter. A first step forward is to discuss which effect sizes are considered meaningful in a specific research line with your peers. Researchers will differ in the effect sizes they consider large enough to be worthwhile (Murphy et al., 2014) . Just as not every scientist will find every research question interesting enough to study, not every scientist will consider the same effect sizes interesting enough to study, and different stakeholders will differ in which effect sizes are considered meaningful (Kelley & Preacher, 2012) .

Even though it might be challenging, there are important benefits of being able to specify a smallest effect size of interest. The population effect size is always uncertain (indeed, estimating this is typically one of the goals of the study), and therefore whenever a study is powered for an expected effect size, there is considerable uncertainty about whether the statistical power is high enough to detect the true effect in the population. However, if the smallest effect size of interest can be specified and agreed upon after careful deliberation, it becomes possible to design a study that has sufficient power (given the inferential goal to detect or reject the smallest effect size of interest with a certain error rate). A smallest effect of interest may be subjective (one researcher might find effect sizes smaller than d = 0.3 meaningless, while another researcher might still be interested in effects larger than d = 0.1), and there might be uncertainty about the parameters required to specify the smallest effect size of interest (e.g., when performing a cost-benefit analysis), but after a smallest effect size of interest has been determined, a study can be designed with a known Type II error rate to detect or reject this value. For this reason an a-priori power based on a smallest effect size of interest is generally preferred, whenever researchers are able to specify one (Aberson, 2019; Albers & Lakens, 2018; Brown, 1983; Cascio & Zedeck, 1983; Dienes, 2014; Lenth, 2001) .

## The Minimal Statistically Detectable Effect

The minimal statistically detectable effect, or the critical effect size, provides information about the smallest effect size that, if observed, would be statistically significant given a specified alpha level and sample size (Cook et al., 2014) . For any critical t value (e.g., t = 1.96 for α = 0.05, for large sample sizes) we can compute a critical mean difference (Phillips et al., 2001) , or a critical standardized effect size. For a two-sided independent t test the critical mean difference is:

and the critical standardized mean difference is:

In Figure 4 the distribution of Cohen’s d is plotted for 15 participants per group when the true effect size is either d = 0 or d = 0.5. This figure is similar to Figure 2 , with the addition that the critical d is indicated. We see that with such a small number of observations in each group only observed effects larger than d = 0.75 will be statistically significant. Whether such effect sizes are interesting, and can realistically be expected, should be carefully considered and justified.

G*Power provides the critical test statistic (such as the critical t value) when performing a power analysis. For example, Figure 5 shows that for a correlation based on a two-sided test, with α = 0.05, and N = 30, only effects larger than r = 0.361 or smaller than r = -0.361 can be statistically significant. This reveals that when the sample size is relatively small, the observed effect needs to be quite substantial to be statistically significant.

It is important to realize that due to random variation each study has a probability to yield effects larger than the critical effect size, even if the true effect size is small (or even when the true effect size is 0, in which case each significant effect is a Type I error). Computing a minimal statistically detectable effect is useful for a study where no a-priori power analysis is performed, both for studies in the published literature that do not report a sample size justification (Lakens, Scheel, et al., 2018) , as for researchers who rely on heuristics for their sample size justification.

It can be informative to ask yourself whether the critical effect size for a study design is within the range of effect sizes that can realistically be expected. If not, then whenever a significant effect is observed in a published study, either the effect size is surprisingly larger than expected, or more likely, it is an upwardly biased effect size estimate. In the latter case, given publication bias, published studies will lead to biased effect size estimates. If it is still possible to increase the sample size, for example by ignoring rules of thumb and instead performing an a-priori power analysis, then do so. If it is not possible to increase the sample size, for example due to resource constraints, then reflecting on the minimal statistically detectable effect should make it clear that an analysis of the data should not focus on p values, but on the effect size and the confidence interval (see Table 3 ).

It is also useful to compute the minimal statistically detectable effect if an ‘optimistic’ power analysis is performed. For example, if you believe a best case scenario for the true effect size is d = 0.57 and use this optimistic expectation in an a-priori power analysis, effects smaller than d = 0.4 will not be statistically significant when you collect 50 observations in a two independent group design. If your worst case scenario for the alternative hypothesis is a true effect size of d = 0.35 your design would not allow you to declare a significant effect if effect size estimates close to the worst case scenario are observed. Taking into account the minimal statistically detectable effect size should make you reflect on whether a hypothesis test will yield an informative answer, and whether your current approach to sample size justification (e.g., the use of rules of thumb, or letting resource constraints determine the sample size) leads to an informative study, or not.

## What is the Expected Effect Size?

Although the true population effect size is always unknown, there are situations where researchers have a reasonable expectation of the effect size in a study, and want to use this expected effect size in an a-priori power analysis. Even if expectations for the observed effect size are largely a guess, it is always useful to explicitly consider which effect sizes are expected. A researcher can justify a sample size based on the effect size they expect, even if such a study would not be very informative with respect to the smallest effect size of interest. In such cases a study is informative for one inferential goal (testing whether the expected effect size is present or absent), but not highly informative for the second goal (testing whether the smallest effect size of interest is present or absent).

There are typically three sources for expectations about the population effect size: a meta-analysis, a previous study, or a theoretical model. It is tempting for researchers to be overly optimistic about the expected effect size in an a-priori power analysis, as higher effect size estimates yield lower sample sizes, but being too optimistic increases the probability of observing a false negative result. When reviewing a sample size justification based on an a-priori power analysis, it is important to critically evaluate the justification for the expected effect size used in power analyses.

## Using an Estimate from a Meta-Analysis

In a perfect world effect size estimates from a meta-analysis would provide researchers with the most accurate information about which effect size they could expect. Due to widespread publication bias in science, effect size estimates from meta-analyses are regrettably not always accurate. They can be biased, sometimes substantially so. Furthermore, meta-analyses typically have considerable heterogeneity, which means that the meta-analytic effect size estimate differs for subsets of studies that make up the meta-analysis. So, although it might seem useful to use a meta-analytic effect size estimate of the effect you are studying in your power analysis, you need to take great care before doing so.

If a researcher wants to enter a meta-analytic effect size estimate in an a-priori power analysis, they need to consider three things (see Table 5 ). First, the studies included in the meta-analysis should be similar enough to the study they are performing that it is reasonable to expect a similar effect size. In essence, this requires evaluating the generalizability of the effect size estimate to the new study. It is important to carefully consider differences between the meta-analyzed studies and the planned study, with respect to the manipulation, the measure, the population, and any other relevant variables.

Second, researchers should check whether the effect sizes reported in the meta-analysis are homogeneous. If not, and there is considerable heterogeneity in the meta-analysis, it means not all included studies can be expected to have the same true effect size estimate. A meta-analytic estimate should be used based on the subset of studies that most closely represent the planned study. Note that heterogeneity remains a possibility (even direct replication studies can show heterogeneity when unmeasured variables moderate the effect size in each sample (Kenny & Judd, 2019; Olsson-Collentine et al., 2020) ), so the main goal of selecting similar studies is to use existing data to increase the probability that your expectation is accurate, without guaranteeing it will be.

Third, the meta-analytic effect size estimate should not be biased. Check if the bias detection tests that are reported in the meta-analysis are state-of-the-art, or perform multiple bias detection tests yourself (Carter et al., 2019) , and consider bias corrected effect size estimates (even though these estimates might still be biased, and do not necessarily reflect the true population effect size).

## Using an Estimate from a Previous Study

If a meta-analysis is not available, researchers often rely on an effect size from a previous study in an a-priori power analysis. The first issue that requires careful attention is whether the two studies are sufficiently similar. Just as when using an effect size estimate from a meta-analysis, researchers should consider if there are differences between the studies in terms of the population, the design, the manipulations, the measures, or other factors that should lead one to expect a different effect size. For example, intra-individual reaction time variability increases with age, and therefore a study performed on older participants should expect a smaller standardized effect size than a study performed on younger participants. If an earlier study used a very strong manipulation, and you plan to use a more subtle manipulation, a smaller effect size should be expected. Finally, effect sizes do not generalize to studies with different designs. For example, the effect size for a comparison between two groups is most often not similar to the effect size for an interaction in a follow-up study where a second factor is added to the original design (Lakens & Caldwell, 2021) .

Even if a study is sufficiently similar, statisticians have warned against using effect size estimates from small pilot studies in power analyses. Leon, Davis, and Kraemer (2011) write:

Contrary to tradition, a pilot study does not provide a meaningful effect size estimate for planning subsequent studies due to the imprecision inherent in data from small samples.

The two main reasons researchers should be careful when using effect sizes from studies in the published literature in power analyses is that effect size estimates from studies can differ from the true population effect size due to random variation, and that publication bias inflates effect sizes. Figure 6 shows the distribution for η p 2 for a study with three conditions with 25 participants in each condition when the null hypothesis is true and when there is a ‘medium’ true effect of η p 2 = 0.0588 (Richardson, 2011) . As in Figure 4 the critical effect size is indicated, which shows observed effects smaller than η p 2 = 0.08 will not be significant with the given sample size. If the null hypothesis is true effects larger than η p 2 = 0.08 will be a Type I error (the dark grey area), and when the alternative hypothesis is true effects smaller than η p 2 = 0.08 will be a Type II error (light grey area). It is clear all significant effects are larger than the true effect size ( η p 2 = 0.0588), so power analyses based on a significant finding (e.g., because only significant results are published in the literature) will be based on an overestimate of the true effect size, introducing bias.

But even if we had access to all effect sizes (e.g., from pilot studies you have performed yourself) due to random variation the observed effect size will sometimes be quite small. Figure 6 shows it is quite likely to observe an effect of η p 2 = 0.01 in a small pilot study, even when the true effect size is 0.0588. Entering an effect size estimate of η p 2 = 0.01 in an a-priori power analysis would suggest a total sample size of 957 observations to achieve 80% power in a follow-up study. If researchers only follow up on pilot studies when they observe an effect size in the pilot study that, when entered into a power analysis, yields a sample size that is feasible to collect for the follow-up study, these effect size estimates will be upwardly biased, and power in the follow-up study will be systematically lower than desired (Albers & Lakens, 2018) .

In essence, the problem with using small studies to estimate the effect size that will be entered into an a-priori power analysis is that due to publication bias or follow-up bias the effect sizes researchers end up using for their power analysis do not come from a full F distribution, but from what is known as a truncated F distribution (Taylor & Muller, 1996) . For example, imagine if there is extreme publication bias in the situation illustrated in Figure 6 . The only studies that would be accessible to researchers would come from the part of the distribution where η p 2 > 0.08, and the test result would be statistically significant. It is possible to compute an effect size estimate that, based on certain assumptions, corrects for bias. For example, imagine we observe a result in the literature for a One-Way ANOVA with 3 conditions, reported as F (2, 42) = 0.017, η p 2 = 0.176. If we would take this effect size at face value and enter it as our effect size estimate in an a-priori power analysis, the suggested sample size to achieve 80% power would suggest we need to collect 17 observations in each condition.

However, if we assume bias is present, we can use the BUCSS R package (S. F. Anderson et al., 2017) to perform a power analysis that attempts to correct for bias. A power analysis that takes bias into account (under a specific model of publication bias, based on a truncated F distribution where only significant results are published) suggests collecting 73 participants in each condition. It is possible that the bias corrected estimate of the non-centrality parameter used to compute power is zero, in which case it is not possible to correct for bias using this method. As an alternative to formally modeling a correction for publication bias whenever researchers assume an effect size estimate is biased, researchers can simply use a more conservative effect size estimate, for example by computing power based on the lower limit of a 60% two-sided confidence interval around the effect size estimate, which Perugini, Gallucci, and Costantini (2014) refer to as safeguard power . Both these approaches lead to a more conservative power analysis, but not necessarily a more accurate power analysis. It is simply not possible to perform an accurate power analysis on the basis of an effect size estimate from a study that might be biased and/or had a small sample size (Teare et al., 2014) . If it is not possible to specify a smallest effect size of interest, and there is great uncertainty about which effect size to expect, it might be more efficient to perform a study with a sequential design (discussed below).

To summarize, an effect size from a previous study in an a-priori power analysis can be used if three conditions are met (see Table 6 ). First, the previous study is sufficiently similar to the planned study. Second, there was a low risk of bias (e.g., the effect size estimate comes from a Registered Report, or from an analysis for which results would not have impacted the likelihood of publication). Third, the sample size is large enough to yield a relatively accurate effect size estimate, based on the width of a 95% CI around the observed effect size estimate. There is always uncertainty around the effect size estimate, and entering the upper and lower limit of the 95% CI around the effect size estimate might be informative about the consequences of the uncertainty in the effect size estimate for an a-priori power analysis.

## Using an Estimate from a Theoretical Model

When your theoretical model is sufficiently specific such that you can build a computational model, and you have knowledge about key parameters in your model that are relevant for the data you plan to collect, it is possible to estimate an effect size based on the effect size estimate derived from a computational model. For example, if one had strong ideas about the weights for each feature stimuli share and differ on, it could be possible to compute predicted similarity judgments for pairs of stimuli based on Tversky’s contrast model (Tversky, 1977) , and estimate the predicted effect size for differences between experimental conditions. Although computational models that make point predictions are relatively rare, whenever they are available, they provide a strong justification of the effect size a researcher expects.

## Compute the Width of the Confidence Interval around the Effect Size

If a researcher can estimate the standard deviation of the observations that will be collected, it is possible to compute an a-priori estimate of the width of the 95% confidence interval around an effect size (Kelley, 2007) . Confidence intervals represent a range around an estimate that is wide enough so that in the long run the true population parameter will fall inside the confidence intervals 100 - α percent of the time. In any single study the true population effect either falls in the confidence interval, or it doesn’t, but in the long run one can act as if the confidence interval includes the true population effect size (while keeping the error rate in mind). Cumming (2013) calls the difference between the observed effect size and the upper bound of the 95% confidence interval (or the lower bound of the 95% confidence interval) the margin of error.

If we compute the 95% CI for an effect size of d = 0 based on the t statistic and sample size (Smithson, 2003) , we see that with 15 observations in each condition of an independent t test the 95% CI ranges from d = -0.72 to d = 0.72 5 . The margin of error is half the width of the 95% CI, 0.72. A Bayesian estimator who uses an uninformative prior would compute a credible interval with the same (or a very similar) upper and lower bound (Albers et al., 2018; Kruschke, 2011) , and might conclude that after collecting the data they would be left with a range of plausible values for the population effect that is too large to be informative. Regardless of the statistical philosophy you plan to rely on when analyzing the data, the evaluation of what we can conclude based on the width of our interval tells us that with 15 observation per group we will not learn a lot.

One useful way of interpreting the width of the confidence interval is based on the effects you would be able to reject if the true effect size is 0. In other words, if there is no effect, which effects would you have been able to reject given the collected data, and which effect sizes would not be rejected, if there was no effect? Effect sizes in the range of d = 0.7 are findings such as “People become aggressive when they are provoked”, “People prefer their own group to other groups”, and “Romantic partners resemble one another in physical attractiveness” (Richard et al., 2003) . The width of the confidence interval tells you that you can only reject the presence of effects that are so large, if they existed, you would probably already have noticed them. If it is true that most effects that you study are realistically much smaller than d = 0.7, there is a good possibility that we do not learn anything we didn’t already know by performing a study with n = 15. Even without data, in most research lines we would not consider certain large effects plausible (although the effect sizes that are plausible differ between fields, as discussed below). On the other hand, in large samples where researchers can for example reject the presence of effects larger than d = 0.2, if the null hypothesis was true, this analysis of the width of the confidence interval would suggest that peers in many research lines would likely consider the study to be informative.

We see that the margin of error is almost, but not exactly, the same as the minimal statistically detectable effect ( d = 0.75). The small variation is due to the fact that the 95% confidence interval is calculated based on the t distribution. If the true effect size is not zero, the confidence interval is calculated based on the non-central t distribution, and the 95% CI is asymmetric. Figure 7 visualizes three t distributions, one symmetric at 0, and two asymmetric distributions with a noncentrality parameter (the normalized difference between the means) of 2 and 3. The asymmetry is most clearly visible in very small samples (the distributions in the plot have 5 degrees of freedom) but remains noticeable in larger samples when calculating confidence intervals and statistical power. For example, for a true effect size of d = 0.5 observed with 15 observations per group would yield d s = 0.50, 95% CI [-0.23, 1.22]. If we compute the 95% CI around the critical effect size, we would get d s = 0.75, 95% CI [0.00, 1.48]. We see the 95% CI ranges from exactly 0.00 to 1.48, in line with the relation between a confidence interval and a p value, where the 95% CI excludes zero if the test is statistically significant. As noted before, the different approaches recommended here to evaluate how informative a study is are often based on the same information.

## Plot a Sensitivity Power Analysis

A sensitivity power analysis fixes the sample size, desired power, and alpha level, and answers the question which effect size a study could detect with a desired power. A sensitivity power analysis is therefore performed when the sample size is already known. Sometimes data has already been collected to answer a different research question, or the data is retrieved from an existing database, and you want to perform a sensitivity power analysis for a new statistical analysis. Other times, you might not have carefully considered the sample size when you initially collected the data, and want to reflect on the statistical power of the study for (ranges of) effect sizes of interest when analyzing the results. Finally, it is possible that the sample size will be collected in the future, but you know that due to resource constraints the maximum sample size you can collect is limited, and you want to reflect on whether the study has sufficient power for effects that you consider plausible and interesting (such as the smallest effect size of interest, or the effect size that is expected).

Assume a researcher plans to perform a study where 30 observations will be collected in total, 15 in each between participant condition. Figure 8 shows how to perform a sensitivity power analysis in G*Power for a study where we have decided to use an alpha level of 5%, and desire 90% power. The sensitivity power analysis reveals the designed study has 90% power to detect effects of at least d = 1.23. Perhaps a researcher believes that a desired power of 90% is quite high, and is of the opinion that it would still be interesting to perform a study if the statistical power was lower. It can then be useful to plot a sensitivity curve across a range of smaller effect sizes.

The two dimensions of interest in a sensitivity power analysis are the effect sizes, and the power to observe a significant effect assuming a specific effect size. These two dimensions can be plotted against each other to create a sensitivity curve. For example, a sensitivity curve can be plotted in G*Power by clicking the ‘X-Y plot for a range of values’ button, as illustrated in Figure 9 . Researchers can examine which power they would have for an a-priori plausible range of effect sizes, or they can examine which effect sizes would provide reasonable levels of power. In simulation-based approaches to power analysis, sensitivity curves can be created by performing the power analysis for a range of possible effect sizes. Even if 50% power is deemed acceptable (in which case deciding to act as if the null hypothesis is true after a non-significant result is a relatively noisy decision procedure), Figure 9 shows a study design where power is extremely low for a large range of effect sizes that are reasonable to expect in most fields. Thus, a sensitivity power analysis provides an additional approach to evaluate how informative the planned study is, and can inform researchers that a specific design is unlikely to yield a significant effect for a range of effects that one might realistically expect.

If the number of observations per group had been larger, the evaluation might have been more positive. We might not have had any specific effect size in mind, but if we had collected 150 observations per group, a sensitivity analysis could have shown that power was sufficient for a range of effects we believe is most interesting to examine, and we would still have approximately 50% power for quite small effects. For a sensitivity analysis to be meaningful, the sensitivity curve should be compared against a smallest effect size of interest, or a range of effect sizes that are expected. A sensitivity power analysis has no clear cut-offs to examine (Bacchetti, 2010) . Instead, the idea is to make a holistic trade-off between different effect sizes one might observe or care about, and their associated statistical power.

## The Distribution of Effect Sizes in a Research Area

In my personal experience the most commonly entered effect size estimate in an a-priori power analysis for an independent t test is Cohen’s benchmark for a ‘medium’ effect size, because of what is known as the default effect . When you open G*Power, a ‘medium’ effect is the default option for an a-priori power analysis. Cohen’s benchmarks for small, medium, and large effects should not be used in an a-priori power analysis (Cook et al., 2014; Correll et al., 2020) , and Cohen regretted having proposed these benchmarks (Funder & Ozer, 2019) . The large variety in research topics means that any ‘default’ or ‘heuristic’ that is used to compute statistical power is not just unlikely to correspond to your actual situation, but it is also likely to lead to a sample size that is substantially misaligned with the question you are trying to answer with the collected data.

Some researchers have wondered what a better default would be, if researchers have no other basis to decide upon an effect size for an a-priori power analysis. Brysbaert (2019) recommends d = 0.4 as a default in psychology, which is the average observed in replication projects and several meta-analyses. It is impossible to know if this average effect size is realistic, but it is clear there is huge heterogeneity across fields and research questions. Any average effect size will often deviate substantially from the effect size that should be expected in a planned study. Some researchers have suggested to change Cohen’s benchmarks based on the distribution of effect sizes in a specific field (Bosco et al., 2015; Funder & Ozer, 2019; Hill et al., 2008; Kraft, 2020; Lovakov & Agadullina, 2017) . As always, when effect size estimates are based on the published literature, one needs to evaluate the possibility that the effect size estimates are inflated due to publication bias. Due to the large variation in effect sizes within a specific research area, there is little use in choosing a large, medium, or small effect size benchmark based on the empirical distribution of effect sizes in a field to perform a power analysis.

Having some knowledge about the distribution of effect sizes in the literature can be useful when interpreting the confidence interval around an effect size. If in a specific research area almost no effects are larger than the value you could reject in an equivalence test (e.g., if the observed effect size is 0, the design would only reject effects larger than for example d = 0.7), then it is a-priori unlikely that collecting the data would tell you something you didn’t already know.

It is more difficult to defend the use of a specific effect size derived from an empirical distribution of effect sizes as a justification for the effect size used in an a-priori power analysis. One might argue that the use of an effect size benchmark based on the distribution of effects in the literature will outperform a wild guess, but this is not a strong enough argument to form the basis of a sample size justification. There is a point where researchers need to admit they are not ready to perform an a-priori power analysis due to a lack of clear expectations (Scheel et al., 2020) . Alternative sample size justifications, such as a justification of the sample size based on resource constraints, perhaps in combination with a sequential study design, might be more in line with the actual inferential goals of a study.

So far, the focus has been on justifying the sample size for quantitative studies. There are a number of related topics that can be useful to design an informative study. First, in addition to a-priori or prospective power analysis and sensitivity power analysis, it is important to discuss compromise power analysis (which is useful) and post-hoc or retrospective power analysis (which is not useful, e.g., Zumbo and Hubley (1998) , Lenth (2007) ). When sample sizes are justified based on an a-priori power analysis it can be very efficient to collect data in sequential designs where data collection is continued or terminated based on interim analyses of the data. Furthermore, it is worthwhile to consider ways to increase the power of a test without increasing the sample size. An additional point of attention is to have a good understanding of your dependent variable, especially it’s standard deviation. Finally, sample size justification is just as important in qualitative studies, and although there has been much less work on sample size justification in this domain, some proposals exist that researchers can use to design an informative study. Each of these topics is discussed in turn.

## Compromise Power Analysis

In a compromise power analysis the sample size and the effect are fixed, and the error rates of the test are calculated, based on a desired ratio between the Type I and Type II error rate. A compromise power analysis is useful both when a very large number of observations will be collected, as when only a small number of observations can be collected.

In the first situation a researcher might be fortunate enough to be able to collect so many observations that the statistical power for a test is very high for all effect sizes that are deemed interesting. For example, imagine a researcher has access to 2000 employees who are all required to answer questions during a yearly evaluation in a company where they are testing an intervention that should reduce subjectively reported stress levels. You are quite confident that an effect smaller than d = 0.2 is not large enough to be subjectively noticeable for individuals (Jaeschke et al., 1989) . With an alpha level of 0.05 the researcher would have a statistical power of 0.994, or a Type II error rate of 0.006. This means that for a smallest effect size of interest of d = 0.2 the researcher is 8.30 times more likely to make a Type I error than a Type II error.

Although the original idea of designing studies that control Type I and Type II error rates was that researchers would need to justify their error rates (Neyman & Pearson, 1933) , a common heuristic is to set the Type I error rate to 0.05 and the Type II error rate to 0.20, meaning that a Type I error is 4 times as unlikely as a Type II error. The default use of 80% power (or a 20% Type II or β error) is based on a personal preference of Cohen (1988) , who writes:

It is proposed here as a convention that, when the investigator has no other basis for setting the desired power value, the value .80 be used. This means that β is set at .20. This arbitrary but reasonable value is offered for several reasons (Cohen, 1965, pp. 98-99). The chief among them takes into consideration the implicit convention for α of .05. The β of .20 is chosen with the idea that the general relative seriousness of these two kinds of errors is of the order of .20/.05, i.e., that Type I errors are of the order of four times as serious as Type II errors. This .80 desired power convention is offered with the hope that it will be ignored whenever an investigator can find a basis in his substantive concerns in his specific research investigation to choose a value ad hoc.

We see that conventions are built on conventions: the norm to aim for 80% power is built on the norm to set the alpha level at 5%. What we should take away from Cohen is not that we should aim for 80% power, but that we should justify our error rates based on the relative seriousness of each error. This is where compromise power analysis comes in. If you share Cohen’s belief that a Type I error is 4 times as serious as a Type II error, and building on our earlier study on 2000 employees, it makes sense to adjust the Type I error rate when the Type II error rate is low for all effect sizes of interest (Cascio & Zedeck, 1983) . Indeed, Erdfelder, Faul, and Buchner (1996) created the G*Power software in part to give researchers a tool to perform compromise power analysis.

Figure 10 illustrates how a compromise power analysis is performed in G*Power when a Type I error is deemed to be equally costly as a Type II error, which for a study with 1000 observations per condition would lead to a Type I error and a Type II error of 0.0179. As Faul, Erdfelder, Lang, and Buchner (2007) write:

Of course, compromise power analyses can easily result in unconventional significance levels greater than α = .05 (in the case of small samples or effect sizes) or less than α = .001 (in the case of large samples or effect sizes). However, we believe that the benefit of balanced Type I and Type II error risks often offsets the costs of violating significance level conventions.

This brings us to the second situation where a compromise power analysis can be useful, which is when we know the statistical power in our study is low. Although it is highly undesirable to make decisions when error rates are high, if one finds oneself in a situation where a decision must be made based on little information, Winer (1962) writes:

The frequent use of the .05 and .01 levels of significance is a matter of convention having little scientific or logical basis. When the power of tests is likely to be low under these levels of significance, and when Type I and Type II errors are of approximately equal importance, the .30 and .20 levels of significance may be more appropriate than the .05 and .01 levels.

For example, if we plan to perform a two-sided t test, can feasibly collect at most 50 observations in each independent group, and expect a population effect size of 0.5, we would have 70% power if we set our alpha level to 0.05. We can choose to weigh both types of error equally, and set the alpha level to 0.149, to end up with a statistical power for an effect of d = 0.5 of 0.851 (given a 0.149 Type II error rate). The choice of α and β in a compromise power analysis can be extended to take prior probabilities of the null and alternative hypothesis into account (Maier & Lakens, 2022; Miller & Ulrich, 2019; Murphy et al., 2014) .

A compromise power analysis requires a researcher to specify the sample size. This sample size itself requires a justification, so a compromise power analysis will typically be performed together with a resource constraint justification for a sample size. It is especially important to perform a compromise power analysis if your resource constraint justification is strongly based on the need to make a decision, in which case a researcher should think carefully about the Type I and Type II error rates stakeholders are willing to accept. However, a compromise power analysis also makes sense if the sample size is very large, but a researcher did not have the freedom to set the sample size. This might happen if, for example, data collection is part of a larger international study and the sample size is based on other research questions. In designs where the Type II error rate is very small (and power is very high) some statisticians have also recommended to lower the alpha level to prevent Lindley’s paradox, a situation where a significant effect ( p < α ) is evidence for the null hypothesis (Good, 1992; Jeffreys, 1939) . Lowering the alpha level as a function of the statistical power of the test can prevent this paradox, providing another argument for a compromise power analysis when sample sizes are large (Maier & Lakens, 2022) . Finally, a compromise power analysis needs a justification for the effect size, either based on a smallest effect size of interest or an effect size that is expected. Table 7 lists three aspects that should be discussed alongside a reported compromise power analysis.

## What to do if Your Editor Asks for Post-hoc Power?

Post-hoc, retrospective, or observed power is used to describe the statistical power of the test that is computed assuming the effect size that has been estimated from the collected data is the true effect size (Lenth, 2007; Zumbo & Hubley, 1998) . Post-hoc power is therefore not performed before looking at the data, based on effect sizes that are deemed interesting, as in an a-priori power analysis, and it is unlike a sensitivity power analysis where a range of interesting effect sizes is evaluated. Because a post-hoc or retrospective power analysis is based on the effect size observed in the data that has been collected, it does not add any information beyond the reported p value, but it presents the same information in a different way. Despite this fact, editors and reviewers often ask authors to perform post-hoc power analysis to interpret non-significant results. This is not a sensible request, and whenever it is made, you should not comply with it. Instead, you should perform a sensitivity power analysis, and discuss the power for the smallest effect size of interest and a realistic range of expected effect sizes.

Post-hoc power is directly related to the p value of the statistical test (Hoenig & Heisey, 2001) . For a z test where the p value is exactly 0.05, post-hoc power is always 50%. The reason for this relationship is that when a p value is observed that equals the alpha level of the test (e.g., 0.05), the observed z score of the test is exactly equal to the critical value of the test (e.g., z = 1.96 in a two-sided test with a 5% alpha level). Whenever the alternative hypothesis is centered on the critical value half the values we expect to observe if this alternative hypothesis is true fall below the critical value, and half fall above the critical value. Therefore, a test where we observed a p value identical to the alpha level will have exactly 50% power in a post-hoc power analysis, as the analysis assumes the observed effect size is true.

For other statistical tests, where the alternative distribution is not symmetric (such as for the t test, where the alternative hypothesis follows a non-central t distribution, see Figure 7 ), a p = 0.05 does not directly translate to an observed power of 50%, but by plotting post-hoc power against the observed p value we see that the two statistics are always directly related. As Figure 11 shows, if the p value is non-significant (i.e., larger than 0.05) the observed power will be less than approximately 50% in a t test. Lenth (2007) explains how observed power is also completely determined by the observed p value for F tests, although the statement that a non-significant p value implies a power less than 50% no longer holds.

When editors or reviewers ask researchers to report post-hoc power analyses they would like to be able to distinguish between true negatives (concluding there is no effect, when there is no effect) and false negatives (a Type II error, concluding there is no effect, when there actually is an effect). Since reporting post-hoc power is just a different way of reporting the p value, reporting the post-hoc power will not provide an answer to the question editors are asking (Hoenig & Heisey, 2001; Lenth, 2007; Schulz & Grimes, 2005; Yuan & Maxwell, 2005) . To be able to draw conclusions about the absence of a meaningful effect, one should perform an equivalence test, and design a study with high power to reject the smallest effect size of interest (Lakens, Scheel, et al., 2018) . Alternatively, if no smallest effect size of interest was specified when designing the study, researchers can report a sensitivity power analysis.

## Sequential Analyses

Whenever the sample size is justified based on an a-priori power analysis it can be very efficient to collect data in a sequential design. Sequential designs control error rates across multiple looks at the data (e.g., after 50, 100, and 150 observations have been collected) and can reduce the average expected sample size that is collected compared to a fixed design where data is only analyzed after the maximum sample size is collected (Proschan et al., 2006; Wassmer & Brannath, 2016) . Sequential designs have a long history (Dodge & Romig, 1929) , and exist in many variations, such as the Sequential Probability Ratio Test (Wald, 1945) , combining independent statistical tests (Westberg, 1985) , group sequential designs (Jennison & Turnbull, 2000) , sequential Bayes factors (Schönbrodt et al., 2017) , and safe testing (Grünwald et al., 2019) . Of these approaches, the Sequential Probability Ratio Test is most efficient if data can be analyzed after every observation (Schnuerch & Erdfelder, 2020) . Group sequential designs, where data is collected in batches, provide more flexibility in data collection, error control, and corrections for effect size estimates (Wassmer & Brannath, 2016) . Safe tests provide optimal flexibility if there are dependencies between observations (ter Schure & Grünwald, 2019) .

Sequential designs are especially useful when there is considerable uncertainty about the effect size, or when it is plausible that the true effect size is larger than the smallest effect size of interest the study is designed to detect (Lakens, 2014) . In such situations data collection has the possibility to terminate early if the effect size is larger than the smallest effect size of interest, but data collection can continue to the maximum sample size if needed. Sequential designs can prevent waste when testing hypotheses, both by stopping early when the null hypothesis can be rejected, as by stopping early if the presence of a smallest effect size of interest can be rejected (i.e., stopping for futility). Group sequential designs are currently the most widely used approach to sequential analyses, and can be planned and analyzed using rpact (Wassmer & Pahlke, 2019) or gsDesign (K. M. Anderson, 2014) . 6

## Increasing Power Without Increasing the Sample Size

The most straightforward approach to increase the informational value of studies is to increase the sample size. Because resources are often limited, it is also worthwhile to explore different approaches to increasing the power of a test without increasing the sample size. The first option is to use directional tests where relevant. Researchers often make directional predictions, such as ‘we predict X is larger than Y’. The statistical test that logically follows from this prediction is a directional (or one-sided) t test. A directional test moves the Type I error rate to one side of the tail of the distribution, which lowers the critical value, and therefore requires less observations to achieve the same statistical power.

Although there is some discussion about when directional tests are appropriate, they are perfectly defensible from a Neyman-Pearson perspective on hypothesis testing (Cho & Abe, 2013) , which makes a (preregistered) directional test a straightforward approach to both increase the power of a test, as the riskiness of the prediction. However, there might be situations where you do not want to ask a directional question. Sometimes, especially in research with applied consequences, it might be important to examine if a null effect can be rejected, even if the effect is in the opposite direction as predicted. For example, when you are evaluating a recently introduced educational intervention, and you predict the intervention will increase the performance of students, you might want to explore the possibility that students perform worse, to be able to recommend abandoning the new intervention. In such cases it is also possible to distribute the error rate in a ‘lop-sided’ manner, for example assigning a stricter error rate to effects in the negative than in the positive direction (Rice & Gaines, 1994) .

Another approach to increase the power without increasing the sample size, is to increase the alpha level of the test, as explained in the section on compromise power analysis. Obviously, this comes at an increased probability of making a Type I error. The risk of making either type of error should be carefully weighed, which typically requires taking into account the prior probability that the null-hypothesis is true (Cascio & Zedeck, 1983; Miller & Ulrich, 2019; Mudge et al., 2012; Murphy et al., 2014) . If you have to make a decision, or want to make a claim, and the data you can feasibly collect is limited, increasing the alpha level is justified, either based on a compromise power analysis, or based on a cost-benefit analysis (Baguley, 2004; Field et al., 2004) .

Another widely recommended approach to increase the power of a study is use a within participant design where possible. In almost all cases where a researcher is interested in detecting a difference between groups, a within participant design will require collecting less participants than a between participant design. The reason for the decrease in the sample size is explained by the equation below from Maxwell, Delaney, and Kelley (2017) . The number of participants needed in a two group within-participants design (NW) relative to the number of participants needed in a two group between-participants design (NB), assuming normal distributions, is:

The required number of participants is divided by two because in a within-participants design with two conditions every participant provides two data points. The extent to which this reduces the sample size compared to a between-participants design also depends on the correlation between the dependent variables (e.g., the correlation between the measure collected in a control task and an experimental task), as indicated by the (1- ρ ) part of the equation. If the correlation is 0, a within-participants design simply needs half as many participants as a between participant design (e.g., 64 instead 128 participants). The higher the correlation, the larger the relative benefit of within-participants designs, and whenever the correlation is negative (up to -1) the relative benefit disappears. Especially when dependent variables in within-participants designs are positively correlated, within-participants designs will greatly increase the power you can achieve given the sample size you have available. Use within-participants designs when possible, but weigh the benefits of higher power against the downsides of order effects or carryover effects that might be problematic in a within-participants design (Maxwell et al., 2017) . 7 For designs with multiple factors with multiple levels it can be difficult to specify the full correlation matrix that specifies the expected population correlation for each pair of measurements (Lakens & Caldwell, 2021) . In these cases sequential analyses might provide a solution.

In general, the smaller the variation, the larger the standardized effect size (because we are dividing the raw effect by a smaller standard deviation) and thus the higher the power given the same number of observations. Some additional recommendations are provided in the literature (Allison et al., 1997; Bausell & Li, 2002; Hallahan & Rosenthal, 1996) , such as:

Use better ways to screen participants for studies where participants need to be screened before participation.

Assign participants unequally to conditions (if data in the control condition is much cheaper to collect than data in the experimental condition, for example).

Use reliable measures that have low error variance (Williams et al., 1995) .

Smart use of preregistered covariates (Meyvis & Van Osselaer, 2018) .

It is important to consider if these ways to reduce the variation in the data do not come at too large a cost for external validity. For example, in an intention-to-treat analysis in randomized controlled trials participants who do not comply with the protocol are maintained in the analysis such that the effect size from the study accurately represents the effect of implementing the intervention in the population, and not the effect of the intervention only on those people who perfectly follow the protocol (Gupta, 2011) . Similar trade-offs between reducing the variance and external validity exist in other research areas.

## Know Your Measure

Although it is convenient to talk about standardized effect sizes, it is generally preferable if researchers can interpret effects in the raw (unstandardized) scores, and have knowledge about the standard deviation of their measures (Baguley, 2009; Lenth, 2001) . To make it possible for a research community to have realistic expectations about the standard deviation of measures they collect, it is beneficial if researchers within a research area use the same validated measures. This provides a reliable knowledge base that makes it easier to plan for a desired accuracy, and to use a smallest effect size of interest on the unstandardized scale in an a-priori power analysis.

In addition to knowledge about the standard deviation it is important to have knowledge about the correlations between dependent variables (for example because Cohen’s d z for a dependent t test relies on the correlation between means). The more complex the model, the more aspects of the data-generating process need to be known to make predictions. For example, in hierarchical models researchers need knowledge about variance components to be able to perform a power analysis (DeBruine & Barr, 2019; Westfall et al., 2014) . Finally, it is important to know the reliability of your measure (Parsons et al., 2019) , especially when relying on an effect size from a published study that used a measure with different reliability, or when the same measure is used in different populations, in which case it is possible that measurement reliability differs between populations. With the increasing availability of open data, it will hopefully become easier to estimate these parameters using data from earlier studies.

If we calculate a standard deviation from a sample, this value is an estimate of the true value in the population. In small samples, our estimate can be quite far off, while due to the law of large numbers, as our sample size increases, we will be measuring the standard deviation more accurately. Since the sample standard deviation is an estimate with uncertainty, we can calculate a confidence interval around the estimate (Smithson, 2003) , and design pilot studies that will yield a sufficiently reliable estimate of the standard deviation. The confidence interval for the variance σ 2 is provided in the following formula, and the confidence for the standard deviation is the square root of these limits:

Whenever there is uncertainty about parameters, researchers can use sequential designs to perform an internal pilot study (Wittes & Brittain, 1990) . The idea behind an internal pilot study is that researchers specify a tentative sample size for the study, perform an interim analysis, use the data from the internal pilot study to update parameters such as the variance of the measure, and finally update the final sample size that will be collected. As long as interim looks at the data are blinded (e.g., information about the conditions is not taken into account) the sample size can be adjusted based on an updated estimate of the variance without any practical consequences for the Type I error rate (Friede & Kieser, 2006; Proschan, 2005) . Therefore, if researchers are interested in designing an informative study where the Type I and Type II error rates are controlled, but they lack information about the standard deviation, an internal pilot study might be an attractive approach to consider (Chang, 2016) .

## Conventions as meta-heuristics

Even when a researcher might not use a heuristic to directly determine the sample size in a study, there is an indirect way in which heuristics play a role in sample size justifications. Sample size justifications based on inferential goals such as a power analysis, accuracy, or a decision all require researchers to choose values for a desired Type I and Type II error rate, a desired accuracy, or a smallest effect size of interest. Although it is sometimes possible to justify these values as described above (e.g., based on a cost-benefit analysis), a solid justification of these values might require dedicated research lines. Performing such research lines will not always be possible, and these studies might themselves not be worth the costs (e.g., it might require less resources to perform a study with an alpha level that most peers would consider conservatively low, than to collect all the data that would be required to determine the alpha level based on a cost-benefit analysis). In these situations, researchers might use values based on a convention.

When it comes to a desired width of a confidence interval, a desired power, or any other input values required to perform a sample size computation, it is important to transparently report the use of a heuristic or convention (for example by using the accompanying online Shiny app). A convention such as the use of a 5% Type 1 error rate and 80% power practically functions as a lower threshold of the minimum informational value peers are expected to accept without any justification (whereas with a justification, higher error rates can also be deemed acceptable by peers). It is important to realize that none of these values are set in stone. Journals are free to specify that they desire a higher informational value in their author guidelines (e.g., Nature Human Behavior requires registered reports to be designed to achieve 95% statistical power, and my own department has required staff to submit ERB proposals where, whenever possible, the study was designed to achieve 90% power). Researchers who choose to design studies with a higher informational value than a conventional minimum should receive credit for doing so.

In the past some fields have changed conventions, such as the 5 sigma threshold now used in physics to declare a discovery instead of a 5% Type I error rate. In other fields such attempts have been unsuccessful (e.g., Johnson (2013) ). Improved conventions should be context dependent, and it seems sensible to establish them through consensus meetings (Mullan & Jacoby, 1985) . Consensus meetings are common in medical research, and have been used to decide upon a smallest effect size of interest (for an example, see Fried, Boers, and Baker (1993) ). In many research areas current conventions can be improved. For example, it seems peculiar to have a default alpha level of 5% both for single studies and for meta-analyses, and one could imagine a future where the default alpha level in meta-analyses is much lower than 5%. Hopefully, making the lack of an adequate justification for certain input values in specific situations more transparent will motivate fields to start a discussion about how to improve current conventions. The online Shiny app links to good examples of justifications where possible, and will continue to be updated as better justifications are developed in the future.

## Sample Size Justification in Qualitative Research

A value of information perspective to sample size justification also applies to qualitative research. A sample size justification in qualitative research should be based on the consideration that the cost of collecting data from additional participants does not yield new information that is valuable enough given the inferential goals. One widely used application of this idea is known as saturation and is indicated by the observation that new data replicates earlier observations, without adding new information (Morse, 1995) . For example, let’s imagine we ask people why they have a pet. Interviews might reveal reasons that are grouped into categories, but after interviewing 20 people, no new categories emerge, at which point saturation has been reached. Alternative philosophies to qualitative research exist, and not all value planning for saturation. Regrettably, principled approaches to justify sample sizes have not been developed for these alternative philosophies (Marshall et al., 2013) .

When sampling, the goal is often not to pick a representative sample, but a sample that contains a sufficiently diverse number of subjects such that saturation is reached efficiently. Fugard and Potts (2015) show how to move towards a more informed justification for the sample size in qualitative research based on 1) the number of codes that exist in the population (e.g., the number of reasons people have pets), 2) the probability a code can be observed in a single information source (e.g., the probability that someone you interview will mention each possible reason for having a pet), and 3) the number of times you want to observe each code. They provide an R formula based on binomial probabilities to compute a required sample size to reach a desired probability of observing codes.

A more advanced approach is used in Rijnsoever (2017) , which also explores the importance of different sampling strategies. In general, purposefully sampling information from sources you expect will yield novel information is much more efficient than random sampling, but this also requires a good overview of the expected codes, and the sub-populations in which each code can be observed. Sometimes, it is possible to identify information sources that, when interviewed, would at least yield one new code (e.g., based on informal communication before an interview). A good sample size justification in qualitative research is based on 1) an identification of the populations, including any sub-populations, 2) an estimate of the number of codes in the (sub-)population, 3) the probability a code is encountered in an information source, and 4) the sampling strategy that is used.

Providing a coherent sample size justification is an essential step in designing an informative study. There are multiple approaches to justifying the sample size in a study, depending on the goal of the data collection, the resources that are available, and the statistical approach that is used to analyze the data. An overarching principle in all these approaches is that researchers consider the value of the information they collect in relation to their inferential goals.

The process of justifying a sample size when designing a study should sometimes lead to the conclusion that it is not worthwhile to collect the data, because the study does not have sufficient informational value to justify the costs. There will be cases where it is unlikely there will ever be enough data to perform a meta-analysis (for example because of a lack of general interest in the topic), the information will not be used to make a decision or claim, and the statistical tests do not allow you to test a hypothesis with reasonable error rates or to estimate an effect size with sufficient accuracy. If there is no good justification to collect the maximum number of observations that one can feasibly collect, performing the study anyway is a waste of time and/or money (Brown, 1983; Button et al., 2013; S. D. Halpern et al., 2002) .

The awareness that sample sizes in past studies were often too small to meet any realistic inferential goals is growing among psychologists (Button et al., 2013; Fraley & Vazire, 2014; Lindsay, 2015; Sedlmeier & Gigerenzer, 1989) . As an increasing number of journals start to require sample size justifications, some researchers will realize they need to collect larger samples than they were used to. This means researchers will need to request more money for participant payment in grant proposals, or that researchers will need to increasingly collaborate (Moshontz et al., 2018) . If you believe your research question is important enough to be answered, but you are not able to answer the question with your current resources, one approach to consider is to organize a research collaboration with peers, and pursue an answer to this question collectively.

A sample size justification should not be seen as a hurdle that researchers need to pass before they can submit a grant, ethical review board proposal, or manuscript for publication. When a sample size is simply stated, instead of carefully justified, it can be difficult to evaluate whether the value of the information a researcher aims to collect outweighs the costs of data collection. Being able to report a solid sample size justification means a researcher knows what they want to learn from a study, and makes it possible to design a study that can provide an informative answer to a scientific question.

This work was funded by VIDI Grant 452-17-013 from the Netherlands Organisation for Scientific Research. I would like to thank Shilaan Alzahawi, José Biurrun, Aaron Caldwell, Gordon Feld, Yaov Kessler, Robin Kok, Maximilian Maier, Matan Mazor, Toni Saari, Andy Siddall, and Jesper Wulff for feedback on an earlier draft. A computationally reproducible version of this manuscript is available at https://github.com/Lakens/sample_size_justification. An interactive online form to complete a sample size justification implementing the recommendations in this manuscript can be found at https://shiny.ieis.tue.nl/sample_size_justification/.

I have no competing interests to declare.

A computationally reproducible version of this manuscript is available at https://github.com/Lakens/sample_size_justification .

The topic of power analysis for meta-analyses is outside the scope of this manuscript, but see Hedges and Pigott (2001) and Valentine, Pigott, and Rothstein (2010) .

It is possible to argue we are still making an inference, even when the entire population is observed, because we have observed a metaphorical population from one of many possible worlds, see Spiegelhalter (2019) .

Power analyses can be performed based on standardized effect sizes or effect sizes expressed on the original scale. It is important to know the standard deviation of the effect (see the ‘Know Your Measure’ section) but I find it slightly more convenient to talk about standardized effects in the context of sample size justifications.

These figures can be reproduced and adapted in an online shiny app: http://shiny.ieis.tue.nl/d_p_power/ .

Confidence intervals around effect sizes can be computed using the MOTE Shiny app: https://www.aggieerin.com/shiny-server/

Shiny apps are available for both rpact: https://rpact.shinyapps.io/public/ and gsDesign: https://gsdesign.shinyapps.io/prod/

You can compare within- and between-participants designs in this Shiny app: http://shiny.ieis.tue.nl/within_between .

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## Writing Research Proposals

The research proposal is your opportunity to show that you—and only you!—are the perfect person to take on your specific project. After reading your research proposal, readers should be confident that…

- You have thoughtfully crafted and designed this project;
- You have the necessary background to complete this project;
- You have the proper support system in place;
- You know exactly what you need to complete this project and how to do so; and
- With this funding in hand, you can be on your way to a meaningful research experience and a significant contribution to your field.

## Research proposals typically include the following components:

- Why is your project important? How does it contribute to the field or to society? What do you hope to prove?
- This section includes the project design, specific methodology, your specific role and responsibilities, steps you will take to execute the project, etc. Here you will show the committee the way that you think by explaining both how you have conceived the project and how you intend to carry it out.
- Please be specific in the project dates/how much time you need to carry out the proposed project. The scope of the project should clearly match the timeframe in which you propose to complete it!
- Funding agencies like to know how their funding will be used. Including this information will demonstrate that you have thoughtfully designed the project and know of all of the anticipated expenses required to see it through to completion.
- It is important that you have a support system on hand when conducting research, especially as an undergraduate. There are often surprises and challenges when working on a long-term research project and the selection committee wants to be sure that you have the support system you need to both be successful in your project and also have a meaningful research experience.
- Some questions to consider are: How often do you intend to meet with your advisor(s)? (This may vary from project to project based on the needs of the student and the nature of the research.) What will your mode of communication be? Will you be attending (or even presenting at) lab meetings?

Don’t be afraid to also include relevant information about your background and advocate for yourself! Do you have skills developed in a different research experience (or leadership position, job, coursework, etc.) that you could apply to the project in question? Have you already learned about and experimented with a specific method of analysis in class and are now ready to apply it to a different situation? If you already have experience with this professor/lab, please be sure to include those details in your proposal! That will show the selection committee that you are ready to hit the ground running!

Lastly, be sure to know who your readers are so that you can tailor the field-specific language of your proposal accordingly. If the selection committee are specialists in your field, you can feel free to use the jargon of that field; but if your proposal will be evaluated by an interdisciplinary committee (this is common), you might take a bit longer explaining the state of the field, specific concepts, and certainly spelling out any acronyms.

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## How to Determine Sample Size for a Research Study

Frankline kibuacha | apr. 06, 2021 | 3 min. read.

This article will discuss considerations to put in place when determining your sample size and how to calculate the sample size.

## Confidence Interval and Confidence Level

As we have noted before, when selecting a sample there are multiple factors that can impact the reliability and validity of results, including sampling and non-sampling errors . When thinking about sample size, the two measures of error that are almost always synonymous with sample sizes are the confidence interval and the confidence level.

## Confidence Interval (Margin of Error)

Confidence intervals measure the degree of uncertainty or certainty in a sampling method and how much uncertainty there is with any particular statistic. In simple terms, the confidence interval tells you how confident you can be that the results from a study reflect what you would expect to find if it were possible to survey the entire population being studied. The confidence interval is usually a plus or minus (±) figure. For example, if your confidence interval is 6 and 60% percent of your sample picks an answer, you can be confident that if you had asked the entire population, between 54% (60-6) and 66% (60+6) would have picked that answer.

## Confidence Level

The confidence level refers to the percentage of probability, or certainty that the confidence interval would contain the true population parameter when you draw a random sample many times. It is expressed as a percentage and represents how often the percentage of the population who would pick an answer lies within the confidence interval. For example, a 99% confidence level means that should you repeat an experiment or survey over and over again, 99 percent of the time, your results will match the results you get from a population.

The larger your sample size, the more confident you can be that their answers truly reflect the population. In other words, the larger your sample for a given confidence level, the smaller your confidence interval.

## Standard Deviation

Another critical measure when determining the sample size is the standard deviation, which measures a data set’s distribution from its mean. In calculating the sample size, the standard deviation is useful in estimating how much the responses you receive will vary from each other and from the mean number, and the standard deviation of a sample can be used to approximate the standard deviation of a population.

The higher the distribution or variability, the greater the standard deviation and the greater the magnitude of the deviation. For example, once you have already sent out your survey, how much variance do you expect in your responses? That variation in responses is the standard deviation.

## Population Size

As demonstrated through the calculation below, a sample size of about 385 will give you a sufficient sample size to draw assumptions of nearly any population size at the 95% confidence level with a 5% margin of error, which is why samples of 400 and 500 are often used in research. However, if you are looking to draw comparisons between different sub-groups, for example, provinces within a country, a larger sample size is required. GeoPoll typically recommends a sample size of 400 per country as the minimum viable sample for a research project, 800 per country for conducting a study with analysis by a second-level breakdown such as females versus males, and 1200+ per country for doing third-level breakdowns such as males aged 18-24 in Nairobi.

## How to Calculate Sample Size

As we have defined all the necessary terms, let us briefly learn how to determine the sample size using a sample calculation formula known as Andrew Fisher’s Formula.

- Determine the population size (if known).
- Determine the confidence interval.
- Determine the confidence level.
- Determine the standard deviation ( a standard deviation of 0.5 is a safe choice where the figure is unknown )
- Convert the confidence level into a Z-Score. This table shows the z-scores for the most common confidence levels:
- Put these figures into the sample size formula to get your sample size.

Here is an example calculation:

Say you choose to work with a 95% confidence level, a standard deviation of 0.5, and a confidence interval (margin of error) of ± 5%, you just need to substitute the values in the formula:

((1.96)2 x .5(.5)) / (.05)2

(3.8416 x .25) / .0025

.9604 / .0025

Your sample size should be 385.

Fortunately, there are several available online tools to help you with this calculation. Here’s an online sample calculator from Easy Calculation. Just put in the confidence level, population size, the confidence interval, and the perfect sample size is calculated for you.

GeoPoll’s Sampling Techniques

With the largest mobile panel in Africa, Asia, and Latin America, and reliable mobile technologies, GeoPoll develops unique samples that accurately represent any population. See our country coverage here , or contact our team to discuss your upcoming project.

Related Posts

Sample Frame and Sample Error

Probability and Non-Probability Samples

How GeoPoll Conducts Nationally Representative Surveys

- Tags market research , Market Research Methods , sample size , survey methodology

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## How to write a research proposal?

Department of Anaesthesiology, Bangalore Medical College and Research Institute, Bengaluru, Karnataka, India

## Devika Rani Duggappa

Writing the proposal of a research work in the present era is a challenging task due to the constantly evolving trends in the qualitative research design and the need to incorporate medical advances into the methodology. The proposal is a detailed plan or ‘blueprint’ for the intended study, and once it is completed, the research project should flow smoothly. Even today, many of the proposals at post-graduate evaluation committees and application proposals for funding are substandard. A search was conducted with keywords such as research proposal, writing proposal and qualitative using search engines, namely, PubMed and Google Scholar, and an attempt has been made to provide broad guidelines for writing a scientifically appropriate research proposal.

## INTRODUCTION

A clean, well-thought-out proposal forms the backbone for the research itself and hence becomes the most important step in the process of conduct of research.[ 1 ] The objective of preparing a research proposal would be to obtain approvals from various committees including ethics committee [details under ‘Research methodology II’ section [ Table 1 ] in this issue of IJA) and to request for grants. However, there are very few universally accepted guidelines for preparation of a good quality research proposal. A search was performed with keywords such as research proposal, funding, qualitative and writing proposals using search engines, namely, PubMed, Google Scholar and Scopus.

Five ‘C’s while writing a literature review

## BASIC REQUIREMENTS OF A RESEARCH PROPOSAL

A proposal needs to show how your work fits into what is already known about the topic and what new paradigm will it add to the literature, while specifying the question that the research will answer, establishing its significance, and the implications of the answer.[ 2 ] The proposal must be capable of convincing the evaluation committee about the credibility, achievability, practicality and reproducibility (repeatability) of the research design.[ 3 ] Four categories of audience with different expectations may be present in the evaluation committees, namely academic colleagues, policy-makers, practitioners and lay audiences who evaluate the research proposal. Tips for preparation of a good research proposal include; ‘be practical, be persuasive, make broader links, aim for crystal clarity and plan before you write’. A researcher must be balanced, with a realistic understanding of what can be achieved. Being persuasive implies that researcher must be able to convince other researchers, research funding agencies, educational institutions and supervisors that the research is worth getting approval. The aim of the researcher should be clearly stated in simple language that describes the research in a way that non-specialists can comprehend, without use of jargons. The proposal must not only demonstrate that it is based on an intelligent understanding of the existing literature but also show that the writer has thought about the time needed to conduct each stage of the research.[ 4 , 5 ]

## CONTENTS OF A RESEARCH PROPOSAL

The contents or formats of a research proposal vary depending on the requirements of evaluation committee and are generally provided by the evaluation committee or the institution.

In general, a cover page should contain the (i) title of the proposal, (ii) name and affiliation of the researcher (principal investigator) and co-investigators, (iii) institutional affiliation (degree of the investigator and the name of institution where the study will be performed), details of contact such as phone numbers, E-mail id's and lines for signatures of investigators.

The main contents of the proposal may be presented under the following headings: (i) introduction, (ii) review of literature, (iii) aims and objectives, (iv) research design and methods, (v) ethical considerations, (vi) budget, (vii) appendices and (viii) citations.[ 4 ]

## Introduction

It is also sometimes termed as ‘need for study’ or ‘abstract’. Introduction is an initial pitch of an idea; it sets the scene and puts the research in context.[ 6 ] The introduction should be designed to create interest in the reader about the topic and proposal. It should convey to the reader, what you want to do, what necessitates the study and your passion for the topic.[ 7 ] Some questions that can be used to assess the significance of the study are: (i) Who has an interest in the domain of inquiry? (ii) What do we already know about the topic? (iii) What has not been answered adequately in previous research and practice? (iv) How will this research add to knowledge, practice and policy in this area? Some of the evaluation committees, expect the last two questions, elaborated under a separate heading of ‘background and significance’.[ 8 ] Introduction should also contain the hypothesis behind the research design. If hypothesis cannot be constructed, the line of inquiry to be used in the research must be indicated.

## Review of literature

It refers to all sources of scientific evidence pertaining to the topic in interest. In the present era of digitalisation and easy accessibility, there is an enormous amount of relevant data available, making it a challenge for the researcher to include all of it in his/her review.[ 9 ] It is crucial to structure this section intelligently so that the reader can grasp the argument related to your study in relation to that of other researchers, while still demonstrating to your readers that your work is original and innovative. It is preferable to summarise each article in a paragraph, highlighting the details pertinent to the topic of interest. The progression of review can move from the more general to the more focused studies, or a historical progression can be used to develop the story, without making it exhaustive.[ 1 ] Literature should include supporting data, disagreements and controversies. Five ‘C's may be kept in mind while writing a literature review[ 10 ] [ Table 1 ].

## Aims and objectives

The research purpose (or goal or aim) gives a broad indication of what the researcher wishes to achieve in the research. The hypothesis to be tested can be the aim of the study. The objectives related to parameters or tools used to achieve the aim are generally categorised as primary and secondary objectives.

## Research design and method

The objective here is to convince the reader that the overall research design and methods of analysis will correctly address the research problem and to impress upon the reader that the methodology/sources chosen are appropriate for the specific topic. It should be unmistakably tied to the specific aims of your study.

In this section, the methods and sources used to conduct the research must be discussed, including specific references to sites, databases, key texts or authors that will be indispensable to the project. There should be specific mention about the methodological approaches to be undertaken to gather information, about the techniques to be used to analyse it and about the tests of external validity to which researcher is committed.[ 10 , 11 ]

The components of this section include the following:[ 4 ]

## Population and sample

Population refers to all the elements (individuals, objects or substances) that meet certain criteria for inclusion in a given universe,[ 12 ] and sample refers to subset of population which meets the inclusion criteria for enrolment into the study. The inclusion and exclusion criteria should be clearly defined. The details pertaining to sample size are discussed in the article “Sample size calculation: Basic priniciples” published in this issue of IJA.

## Data collection

The researcher is expected to give a detailed account of the methodology adopted for collection of data, which include the time frame required for the research. The methodology should be tested for its validity and ensure that, in pursuit of achieving the results, the participant's life is not jeopardised. The author should anticipate and acknowledge any potential barrier and pitfall in carrying out the research design and explain plans to address them, thereby avoiding lacunae due to incomplete data collection. If the researcher is planning to acquire data through interviews or questionnaires, copy of the questions used for the same should be attached as an annexure with the proposal.

## Rigor (soundness of the research)

This addresses the strength of the research with respect to its neutrality, consistency and applicability. Rigor must be reflected throughout the proposal.

It refers to the robustness of a research method against bias. The author should convey the measures taken to avoid bias, viz. blinding and randomisation, in an elaborate way, thus ensuring that the result obtained from the adopted method is purely as chance and not influenced by other confounding variables.

## Consistency

Consistency considers whether the findings will be consistent if the inquiry was replicated with the same participants and in a similar context. This can be achieved by adopting standard and universally accepted methods and scales.

## Applicability

Applicability refers to the degree to which the findings can be applied to different contexts and groups.[ 13 ]

## Data analysis

This section deals with the reduction and reconstruction of data and its analysis including sample size calculation. The researcher is expected to explain the steps adopted for coding and sorting the data obtained. Various tests to be used to analyse the data for its robustness, significance should be clearly stated. Author should also mention the names of statistician and suitable software which will be used in due course of data analysis and their contribution to data analysis and sample calculation.[ 9 ]

## Ethical considerations

Medical research introduces special moral and ethical problems that are not usually encountered by other researchers during data collection, and hence, the researcher should take special care in ensuring that ethical standards are met. Ethical considerations refer to the protection of the participants' rights (right to self-determination, right to privacy, right to autonomy and confidentiality, right to fair treatment and right to protection from discomfort and harm), obtaining informed consent and the institutional review process (ethical approval). The researcher needs to provide adequate information on each of these aspects.

Informed consent needs to be obtained from the participants (details discussed in further chapters), as well as the research site and the relevant authorities.

When the researcher prepares a research budget, he/she should predict and cost all aspects of the research and then add an additional allowance for unpredictable disasters, delays and rising costs. All items in the budget should be justified.

Appendices are documents that support the proposal and application. The appendices will be specific for each proposal but documents that are usually required include informed consent form, supporting documents, questionnaires, measurement tools and patient information of the study in layman's language.

As with any scholarly research paper, you must cite the sources you used in composing your proposal. Although the words ‘references and bibliography’ are different, they are used interchangeably. It refers to all references cited in the research proposal.

Successful, qualitative research proposals should communicate the researcher's knowledge of the field and method and convey the emergent nature of the qualitative design. The proposal should follow a discernible logic from the introduction to presentation of the appendices.

## Financial support and sponsorship

Conflicts of interest.

There are no conflicts of interest.

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How to determine the correct sample size.

6 min read Finding the perfect sample size for statistically sound results is an age old problem. Here we shed light on some methods and tools for sample size determination.

## What is sample size?

Sample size is a frequently-used term in statistics and market research , and one that inevitably comes up whenever you’re surveying a large population of respondents. It relates to the way research is conducted on large populations.

Free eBook: The ultimate guide to conducting market research

So what is sampling, and why does sample size matter?

When you survey a large population of respondents, you’re interested in the entire group, but it’s not realistically possible to get answers or results from absolutely everyone. So you take a random sample of individuals which represents the population as a whole.

The size of the sample is very important for getting accurate, statistically significant results and running your study successfully.

- If your sample is too small , you may include a disproportionate number of individuals which are outliers and anomalies. These skew the results and you don’t get a fair picture of the whole population.
- If the sample is too big , the whole study becomes complex, expensive and time-consuming to run, and although the results are more accurate, the benefits don’t outweigh the costs.

If you’ve already worked out your variables you can get to the right sample size quickly with the sample size calculator .

If you want to start from scratch in determining the right sample size for your market research , let us walk you through the steps.

## Learn how to determine sample size

To choose the correct sample size, you need to consider a few different factors that affect your research, and gain a basic understanding of the statistics involved. You’ll then be able to use a sample size formula to bring everything together and sample confidently, knowing that there is a high probability that your survey is statistically accurate.

The steps that follow are suitable for finding a sample size for continuous data – i.e. data that is counted numerically. It doesn’t apply to categorical data – i.e. put into categories like green, blue, male, female etc.

## Stage 1: Consider your sample size variables

Before you can calculate a sample size, you need to determine a few things about the target population and the level of accuracy you need:

## 1. Population size

How many people are you talking about in total? To find this out, you need to be clear about who does and doesn’t fit into your group. For example, if you want to know about dog owners, you’ll include everyone who has at some point owned at least one dog. (You may include or exclude those who owned a dog in the past, depending on your research goals.) Don’t worry if you’re unable to calculate the exact number. It’s common to have an unknown number or an estimated range.

## 2. Margin of error (confidence interval)

Errors are inevitable – the question is how much error you’ll allow. The margin of error, AKA confidence interval, is expressed in terms of mean numbers. You can set how much difference you’ll allow between the mean number of your sample and the mean number of your population. If you’ve ever seen a political poll on the news, you’ve seen a confidence interval and how it’s expressed. It will look something like this: “68% of voters said yes to Proposition Z, with a margin of error of +/- 5%.”

## 3. Confidence level

This is a separate step to the similarly-named confidence interval in step 2. It deals with how confident you want to be that the actual mean falls within your margin of error. The most common confidence intervals are 90% confident, 95% confident, and 99% confident.

## 4. Standard deviation

This step asks you to estimate how much the responses you receive will vary from each other and from the mean number. A low standard deviation means that all the values will be clustered around the mean number, whereas a high standard deviation means they are spread out across a much wider range with very small and very large outlying figures. Since you haven’t yet run your survey, a safe choice is a standard deviation of .5 which will help make sure your sample size is large enough.

## Stage 2: Calculate sample size

Now that you’ve got answers for steps 1 – 4, you’re ready to calculate the sample size you need. This can be done using the online sample size calculator above or with paper and pencil.

## 1. Find your Z-score

Next, you need to turn your confidence level into a Z-score. Here are the Z-scores for the most common confidence levels:

- 90% – Z Score = 1.645
- 95% – Z Score = 1.96
- 99% – Z Score = 2.576

If you chose a different confidence level, use our Z-score table to find your score.

## 2. Use the sample size formula

Plug in your Z-score, standard of deviation, and confidence interval into the sample size calculator or use this sample size formula to work it out yourself:

This equation is for an unknown population size or a very large population size. If your population is smaller and known, just use the sample size calculator above, or find it here .

What does that look like in practice?

Here’s a worked example, assuming you chose a 95% confidence level, .5 standard deviation, and a margin of error (confidence interval) of +/- 5%.

((1.96)2 x .5(.5)) / (.05)2

(3.8416 x .25) / .0025

.9604 / .0025

385 respondents are needed

Voila! You’ve just determined your sample size.

## Troubleshooting your sample size results

If the sample size is too big to manage, you can adjust the results by either

- decreasing your confidence level
- increasing your margin of error

This will increase the chance for error in your sampling , but it can greatly decrease the number of responses you need.

Continue the journey with our Guide to Market Research

## Related resources

Simple random sampling 9 min read, sampling techniques 10 min read, sampling and non-sampling errors 10 min read, selection bias 11 min read, systematic random sampling 15 min read, convenience sampling 18 min read, probability sampling 8 min read, request demo.

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## How to Calculate Sample Size

Last Updated: July 21, 2023 Fact Checked

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, 10 people, some anonymous, worked to edit and improve it over time. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 344,265 times. Learn more...

Scientific studies often rely on surveys distributed among a sample of some total population. Your sample will need to include a certain number of people, however, if you want it to accurately reflect the conditions of the overall population it's meant to represent. To calculate your necessary sample size, you'll need to determine several set values and plug them into an appropriate formula.

## Determining Key Values

- Precision has a greater statistical impact when you work with a smaller group. For instance, if you wish to perform a survey among members of a local organization or employees of a small business, the population size should be accurate within a dozen or so people. [1] X Research source
- Larger surveys allow for a greater deviance in the actual population. For example, if your demographic includes everyone living in the United States, you could estimate the size to roughly 320 million people, even though the actual value may vary by hundreds of thousands.

- The margin of error is a percentage the indicates how close your sample results will be to the true value of the overall population discussed in your study.
- Smaller margin of errors will result in more accurate answers, but choosing a smaller margin of error will also require a larger sample.
- In this example, the margin of error essentially indicates that, if the entire population were asked the same poll question, you are “confident” that somewhere between 30% (35 - 5) and 40% (35 + 5) would agree with option A .

- In other words, choosing a confidence level of 95% allows you to claim that you 95% certain that your results accurately fall within your chosen margin of error.
- A larger confidence level indicates a greater degree of accuracy, but it will also require a larger sample. The most common confidence levels are 90% confident, 95% confident, and 99% confident.
- Setting a confidence level of 95% for the example stated in the margin of error step would mean that you are 95% certain that 30% to 40% of the total concerned population would agree with option A of your survey.

- Plainly stated, if 99% of your survey responses answer "Yes" and only 1% answer "No," the sample probably represents the overall population very accurately.
- On the other hand, if 45% answer "Yes" and 55% answer "No," there is a greater chance of error.
- Since this value is difficult to determine you give the actual survey, most researchers set this value at 0.5 (50%). This is the worst case scenario percentage, so sticking with this value will guarantee that your calculated sample size is large enough to accurately represent the overall population within your confidence interval and confidence level.

- You can calculate z-scores by hand, look for an online calculator, or find your z-score on a z-score table. Each of these methods can be fairly complex, however.
- 80% confidence => 1.28 z-score
- 85% confidence => 1.44 z-score
- 90% confidence => 1.65 z-score
- 95% confidence => 1.96 z-score
- 99% confidence => 2.58 z-score

## Using the Standard Formula

- N = population size
- z = z-score
- e = margin of error
- p = standard of deviation

- Example: Determine the ideal survey size for a population size of 425 people. Use a 99% confidence level, a 50% standard of deviation, and a 5% margin of error.
- For 99% confidence, you would have a z-score of 2.58.

- = [2.58 2 * 0.5(1-0.5)] / 0.05 2 / 1 + [2.58 2 * 0.5(1-0.5)] / 0.05 2 * 425 ]
- = [6.6564 * 0.25] / 0.0025 / 1 + [6.6564 * 0.25] / 1.0625 ]
- = 665 / 2.5663
- = 259.39 (final answer)

## Creating a Formula for Unknown or Very Large Populations

- Note that this equation is merely the top half of the full formula.

- Example: Determine the necessary survey size for an unknown population with a 90% confidence level, 50% standard of deviation, a 3% margin of error.
- For 90% confidence, use the z-score would be 1.65.

- = [1.65 2 * 0.5(1-0.5)] / 0.03 2
- = [2.7225 * 0.25] / 0.0009
- = 0.6806 / 0.0009
- = 756.22 (final answer)

## Using Slovin's Formula

- Note that this is the least accurate formula and, as such, the least ideal. You should only use this if circumstances prevent you from determining an appropriate standard of deviation and/or confidence level (thereby preventing you from determining your z-score, as well).

- Example: Calculate the necessary survey size for a population of 240, allowing for a 4% margin of error.

- = 240 / (1 + 240 * 0.04 2 )
- = 240 / (1 + 240 * 0.0016)
- = 240 / (1 + 0.384}
- = 240 / (1.384)
- = 173.41 (final answer)

## Calculator, Practice Problems, and Answers

## Community Q&A

## You Might Also Like

- ↑ https://www.khanacademy.org/math/ap-statistics/gathering-data-ap/sampling-observational-studies/e/identifying-population-sample
- ↑ https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample/estimating-population-proportion/v/margin-of-error-1
- ↑ https://www.khanacademy.org/math/ap-statistics/xfb5d8e68:inference-categorical-proportions/introduction-confidence-intervals/a/interpreting-confidence-levels-and-confidence-intervals
- ↑ https://www.mathsisfun.com/data/standard-deviation-formulas.html
- ↑ https://www.mathsisfun.com/data/standard-deviation.html
- ↑ https://prudencexd.weebly.com/
- ↑ https://www.mathsisfun.com/data/sampling.html

## About This Article

To calculate sample size, first find the population size, or number of people taking your study, and margin of error, which is the amount of error you'll allow in your results. Then, calculate your confidence level, which is how confident you are in percentage terms that your results will fall within your margin of error, and z-score, a constant value linked to your confidence level. Next, specify your standard of deviation, which is the amount of variation you expect in your results. Finally, plug your variables into the standard formula to figure out the sample size. To learn how to create a formula for unknown populations, read on! Did this summary help you? Yes No

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Research Paper Guide

Writing Research Proposal

## Research Proposal Writing - A Step-by-Step Guide

13 min read

Published on: Dec 9, 2017

Last updated on: Oct 30, 2023

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If you're a student, you've probably heard about research proposals, but what are they exactly, and how do you write one without feeling overwhelmed?

A research proposal is a plan of your research paper that shows what you want to study, how you'll do it, and why it's important.

But don't worry if it sounds a bit complicated at first; we're here to make it clear and straightforward.

In this guide, we'll break down research proposals into easy-to-understand steps.

We'll explain how to structure your proposal, share examples, and tell you about common mistakes to avoid.

Let's begin!

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## What is a Research Proposal?

According to the research proposal definition, it is like a roadmap. It's a document that explains what you want to study, why it's important, and how you plan to do it. Think of it as your guidebook for conducting research.

## How Long is a Research Proposal?

A research proposal typically ranges from 1,500 to 3,000 words in length for Bachelor’s or Master’s. However, the proposals for PhD dissertations are long and detailed due to their complexity when developing research strategies.

The precise length can vary depending on the academic institution, funding agency, or specific guidelines provided for your research proposal.

## Why Research Proposal is Important?

A research proposal has different purposes, including:

- Clarifying Intentions: It forces you to think about what you want to investigate and why it matters.
- Seeking Approval: In many academic settings, you need to get approval for your research. A well-written proposal is your ticket to gaining that approval.
- Funding Your Research: If you need financial support for your research, a proposal is often required. It helps funding organizations understand the value of your work.

## Key Questions to Address in Your Research Proposal

When crafting a research proposal, it's essential to address several key questions to ensure that your proposal is comprehensive and well-structured.

Here are the fundamental questions to consider:

- What is Your Research Topic or Problem?

This question asks you to define the central issue or question that your research intends to explore. It's the starting point for your proposal and sets the stage for what you aim to investigate.

- Why is Your Research Important?

Here, you explain the significance of your research. You need to clarify why your study is relevant and what impact it may have on your field of study or on society as a whole.

- What is Your Research Objective or Hypothesis?

You should state what you intend to achieve or discover through your research. If applicable, you can also provide a hypothesis, which is a tentative answer to your research question.

- What Previous Research Exists?

This involves conducting a brief literature review to identify existing research related to your topic. You should outline what has been studied before and highlight any gaps or unanswered questions that your research addresses.

- What is Your Research Methodology?

Here, you describe the methods and techniques you plan to use in your research. You need to be specific about how you will collect and analyze data, which is crucial for evaluating the validity of your study.

- What Are the Expected Outcomes?

This question asks you to outline the expected results or findings of your research. What do you hope to discover, prove, or contribute to your field? It provides a preview of the potential impact of your study.

## Research Proposal Format

The components of a research proposal included in the format are explained in detail below.

In a research proposal, the title page is the very first section, serving as the cover page for your document.

It typically includes the following essential elements:

- Main Title of the Research Work
- Student's Name
- Supervisor's Name
- Institution and Department

## Abstract and Table of Contents

In a research proposal, the abstract is a concise summary of your research proposal, providing a snapshot of its key elements. Keep the abstract brief, typically within 250 words, and include:

- A concise statement of your research topic and its significance.
- A brief overview of your research objectives or questions.
- A summary of your research methodology.
- The expected outcomes or contributions of your study.

The table of contents is a structured outline of your research proposal's contents. It acts as a roadmap, aiding readers in navigating the document efficiently. A well-organized table of contents typically includes:

- Section headings and subheadings.
- Page numbers for each section.
- A clear hierarchy that reflects the document's structure.

A well-structured research proposal outline is a way to organize your ideas before writing a research paper. It will decide what headings and subheadings the research paper will have.

## Research Paper Introduction

The introduction of your research paper serves as a concise and compelling entry point for your readers. It should include:

- Introduction of the Topic: Briefly introduce the subject matter and what readers can expect from your paper.
- Main Research Problem: Clearly state the central research question or problem you are addressing.
- Background of the Issue: Provide context by explaining the importance of the problem and any gaps in existing research.
- Methodology: Mention the research methods you've employed.
- Significance: Explain why your research matters and its potential impact.
- Future Plan: Conclude with an overview of your paper's structure and research objectives.

## Background and Significance

This section provides essential context and rationale for your research:

- State the Problem: Clearly define the research problem and its complexities.
- Rationale of the Study: Explain why your research is important and its relevance within your field.
- Critical Issues Addressed: Specify the key issues your research aims to resolve.
- Research Methodology: Briefly describe your chosen methods and data sources.
- Scope Clarification: Define the research boundaries to outline what you will and won't cover.
- Key Term Definitions: Provide concise explanations of any specialized terms or concepts.

## Literature Review

A comprehensive literature review in a research proposal is important. It is a thorough analysis of literature sources that are relevant to the research topic. A strong review aims to convince readers about the valuable contribution to the existing knowledge by giving information.

## Key Elements of a Literature Review

Here are the 5 C’s that can make up a literature review.

- Cite: Reference relevant sources to acknowledge previous research.
- Contrast: Highlight differences among theories or findings.
- Compare : Identify similarities and shared insights.
- Connect: Explain how your research builds upon prior work.
- Critique: Assess the strengths and weaknesses of previous studies.

By using them, compare and contrast the main theories and methods. Also, identify the strengths and weaknesses of the different approaches while writing a literature review.

## Research Design and Methods

In this section, you outline the overall strategy and research proposal steps you'll take to address your research questions.

The key is not just listing methods but demonstrating why your chosen method is the most suitable approach to answer your questions.

The below table will help you identify the methodology in a research proposal.

## Hypothesis

A hypothesis is a critical initial step in defining the purpose of your research. It not only provides a clear objective for researchers but also helps readers understand the essence of your study.

Well-crafted hypotheses streamline the research process, making it more efficient.

## Key Questions to Consider When Formulating a Hypothesis

- What Will be the Research Outcome Concerning the Theoretical Framework and Assumptions?

Define the expected outcome of your study in relation to the theoretical framework and underlying assumptions you've established.

- What Suggestions Could Arise from Research Outcomes?

Anticipate potential recommendations or suggestions that might emerge from the results of your research.

- How Will Results Contribute to the Natural Workplace Setting?

Consider how the research outcomes might impact or enhance the natural dynamics of a workplace or relevant setting.

- Will the Outcomes Contribute to Social and Economic Issues?

Assess whether your research results have implications for broader social or economic problems.

- How Will the Outcomes Influence Policy Decisions?

Explore how your research findings could inform or influence policy decisions at various levels.

- How Can Research Benefits Extend to Individuals or Groups?

Identify the potential benefits that your research might offer to individuals or specific groups within society.

- What Aspects Can Be Improved as a Result of Your Study?

Determine the areas or practices that could be enhanced based on the findings of your research.

- How Will Study Outcomes be Implemented in the Future?

Consider the practical application and implementation of your research outcomes in the future.

The primary purpose of the discussion is to analyze the significance of your findings in the context of the research problem.

Additionally, this section explores new and promising insights that can guide future research studies.

- Highlight Frameworks: Emphasize the frameworks that guided your study.
- Examine Significance: Analyze the importance of your findings in addressing the research problem.
- Connect to Introduction: Maintain alignment with your research's purpose and introduction.
- Present Fresh Insights: Share new insights that emerged from your study.
- Propose Future Research: Suggest directions for future studies based on your research.

## Research Paper Conclusion

The conclusion serves as a concise summary of your entire research study, highlighting its significance and importance.

It should be a brief section, typically comprising one to two paragraphs, focusing on:

- Purpose of the Research Study: Clarify why your research study was conducted and what overarching questions it sought to address.
- Advancement of Existing Knowledge: Emphasize how your research contributes to the current body of knowledge in your field or area of study.
- Relation to Theory or Hypothesis: Discuss how your research aligns with the theoretical framework or hypothesis you proposed earlier in the paper.
- Benefits to Scholars: Consider how your research findings might benefit other scholars, researchers, or practitioners in your field.
- Prospects for Future Implications: Conclude by highlighting the potential implications of your study for future research or practice.

A research proposal formatting must include proper citations for every source that you have used. Similarly, the referencing list should also contain full publication details.

A standard paper proposal has two kinds of citations.

- References - Only list the sources you have used in the proposal.
- Bibliography - List sources used along with other additional citations that you have studied to conduct the research.

Always choose the specific citation formats required by the professors. It includes APA, MLA, and Chicago.

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## Research Proposal Examples

Have a look at the sample research proposal for a better understanding.

APA Research Proposal

Student Research Proposal Example

Research Proposal Sample

Research Proposal Template

Research Proposal on Covid 19

## Research Proposal Topics

Here are some considerations and examples to inspire your research proposal topics:

- The Impact of Technology on Remote Work Productivity
- Gender Disparities in STEM Education
- Climate Change Adaptation Strategies for Urban Areas
- Mental Health and Social Media Use
- The Role of Artificial Intelligence in Healthcare
- Cultural Heritage Preservation in the Digital Age
- The Economics of Sustainable Agriculture
- Online Learning and Student Engagement
- Psychological Resilience in the Face of Natural Disasters
- Exploring Ethical Implications of Genetic Engineering

## Mistake to Avoid when Writing a Research Proposal

When crafting your research proposal, it's crucial to steer clear of common pitfalls that can hinder the quality and effectiveness of your proposal. Here are some key mistakes to avoid:

- Lack of Clarity: Failing to clearly articulate your research questions, objectives, and methodology can lead to confusion among readers.
- Insufficient Literature Review: Neglecting a comprehensive review of existing research can result in a lack of context and relevance for your study.
- Overly Ambitious Scope: Trying to tackle too broad a topic within the constraints of a research proposal can lead to unrealistic expectations and an unfocused study.
- Weak or Absent Justification: Failing to explain the significance and relevance of your research can undermine its credibility.
- Inadequate Methodology: A poorly defined research methodology can raise doubts about the validity and reliability of your study.
- Ignoring Ethical Considerations: Neglecting ethical considerations can have serious consequences for your research and its approval.
- Neglecting Proofreading and Editing: Typos, grammatical errors, and formatting issues can detract from the professionalism of your proposal.

In conclusion, crafting a well-structured and compelling research proposal is an essential step in your academic journey. We've provided you with a complete format guide and useful templates to help you get started. Remember, a strong research proposal is the foundation for a successful research project.

If you find yourself needing further assistance in your academic pursuits don't hesitate to reach out to us. Our best paper writing service is here to assist you every step of the way, ensuring your academic success.

So, contact our research proposal writing service and make your academic dreams a reality!

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Nova Allison is a Digital Content Strategist with over eight years of experience. Nova has also worked as a technical and scientific writer. She is majorly involved in developing and reviewing online content plans that engage and resonate with audiences. Nova has a passion for writing that engages and informs her readers.

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## 17 Research Proposal Examples

A research proposal systematically and transparently outlines a proposed research project.

The purpose of a research proposal is to demonstrate a project’s viability and the researcher’s preparedness to conduct an academic study. It serves as a roadmap for the researcher.

The process holds value both externally (for accountability purposes and often as a requirement for a grant application) and intrinsic value (for helping the researcher to clarify the mechanics, purpose, and potential signficance of the study).

Key sections of a research proposal include: the title, abstract, introduction, literature review, research design and methods, timeline, budget, outcomes and implications, references, and appendix. Each is briefly explained below.

## Research Proposal Sample Structure

Title: The title should present a concise and descriptive statement that clearly conveys the core idea of the research projects. Make it as specific as possible. The reader should immediately be able to grasp the core idea of the intended research project. Often, the title is left too vague and does not help give an understanding of what exactly the study looks at.

Abstract: Abstracts are usually around 250-300 words and provide an overview of what is to follow – including the research problem , objectives, methods, expected outcomes, and significance of the study. Use it as a roadmap and ensure that, if the abstract is the only thing someone reads, they’ll get a good fly-by of what will be discussed in the peice.

Introduction: Introductions are all about contextualization. They often set the background information with a statement of the problem. At the end of the introduction, the reader should understand what the rationale for the study truly is. I like to see the research questions or hypotheses included in the introduction and I like to get a good understanding of what the significance of the research will be. It’s often easiest to write the introduction last

Literature Review: The literature review dives deep into the existing literature on the topic, demosntrating your thorough understanding of the existing literature including themes, strengths, weaknesses, and gaps in the literature. It serves both to demonstrate your knowledge of the field and, to demonstrate how the proposed study will fit alongside the literature on the topic. A good literature review concludes by clearly demonstrating how your research will contribute something new and innovative to the conversation in the literature.

Research Design and Methods: This section needs to clearly demonstrate how the data will be gathered and analyzed in a systematic and academically sound manner. Here, you need to demonstrate that the conclusions of your research will be both valid and reliable. Common points discussed in the research design and methods section include highlighting the research paradigm, methodologies, intended population or sample to be studied, data collection techniques, and data analysis procedures . Toward the end of this section, you are encouraged to also address ethical considerations and limitations of the research process , but also to explain why you chose your research design and how you are mitigating the identified risks and limitations.

Timeline: Provide an outline of the anticipated timeline for the study. Break it down into its various stages (including data collection, data analysis, and report writing). The goal of this section is firstly to establish a reasonable breakdown of steps for you to follow and secondly to demonstrate to the assessors that your project is practicable and feasible.

Budget: Estimate the costs associated with the research project and include evidence for your estimations. Typical costs include staffing costs, equipment, travel, and data collection tools. When applying for a scholarship, the budget should demonstrate that you are being responsible with your expensive and that your funding application is reasonable.

Expected Outcomes and Implications: A discussion of the anticipated findings or results of the research, as well as the potential contributions to the existing knowledge, theory, or practice in the field. This section should also address the potential impact of the research on relevant stakeholders and any broader implications for policy or practice.

References: A complete list of all the sources cited in the research proposal, formatted according to the required citation style. This demonstrates the researcher’s familiarity with the relevant literature and ensures proper attribution of ideas and information.

Appendices (if applicable): Any additional materials, such as questionnaires, interview guides, or consent forms, that provide further information or support for the research proposal. These materials should be included as appendices at the end of the document.

## Research Proposal Examples

Research proposals often extend anywhere between 2,000 and 15,000 words in length. The following snippets are samples designed to briefly demonstrate what might be discussed in each section.

## 1. Education Studies Research Proposals

See some real sample pieces:

- Assessment of the perceptions of teachers towards a new grading system
- Does ICT use in secondary classrooms help or hinder student learning?
- Digital technologies in focus project
- Urban Middle School Teachers’ Experiences of the Implementation of
- Restorative Justice Practices
- Experiences of students of color in service learning

Consider this hypothetical education research proposal:

The Impact of Game-Based Learning on Student Engagement and Academic Performance in Middle School Mathematics

Abstract: The proposed study will explore multiplayer game-based learning techniques in middle school mathematics curricula and their effects on student engagement. The study aims to contribute to the current literature on game-based learning by examining the effects of multiplayer gaming in learning.

Introduction: Digital game-based learning has long been shunned within mathematics education for fears that it may distract students or lower the academic integrity of the classrooms. However, there is emerging evidence that digital games in math have emerging benefits not only for engagement but also academic skill development. Contributing to this discourse, this study seeks to explore the potential benefits of multiplayer digital game-based learning by examining its impact on middle school students’ engagement and academic performance in a mathematics class.

Literature Review: The literature review has identified gaps in the current knowledge, namely, while game-based learning has been extensively explored, the role of multiplayer games in supporting learning has not been studied.

Research Design and Methods: This study will employ a mixed-methods research design based upon action research in the classroom. A quasi-experimental pre-test/post-test control group design will first be used to compare the academic performance and engagement of middle school students exposed to game-based learning techniques with those in a control group receiving instruction without the aid of technology. Students will also be observed and interviewed in regard to the effect of communication and collaboration during gameplay on their learning.

Timeline: The study will take place across the second term of the school year with a pre-test taking place on the first day of the term and the post-test taking place on Wednesday in Week 10.

Budget: The key budgetary requirements will be the technologies required, including the subscription cost for the identified games and computers.

Expected Outcomes and Implications: It is expected that the findings will contribute to the current literature on game-based learning and inform educational practices, providing educators and policymakers with insights into how to better support student achievement in mathematics.

## 2. Psychology Research Proposals

See some real examples:

- A situational analysis of shared leadership in a self-managing team
- The effect of musical preference on running performance
- Relationship between self-esteem and disordered eating amongst adolescent females

Consider this hypothetical psychology research proposal:

The Effects of Mindfulness-Based Interventions on Stress Reduction in College Students

Abstract: This research proposal examines the impact of mindfulness-based interventions on stress reduction among college students, using a pre-test/post-test experimental design with both quantitative and qualitative data collection methods .

Introduction: College students face heightened stress levels during exam weeks. This can affect both mental health and test performance. This study explores the potential benefits of mindfulness-based interventions such as meditation as a way to mediate stress levels in the weeks leading up to exam time.

Literature Review: Existing research on mindfulness-based meditation has shown the ability for mindfulness to increase metacognition, decrease anxiety levels, and decrease stress. Existing literature has looked at workplace, high school and general college-level applications. This study will contribute to the corpus of literature by exploring the effects of mindfulness directly in the context of exam weeks.

Research Design and Methods: Participants ( n= 234 ) will be randomly assigned to either an experimental group, receiving 5 days per week of 10-minute mindfulness-based interventions, or a control group, receiving no intervention. Data will be collected through self-report questionnaires, measuring stress levels, semi-structured interviews exploring participants’ experiences, and students’ test scores.

Timeline: The study will begin three weeks before the students’ exam week and conclude after each student’s final exam. Data collection will occur at the beginning (pre-test of self-reported stress levels) and end (post-test) of the three weeks.

Expected Outcomes and Implications: The study aims to provide evidence supporting the effectiveness of mindfulness-based interventions in reducing stress among college students in the lead up to exams, with potential implications for mental health support and stress management programs on college campuses.

## 3. Sociology Research Proposals

- Understanding emerging social movements: A case study of ‘Jersey in Transition’
- The interaction of health, education and employment in Western China
- Can we preserve lower-income affordable neighbourhoods in the face of rising costs?

Consider this hypothetical sociology research proposal:

The Impact of Social Media Usage on Interpersonal Relationships among Young Adults

Abstract: This research proposal investigates the effects of social media usage on interpersonal relationships among young adults, using a longitudinal mixed-methods approach with ongoing semi-structured interviews to collect qualitative data.

Introduction: Social media platforms have become a key medium for the development of interpersonal relationships, particularly for young adults. This study examines the potential positive and negative effects of social media usage on young adults’ relationships and development over time.

Literature Review: A preliminary review of relevant literature has demonstrated that social media usage is central to development of a personal identity and relationships with others with similar subcultural interests. However, it has also been accompanied by data on mental health deline and deteriorating off-screen relationships. The literature is to-date lacking important longitudinal data on these topics.

Research Design and Methods: Participants ( n = 454 ) will be young adults aged 18-24. Ongoing self-report surveys will assess participants’ social media usage, relationship satisfaction, and communication patterns. A subset of participants will be selected for longitudinal in-depth interviews starting at age 18 and continuing for 5 years.

Timeline: The study will be conducted over a period of five years, including recruitment, data collection, analysis, and report writing.

Expected Outcomes and Implications: This study aims to provide insights into the complex relationship between social media usage and interpersonal relationships among young adults, potentially informing social policies and mental health support related to social media use.

## 4. Nursing Research Proposals

- Does Orthopaedic Pre-assessment clinic prepare the patient for admission to hospital?
- Nurses’ perceptions and experiences of providing psychological care to burns patients
- Registered psychiatric nurse’s practice with mentally ill parents and their children

Consider this hypothetical nursing research proposal:

The Influence of Nurse-Patient Communication on Patient Satisfaction and Health Outcomes following Emergency Cesarians

Abstract: This research will examines the impact of effective nurse-patient communication on patient satisfaction and health outcomes for women following c-sections, utilizing a mixed-methods approach with patient surveys and semi-structured interviews.

Introduction: It has long been known that effective communication between nurses and patients is crucial for quality care. However, additional complications arise following emergency c-sections due to the interaction between new mother’s changing roles and recovery from surgery.

Literature Review: A review of the literature demonstrates the importance of nurse-patient communication, its impact on patient satisfaction, and potential links to health outcomes. However, communication between nurses and new mothers is less examined, and the specific experiences of those who have given birth via emergency c-section are to date unexamined.

Research Design and Methods: Participants will be patients in a hospital setting who have recently had an emergency c-section. A self-report survey will assess their satisfaction with nurse-patient communication and perceived health outcomes. A subset of participants will be selected for in-depth interviews to explore their experiences and perceptions of the communication with their nurses.

Timeline: The study will be conducted over a period of six months, including rolling recruitment, data collection, analysis, and report writing within the hospital.

Expected Outcomes and Implications: This study aims to provide evidence for the significance of nurse-patient communication in supporting new mothers who have had an emergency c-section. Recommendations will be presented for supporting nurses and midwives in improving outcomes for new mothers who had complications during birth.

## 5. Social Work Research Proposals

- Experiences of negotiating employment and caring responsibilities of fathers post-divorce
- Exploring kinship care in the north region of British Columbia

Consider this hypothetical social work research proposal:

The Role of a Family-Centered Intervention in Preventing Homelessness Among At-Risk Youthin a working-class town in Northern England

Abstract: This research proposal investigates the effectiveness of a family-centered intervention provided by a local council area in preventing homelessness among at-risk youth. This case study will use a mixed-methods approach with program evaluation data and semi-structured interviews to collect quantitative and qualitative data .

Introduction: Homelessness among youth remains a significant social issue. This study aims to assess the effectiveness of family-centered interventions in addressing this problem and identify factors that contribute to successful prevention strategies.

Literature Review: A review of the literature has demonstrated several key factors contributing to youth homelessness including lack of parental support, lack of social support, and low levels of family involvement. It also demonstrates the important role of family-centered interventions in addressing this issue. Drawing on current evidence, this study explores the effectiveness of one such intervention in preventing homelessness among at-risk youth in a working-class town in Northern England.

Research Design and Methods: The study will evaluate a new family-centered intervention program targeting at-risk youth and their families. Quantitative data on program outcomes, including housing stability and family functioning, will be collected through program records and evaluation reports. Semi-structured interviews with program staff, participants, and relevant stakeholders will provide qualitative insights into the factors contributing to program success or failure.

Timeline: The study will be conducted over a period of six months, including recruitment, data collection, analysis, and report writing.

Budget: Expenses include access to program evaluation data, interview materials, data analysis software, and any related travel costs for in-person interviews.

Expected Outcomes and Implications: This study aims to provide evidence for the effectiveness of family-centered interventions in preventing youth homelessness, potentially informing the expansion of or necessary changes to social work practices in Northern England.

## Research Proposal Template

This is a template for a 2500-word research proposal. You may find it difficult to squeeze everything into this wordcount, but it’s a common wordcount for Honors and MA-level dissertations.

Your research proposal is where you really get going with your study. I’d strongly recommend working closely with your teacher in developing a research proposal that’s consistent with the requirements and culture of your institution, as in my experience it varies considerably. The above template is from my own courses that walk students through research proposals in a British School of Education.

Chris Drew (PhD)

Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

- Chris Drew (PhD) https://helpfulprofessor.com/author/admin/ 102 Examples of Social Norms (List)
- Chris Drew (PhD) https://helpfulprofessor.com/author/admin/ 15 Social Environment Examples
- Chris Drew (PhD) https://helpfulprofessor.com/author/admin/ 15 Selective Perception Examples
- Chris Drew (PhD) https://helpfulprofessor.com/author/admin/ Field Observation (Research Method): Definition and Examples

## 6 thoughts on “17 Research Proposal Examples”

Very excellent research proposals

very helpful

Very helpful

Dear Sir, I need some help to write an educational research proposal. Thank you.

Hi Levi, use the site search bar to ask a question and I’ll likely have a guide already written for your specific question. Thanks for reading!

very good research proposal

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## IMAGES

## COMMENTS

The sample size for a study needs to be estimated at the time the study is proposed; too large a sample is unnecessary and unethical, and too small a sample is unscientific and also unethical. The necessary sample size can be calculated, using statistical software, based on certain assumptions. If no assumptions can be made, then an arbitrary ...

Similarly, if the sample size is inappropriate it may lead to erroneous conclusions. The aim of this paper is to sensitize our researchers on the importance of proper sampling and sample size ...

Research Proposal Format Example ... B. Sample and Procedures (Chapter 7) 1. Describe your study population and proposed sample (expected size, demographics, etc.) 2. How will the sample be selected? Once they are selected what procedures will they follow as they participate in your study. 2 3. Informed consent: Explain thoroughly how you will ...

Sample size calculation or estimation is an important consideration which necessitate all researchers to pay close attention to when planning a study, which has also become a compulsory consideration for all experimental studies . ... For the development of a research proposal, different institutions may apply different approaches for sample ...

It is the ability of the test to detect a difference in the sample, when it exists in the target population. Calculated as 1-Beta. The greater the power, the larger the required sample size will be. A value between 80%-90% is usually used. Relationship between non-exposed/exposed groups in the sample.

4 How to justify sample size. To justify the sample size for your research proposal or report, you need to explain how you determined the values of the four main elements and how they relate to ...

Research proposal examples. Writing a research proposal can be quite challenging, but a good starting point could be to look at some examples. We've included a few for you below. Example research proposal #1: "A Conceptual Framework for Scheduling Constraint Management" Example research proposal #2: "Medical Students as Mediators of ...

An important step when designing an empirical study is to justify the sample size that will be collected. The key aim of a sample size justification for such studies is to explain how the collected data is expected to provide valuable information given the inferential goals of the researcher. In this overview article six approaches are discussed to justify the sample size in a quantitative ...

Writing Research Proposals. The research proposal is your opportunity to show that you—and only you!—are the perfect person to take on your specific project. After reading your research proposal, readers should be confident that…. You have thoughtfully crafted and designed this project; You have the necessary background to complete this ...

Sample size: The proposal should provide information and justification (basis on which the sample size is calculated) about sample size in the methodology section. 3 A larger sample size than needed to test the research hypothesis increases the cost and duration of the study and will be unethical if it exposes human subjects to any potential unnecessary risk without additional benefit.

If the sample size is low, the research outcome might not be reproducible. 1 Informal guidelines for sample size based on the experience of researchers are used in most research studies and may be sufficient, ... More recent proposals in sample size determination reportedly overcome the design or practical challenges in the field. 7, 59.

2.58. Put these figures into the sample size formula to get your sample size. Here is an example calculation: Say you choose to work with a 95% confidence level, a standard deviation of 0.5, and a confidence interval (margin of error) of ± 5%, you just need to substitute the values in the formula: ( (1.96)2 x .5 (.5)) / (.05)2.

A search was conducted with keywords such as research proposal, writing proposal and qualitative using search engines, namely, PubMed and Google Scholar, and an attempt has been made to provide broad guidelines for writing a scientifically appropriate research proposal. ... The details pertaining to sample size are discussed in the article ...

Introduces your topic. States your problem statement and the questions your research aims to answer. Provides context for your research. In a research proposal, an introduction can be a few paragraphs long. It should be concise, but don't feel like you need to cram all of your information into one paragraph.

Lecture 23 Writing the Sample Size Section for your Proposal 10 28 1. Align power analysis with data analysis 2. Justify the power analysis 3. Account for uncertainty 4. Plan for missing data 5. Demonstrate enrollment feasibility 6. Plan for multiple aims We discussed six components that should be in the sample size section of a grant proposal.

Now that you've got answers for steps 1 - 4, you're ready to calculate the sample size you need. This can be done using the online sample size calculator above or with paper and pencil. Next, you need to turn your confidence level into a Z-score. Here are the Z-scores for the most common confidence levels: Instantly calculate your ideal ...

To calculate your necessary sample size, you'll need to determine several set values and plug them into an appropriate formula. Steps. ... "The formula for calculating sample from large or unknown population met my research needs." Nihi Dosi. Aug 19, 2017 "It helped me a lot in clearing my concepts on estimation of a sample size, thanks a lot!" ...

A Sample Research Proposal with Comments A research project or thesis will take at least two semesters to complete. Prior to starting a research, i.e. enrolling in the first semester research course, students must go through the proposal stage, during which students will develop their proposal and have it reviewed by his/her research advisor. ...

sample size in grant proposals imposes a major handicap on applicants, particularly for those pro-posingworkin anyaspect ofresearchin the health services. Members of research committees need evidencethatastudyis ofadequatesizeforthereto ... effect for a given sample size, effect size, and level of significance.

Keep the abstract brief, typically within 250 words, and include: A concise statement of your research topic and its significance. A brief overview of your research objectives or questions. A summary of your research methodology. The expected outcomes or contributions of your study.

The length of the research proposal template is not long. You'll find most writing requirements demand a word count of 2500 to 3000 words. Depending on spacing requirements, this might be three to five pages of information. This ensures concise writing.

17 Research Proposal Examples. By Chris Drew (PhD) / September 21, 2023. A research proposal systematically and transparently outlines a proposed research project. The purpose of a research proposal is to demonstrate a project's viability and the researcher's preparedness to conduct an academic study. It serves as a roadmap for the researcher.

The fuzzy logic model is proposed to determine the sample size in qualitative research. Design/methodology/approach Considering the structure of the problem in the present study, the proposed ...

A research proposal sample that has been previously downloaded may help the student by giving information such as: The paper format. You will grasp enough knowledge about how the paper should be formatted without making any flimsy errors and how many pages and words should be in the paper like 1000 word essay .