sample size estimation thesis

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• v.28(2); 2021 Apr

A Step-by-Step Process on Sample Size Determination for Medical Research

Determination of a minimum sample size required for a study is a major consideration which all researchers are confronted with at the early stage of developing a research protocol. This is because the researcher will need to have a sound prerequisite knowledge of inferential statistics in order to enable him/her to acquire a thorough understanding of the overall concept of a minimum sample size requirement and its estimation. Besides type I error and power of the study, some estimates for effect sizes will also need to be determined in the process to calculate or estimate the sample size. The appropriateness in calculating or estimating the sample size will enable the researchers to better plan their study especially pertaining to recruitment of subjects. To facilitate a researcher in estimating the appropriate sample size for their study, this article provides some recommendations for researchers on how to determine the appropriate sample size for their studies. In addition, several issues related to sample size determination were also discussed.

Introduction

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 ( 1 ). Moreover, nowadays, the selection of an appropriate sample size is also drawing much attention from researchers who are involved in observational studies when they are developing research proposals as this is now one of the factors that provides a valid justification for the application of a research grant ( 2 ). Sample size must be estimated before a study is conducted because the number of subjects to be recruited for a study will definitely have a bearing on the availability of vital resources such as manpower, time and financial allocation for the study. Nevertheless, a thorough understanding of the need to estimate or calculate an appropriate sample size for a study is crucial for a researcher to appreciate the effort expended in it.

Ideally, one can determine the parameter of a variable from a population through a census study. A census study recruits each and every subject in a population and an analysis is conducted to determine the parameter or in other words, the true value of a specific variable will be calculated in a targeted population. This approach of analysis is known as descriptive analysis. On the other hand, the estimate that is derived from a sample study is termed as a ‘statistic’ because it analyses sample data and subsequently makes inferences and conclusions from the results. This approach of analysis is known as inferential analysis, which is also the most preferred approach in research because drawing a conclusion from the sample data is much easier than performing a census study, due to various constraints especially in terms of cost, time and manpower.

In a census study, the accuracy of the parameters cannot be disputed because the parameters are derived from all subjects in the population. However, when statistics are derived from a sample, it is possible for readers to query to what extent these statistics are representative of the true values in the population. Thus, researchers will need to provide an additional piece of evidence besides the statistics, which is the P -value. The statistical significance or usually termed as ‘ P -value less than 0.05’, and it shall stand as an evidence or justification that the statistics derived from the sample can be inferred to the larger population. Some scholars may argue over the utility and versatility of P -value but it is nevertheless still applicable and acceptable until now ( 3 – 5 ).

Why It is Necessary to Perform a Sample Size Calculation or Estimation?

In order for the analysis to be conducted for addressing a specific objective of a study to be able to generate a statistically-significant result, a particular study must be conducted using a sufficiently large sample size that can detect the target effect sizes with an acceptable margin of error. In brief, a sample size is determined by three elements: i) type I error (alpha); ii) power of the study (1-type II error) and iii) effect size. A proper understanding of the concept of type I error and type II error will require a lengthy discussion. The prerequisite knowledge of statistical inference, probability and distribution function is also required to understand the overall concept ( 6 – 7 ). However, in sample size calculation, the values of both type I and type II errors are usually fixed. Type I error is usually fixed at 0.05 and sometimes 0.01 or 0.10, depending on the researcher. Meanwhile, power is usually set at 80% or 90% indicating 20% or 10% type II error, respectively. Hence, the only one factor that remains unspecified in the calculation of a sample size is the effect size of a study.

Effect size measures the ‘magnitude of effect’ of a test and it is independent of influences by the sample size ( 8 ). In other words, effect size measures the real effect of a test irrespective of its sample size. With reference to statistical tests, it is an expected parameter of a particular association (or correlation or relationship) with other tests in a targeted population. In a real setting, the parameter of a variable in a targeted population is usually unknown and therefore a study will be conducted to test and confirm these effect sizes. However, for the purpose of sample size calculation, it is still necessary to estimate the target effect sizes. By the same token, Cohen ( 9 ) presented in his article that a larger sample size is necessary to estimate small effect sizes and vice versa.

The main advantage of estimating the minimum sample size required is for planning purposes. For example, if the minimum sample size required for a particular study is estimated to be 300 subjects and a researcher already knows that he/she can only recruit 15 subjects in a month from a single centre. Thus, the researchers will need at least 20 months for data collection if there is only one study site. If the plan for data collection period is shorter than 20 months, then the researchers may consider to recruit subjects in more than one centre. In case where the researchers will not be able to recruit 300 subjects within the planned data collection period, the researchers may need to revisit the study objective or plan for a totally different study instead. If the researcher still wishes to pursue the study but is unable to meet the minimum required sample size; then it is likely that the study may not be able to reach a valid conclusion at the end, which will result in a waste of resources because it does not add any scientific contributions.

How to Calculate or Estimate Sample Size?

Sample size calculation serves two important functions. First, it aims to estimate a minimum sample size that can be sufficient for achieving a target level of accuracy in an estimate for a specific population parameter. In this instance, the researcher aims to produce an estimate that is expected to be equally accurate as an actual parameter in the target population. Second, it also aims to determine the level of statistical significance (i.e. P -value < 0.05) attained by these desired effect sizes. In other words, a researcher aims to infer the statistics derived from the sample to that of the larger population. In this case, a specific statistical test will be applied and the P -value will be calculated by using the statistical test (which will determine the level of statistical significance).

For univariate statistical test such as independent sample t -test or Pearson’s chi-square test, these sample size calculations can be done manually using a rather simple formula. However, the manual calculation can still be difficult for researchers who are non-statisticians. Various sample size software have now been introduced which make these sample size calculation easier. Nevertheless, a researcher may still experience some difficulty in using the software if he/she is not familiar with the concept of sample size calculation and the statistical tests. Therefore, various scholars have expended some effort to assist the researchers in the determination of sample sizes for various purposes by publishing sample size tables for various statistical tests ( 10 – 12 ). These sample sizes tables can be used to estimate the minimum sample size that is required for a study. Although such tables may have only a limited capacity for the selection of various effect sizes, and their corresponding sample size requirements; it is nonetheless much more practical and easier to use.

For some study objectives, it is often much easier to estimate the sample size based on a rule-of-thumb instead of manual calculation or sample size software. Taking an example of an objective of a study that needs to be answered using multivariate analysis, the estimation of an association between a set of predictors and an outcome can be very complicated if it involves many independent variables. In addition, the actual ‘effect size’ can range from low to high, which renders it even more difficult to be estimated. Therefore, it is recommended to adopt the conventional rule-of-thumb for estimating these sample sizes in these circumstances. Although some scholars have initially thought that the concept of rule-of-thumb may not be as scientifically robust when compared to actual calculations, it is still considered to be an acceptable approach ( 13 – 15 ). Table 1 illustrates some published articles for various sample size determinations for descriptive studies and statistical tests.

Summary of published articles related to sample size determination for various statistical tests

In brief, the present paper will be proposing five main steps for sample size determination as shown in Figure 1 . The following provides an initial description and then a discussion of each of these five steps:

Recommended steps in sample size determination

Step 1: To Understand the Objective of the Study

The objective of a study has to be measurable or in other words, can be determined by using statistical analysis. Sometimes, a single study may have several objectives. One of the common approaches to achieve this is to estimate the sample size required for every single objective and then the minimum required sample size for the study will be selected to be the highest number of all sample sizes calculated. However, this paper recommends that the minimum sample size be calculated only for the primary objective, which will remain valid as long as the primary objective is more important than all the other objectives. This also means that the calculation of minimum sample size for any other objectives (apart from the primary objective) will only be considered unless they are considered to be equally important as the primary objective. For the development of a research proposal, different institutions may apply different approaches for sample size determinations and hence, it is mandatory to adhere to their specific requirements for sample size determinations.

However, the estimation or calculation of sample size for every study objective can be further complicated by the fact that some of the secondary objectives may require a larger sample size than the primary objective. If the recruitment of a larger number of subjects is not an issue, then it will always be viable to obtain a larger sample size in order to accommodate the sample size requirements for each and every objective of the study. Otherwise, it may be advisable for a researcher to forgo some of the secondary objectives so that they will not be too burdensome for the him/her.

Step 2: To Select the Appropriate Statistical Analysis

Researchers have to decide the appropriate analysis or statistical test to be used to answer the study objective; regardless of whether the aim is to determine a single mean, or a prevalence, or correlation, or association, just to name a few. The formula that will be used to estimate or calculate the sample size will be the same as the formula for performing the statistical test that will be used to answer the objective of study. For example, if an independent sample t -test has to be used for analysis, then its sample size formula should be based on an independent sample t -test. Hence, there is no a single formula for sample size calculation or estimation which can apply universally to all situations and circumstances.

Step 3: To Calculate or Estimate the Sample Size

Estimating or calculating the sample size can be done either by using manual calculation, sample size software, sample size tables from scientific published articles, or by adopting various acceptable rule-of-thumbs. Since both the type I and type II errors are already pre-specified and fixed, hence only the effect size remains to be specified in order for the determination of an appropriate sample size. To illustrate this point, it will be easier to demonstrate by using a case scenario as an example. Say a researcher would like to study an effectiveness of a new diet programme to reduce weight. The researcher believes the new diet programme is better than the conventional diet programme. It was found that the conventional diet programme can reduce on average 1 kg in 1 month. How many subjects are required to prove that the new diet programme is better than the conventional diet programme?

Based on Step 1 and Step 2, a researcher has decided to apply the independent sample t -test to answer the objective of study. Next, the researcher will need to specify the effect size after having both type I error and power set at 0.05% and 80%, respectively (type II error = 20%). What margin of effect size will be appropriate? This shall depend on the condition itself or the underlying research rationale which can then be further classified into two categories. In the first category, the research rationale is to prove that the new diet programme (for reducing weight) is superior to the conventional diet programme. In this case, the researcher should aim for sizeably large effect size. In other words, the difference between means of the weight reduction (which constitutes part of the effect size for independent sample t -test) should be sufficiently large to demonstrate the superiority of the new diet programme over the conventional diet programme.

In the second category, the research rationale is to measure accurately the effectiveness of the new diet programme to reduce weight in comparison with conventional diet programme, irrespective of whether the difference between both programmes is large or small. In this situation, the difference does not matter since the researcher aims to measure an exact difference between them, which means that it can only tolerate a very low margin of difference. In this circumstance, the researcher will therefore only be able to accept the smaller effect sizes. The estimate of effect sizes in this instance can be reviewed either from literatures, pilot study, historical data and rarely by using an educated guess.

The acceptable or desirable effect size that can be found from the literature can vary over a wide range. Thus, one of the better options is to seek for the relevant information from published articles of recent studies (within 5 years) that applied almost similar research design such as used the same treatments and had reported about similar patient characteristics. If none of these published articles can provide a rough estimate of the desired effect size, then the researcher may have to consider conducting a pilot study to obtain a rough estimate of the closest approximation to the actual desired effect size. Besides, historical data or secondary data can also be used to estimate the desired effect size, provided that the researcher has access to the secondary data of the two diet programmes. However, it must be emphasised that deriving the effect size from secondary data may not always be feasible since the performance of the new intervention may still not yet have been assessed.

The last option is to estimate the desired effect size based on a scientifically or a clinically meaningful effect. This means the researcher, through his or her own knowledge and experience, is able to determine an expectation of the difference in effect, and then to set a target difference (namely, effect size) to be achieved. For example, a researcher makes an educated guess about the new diet programme, and requires it to achieve a minimum difference of 3 kg in weight reduction per month in order for it to demonstrate superiority over the conventional diet programme. Although it is always feasible to set a large effect size especially if the new diet programme has proven to be a more rigorous intervention and probably also costlier; however, there is also a risk for the study to might have possibly failed to report a statistically significant result if it has subsequently been found that the actual effect size is much smaller than that adopted by the study, after the analysis has been completed. Therefore, it is usually quite a challenging task to estimate an accurate effect size since the exact value of the effect size is not known until the study is completed. However, the researcher will still have to set the value of effect size for the purpose of sample size calculation or estimation.

Next is to calculate or estimate sample size either based on manual calculation, sample size software, sample size tables or by adopting a conventional rule of thumb. Referring to the example for illustration purposes, the sample size calculation was calculated by using the sample size software as follows; with a study setting of equal sample size for both groups, the mean reduction is set at only 1 kg with within group standard deviation estimated at 0.8 (derived from literature, pilot study or based on a reliable source), type I error at 0.05 and 80% power, a minimum sample size of 11 subjects are required for each group (both for new diet programme and conventional diet programme). The sample size was calculated using Power and Sample Size (PS) software (by William D Dupont and W Dale Plummer, Jr. is licensed under a Creative Commons Attribution-NonCommercial- NoDerivs 3.0 United States License).

Step 4: To Provide an Additional Allowance During Subject Recruitment to Cater for a Certain Proportion of Non-Response

After the minimum required sample size has been identified, it is necessary to provide additional allowances to cater for potential non-response subjects. A minimum required sample size simply means the minimum number of subjects a study must have after recruitment is completed. Thus, researchers must ideally be able to recruit subjects at least beyond the minimum required sample size. To avoid underestimation of sample size, researchers will need to anticipate the problem of non-response and then to make up for it by recruiting more subjects on top of the minimum sample size, usually by 20% to 30%. If, for example, the researcher is expecting a high non-response rate in a self-administered survey, then he/she should provide an allowance for it by adding more than 30% such as 40% to 50%. The occurrence of non-response could also happen in various other scenarios such as dropping out or loss to follow-up in a cohort study and experimental studies. Besides that, missing data or loss of records are also potential problems that can result in attrition in observational studies.

Referring to previous example as an illustration, by adding 20% of non-response rate in each group, 14 subjects are required in each group. The calculation should be done as follow:

Likewise, for a 30% non-response rate, the sample size required in each group will then be increased to 16 subjects (11/0.7 = 15.7 ≈ 16).

Step 5: To Write a Sample Size Statement

The sample size statement is important and it is usually included in the protocol or manuscript. In the existing research literatures, the sample size statement is written in various styles. This paper recommends for the sample size statement to start by reminding the readers or reviewers about the main objective of study. Hence, this paper recommends all the elements from Step 1 until Step 4 (study objective, appropriate statistical analysis, sample size estimation/calculation and non-response rate) should be fully stated in the sample size statement. Therefore, a proposed outline of this sample size statement of the previous example for two weight-losing diet programmes is as follows:

“This study hypothesised that the new diet programme is better than conventional diet programme in terms of weight reduction at a 1-month follow-up. Therefore, the sample size formula is derived from the independent sample t -test. Based on the results of a previous study (cite the appropriate reference), all the response within each subject group are assumed to be normally distributed with a within-group standard deviation (SD) of 0.80 kg. If the true mean difference of the new diet programme versus the conventional diet programme is estimated at 1.0 kg, the study will need to recruit 11 subjects in each group to be able to reject the null hypothesis that the population means of the new diet programme and conventional diet programme are found to be equal with a type I error of 0.05 and with at least 80% power of this study. By providing an additional allowance of 20% in sample recruitment due to possible non-response rate, the required sample size has been increased to 14 subjects in each group. The formula of sample size calculation is based on a study reported by Dupont and Plummer ( 31 ).”

Discussion on Effect Size Planning

Sample size is just an estimate indicating a minimum required number of sample data that is necessary to be collected to derive an accurate estimate for the target population or to obtain statistically significant results based on the desired effect sizes. In order to calculate or estimate sample size, researchers will need to provide some initial estimates for effect sizes. It is usually quite challenging to provide a reasonably accurate value of the effect size because the exact values of these effect sizes are usually not known and can only be derived from the study after the analysis is completed. Hence, the discrepancies of the effect sizes are commonly expected where the researchers will usually either overestimate or underestimate them.

A major problem often arises when the researchers overestimate the effect sizes during sample size estimation, which can lead to a failure of a study to detect a statistically significant result. To avoid such a problem, the researchers are encouraged to recruit more subjects than the minimum required sample size of the study. By referring to the same example previously (new diet programme versus conventional diet programme), if the required sample size is 11 subjects in each group, then researchers may consider recruiting more than 11 subjects such as 18 to 20 subjects in each group. This is possible if the researchers have the capability in terms of manpower and research grant to recruit more subjects and also if there are adequate number of subjects available to be recruited.

After the study is completed, if the difference in mean reduction was found not at least 1 kg after 1 month, then the result might not be statistically significant (depending on the actual value of the within-group SD) for a sample size of 11 subjects in each group. However, if the researchers had recruited 18 subjects in each group, the study will still obtain significant results even though the difference of mean reduction was 0.8 kg (if the within-group SD is estimated to be 0.8, and an equal sample size is planned for both groups, with type I error set at 0.05 and power of at least 80%). In this situation, researcher would still be able to draw a conclusion that the difference in mean reduction after one month was 0.8 kg, and this result was statistically significant. Such a conclusion is perhaps more meaningful than stating a non-significant result ( P > 0.05) for another study with only 11 subjects in each group.

However, it is necessary to always bear in mind that obtaining a larger sample size merely to show that P -value is less than 0.05 is not the right thing to do and it can also result in a waste of resources. Hence, the purpose of increasing the size of the sample from 11 to 18 per group is not merely for obtaining a P -value of less than 0.05; but more importantly, it is now able to draw a valid and clinically-significant conclusion from the smallest acceptable value of the effect size. In other words, the researcher is now able to tolerate a smaller effect size by stating that the difference in mean reduction of 0.8 kg is also considered to be a sizeable effect size because it is clinically significant. However, if the researcher insists that the difference in mean reduction should be at least 1.0 kg, then it will be necessary to maintain a minimum sample of only 11 subjects per group. It is now clear that such a subjective variation in the overall consideration of the magnitude of this effect size sometimes depends on the effectiveness and the cost of the new diet programme and hence, this will always require some degree of clinical judgement.

The concept of setting a desired value of the effect size is almost identical for all types of statistical test. The above example is only describing an analysis using the independent sample t -test. Since each statistical test may require a different effect size in its calculation or estimation of the sample size; thus, it is necessary for the researchers to be familiarised with each of these statistical tests in order to be able to set the desired values of the effect sizes for the study. In addition, further assistance may be sought from statisticians or biostatisticians for the determination of an adequate minimum sample size required for these studies.

Another Example of Sample Size Estimation Using General Rule of Thumb

Say a study aims to determine the association of factors with optimal HbA1c level as determined by its cut-off point of < 6.5% among patients with type 2 diabetes mellitus (T2DM). Previous study had already estimated that several significant factors were identified, and then included as three to four variables in the final model consisting of parameters that were selected from demographic profile of patients and clinical parameters (cite the appropriate reference). Now, the question is: How many T2DM patients should the study recruit in order to answer the study objective?

Step 1: To Understand the Objective of Study

The study aims to determine a set of independent variables that show a significant association with optimal HbA1c level (as determined by its cut-off point of < 6.5%) among T2DM patients.

Step 2: To Decide the Appropriate Statistical Analysis

In this example, the outcome variable is in the categorical and binary form, such as HbA1c level of < 6.5% versus ≥ 6.5%. On the other hand, there are about 3 to 4 independent variables, which can be expressed in both the categorical and numerical form. Therefore, an appropriate statistical analysis shall be logistic regression.

Step 3: To Estimate or Calculate the Sample Size Required

Since this study will require a multivariate regression analysis, thus it is recommended to estimate sample size based on the general rule of thumb. This is because the calculation of sample size for a multivariate regression analysis can be very complicated as the analysis will involve many variables and effect sizes. There are several general rules of thumb available for estimating the sample size for multivariate logistic regression. One of the latest rule of thumb is proposed by Bujang et al. ( 44 ). Two approaches are introduced here, namely: i) sample size estimation based on concept of event per variable (EPV) and ii) sample size estimation based on a simple formula.

• i) Sample size estimation based on a concept EPV 50

For EPV 50, the researcher will need to know the prevalence of the ‘good’ outcome category and the number of subjects in the ‘good’ outcome category to fit the rule of EPV 50 ( 14 , 44 ). Say, the prevalence of ‘good’ outcome category is reported at 70% (cite the appropriate reference). Then, with a total of four independent variables, the minimum sample size required in the ‘poor’ outcome category will be at least 200 subjects in order to fulfil the condition for EPV 50 (i.e. 200/4 = 50). On the other hand, by estimating the prevalence of ‘good’ outcome at 70.0%, this study will therefore need to recruit at least 290 subjects in order to ensure that a minimum 200 subjects will be obtained in the ‘poor’ outcome category (70/100 x 290 = 203, and 203 > 200).

• ii) Sample size estimation based on a formula of n = 100 + 50i (where i represents number of independent variable in the final model)

When using this formula, the researcher will first need to set the total number of independent variables in the final model ( 44 ). As stated in the example, the total number of independent variables were estimated to be about three to four (cite the appropriate reference). Then, with a total of four independent variables, the minimum required sample size will be 300 patients [(i.e. 100 + 50 ( 4 ) = 300].

Step 4: To Provide Additional Allowance for a Certain Proportion of Non-Response Rate

In order to make up for a rough estimate of 20.0% of non-response rate, the minimum sample size requirement is calculated to be 254 patients (i.e. 203/0.8) by estimating the sample size based on the EPV 50, and is calculated to be 375 patients (i.e. 300/0.8) by estimating the sample size based on the formula n = 100 + 50 i.

There were previously two approaches that were introduced to estimate sample size for logistic regression. Say, if the researcher chooses to apply the formula n = 100 + 50 i. Therefore, the sample size statement will be written as follows:

“The main objective of this study is to determine the association of factors with optimal HbA1c level as determined by its cut-off point of < 6.5% among patients with type 2 diabetes mellitus (T2DM). The sample size estimation is derived from the general rule of thumb for logistic regression proposed by Bujang et al. ( 44 ), which had established a simple guideline of sample size determination for logistic regression. In this study, Bujang et al. ( 44 ) suggested to calculate the sample size by basing on a formula n = 100 + 50 i. The estimated total number of independent variables was about three to four (cite the appropriate reference). Thus, with a total of four independent variables, the minimum required sample size will be 300 patients (i.e. 100 + 50 ( 4 ) = 300). By providing an additional allowance to cater for a possible dropout rate of 20%, this study will therefore need at least a sample size of 300/0.8 = 375 patients.”

Other Issues

Previously, there are four different approaches to estimate an effect size such as: i) by deriving it from the literature; ii) by using historical data or secondary data to estimate it; iii) by determining the clinical meaningful effect and last but not least and iv) by deriving it from the results of a pilot study. It is a controversial practice to estimate the effect size from a pilot study because it may not be accurate since the effect size has been derived from a small sample provided by a pilot study ( 52 – 55 ). In reality, many researchers often encounter great difficulties in the estimation of sample size either i) when the required effect size is not reported by the existing literature; or ii) if some new, innovative research proposals which may pose pioneering research questions that have never been addressed; or iii) if the research is examining a new intervention or exploring a new research area in where no similar studies have previously been conducted. Although there are many concerns about validity of using pilot studies for power calculation, further research is still being conducted in pilot studies in order to apply more scientifically robust approaches for using pilot studies in gathering preliminary support for subsequent research. For example, there are now many published studies regarding guidelines for estimating sample size requirements in pilot studies ( 54 – 61 ).

This article has sought to provide a brief but clear guidance on how to determine the minimum sample size requirements for all researchers. Sample size calculation can be a difficult task, especially for the junior researcher. However, the availability of sample size software, and sample size tables for sample size determination based on various statistical tests, and several recommended rules of thumb which can be helpful for guiding the researchers in the determination of an adequate sample size for their studies. For the sake of brevity and convenience, this paper hereby proposes a useful checklist that is presented in Table 2 , which aims to guide and assist all researchers to determine an adequate sample size for their studies.

A step-by-step guide for sample size determination

Acknowledgements

I would like to thank the Director General of Health, Ministry of Health Malaysia for his permission to publish this article. I would also thank Dr Ang Swee Hung and Mr Hoon Yon Khee for proofreading this article.

Conflict of Interest

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How to determine sample size.

12 min read Sample size can make or break your research project. Here’s how to master the delicate art of choosing the right sample size.

Author:  Will Webster

Sample size is the beating heart of any research project. It’s the invisible force that gives life to your data, making your findings robust, reliable and believable.

Sample size is what determines if you see a broad view or a focus on minute details; the art and science of correctly determining it involves a careful balancing act. Finding an appropriate sample size demands a clear understanding of the level of detail you wish to see in your data and the constraints you might encounter along the way.

Remember, whether you’re studying a small group or an entire population, your findings are only ever as good as the sample you choose.

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Let’s delve into the world of sampling and uncover the best practices for determining sample size for your research.

“How much sample do we need?” is one of the most commonly-asked questions and stumbling points in the early stages of research design . Finding the right answer to it requires first understanding and answering two other questions:

How important is statistical significance to you and your stakeholders?

What are your real-world constraints.

At the heart of this question is the goal to confidently differentiate between groups, by describing meaningful differences as statistically significant. Statistical significance isn’t a difficult concept, but it needs to be considered within the unique context of your research and your measures.

First, you should consider when you deem a difference to be meaningful in your area of research. While the standards for statistical significance are universal, the standards for “meaningful difference” are highly contextual.

For example, a 10% difference between groups might not be enough to merit a change in a marketing campaign for a breakfast cereal, but a 10% difference in efficacy of breast cancer treatments might quite literally be the difference between life and death for hundreds of patients. The exact same magnitude of difference has very little meaning in one context, but has extraordinary meaning in another. You ultimately need to determine the level of precision that will help you make your decision.

Within sampling, the lowest amount of magnification – or smallest sample size – could make the most sense, given the level of precision needed, as well as timeline and budgetary constraints.

If you’re able to detect statistical significance at a difference of 10%, and 10% is a meaningful difference, there is no need for a larger sample size, or higher magnification. However, if the study will only be useful if a significant difference is detected for smaller differences – say, a difference of 5% — the sample size must be larger to accommodate this needed precision. Similarly, if 5% is enough, and 3% is unnecessary, there is no need for a larger statistically significant sample size.

You should also consider how much you expect your responses to vary. When there isn’t a lot of variability in response, it takes a lot more sample to be confident that there are statistically significant differences between groups.

For instance, it will take a lot more sample to find statistically significant differences between groups if you are asking, “What month do you think Christmas is in?” than if you are asking, “How many miles are there between the Earth and the moon?”. In the former, nearly everybody is going to give the exact same answer, while the latter will give a lot of variation in responses. Simply put, when your variables do not have a lot of variance, larger sample sizes make sense.

Statistical significance

The likelihood that the results of a study or experiment did not occur randomly or by chance, but are meaningful and indicate a genuine effect or relationship between variables.

Magnitude of difference

The size or extent of the difference between two or more groups or variables, providing a measure of the effect size or practical significance of the results.

Actionable insights

Valuable findings or conclusions drawn from data analysis that can be directly applied or implemented in decision-making processes or strategies to achieve a particular goal or outcome.

It’s crucial to understand the differences between the concepts of “statistical significance”, “magnitude of difference” and “actionable insights” – and how they can influence each other:

• Even if there is a statistically significant difference, it doesn’t mean the magnitude of the difference is large: with a large enough sample, a 3% difference could be statistically significant
• Even if the magnitude of the difference is large, it doesn’t guarantee that this difference is statistically significant: with a small enough sample, an 18% difference might not be statistically significant
• Even if there is a large, statistically significant difference, it doesn’t mean there is a story, or that there are actionable insights

There is no way to guarantee statistically significant differences at the outset of a study – and that is a good thing.

Even with a sample size of a million, there simply may not be any differences – at least, any that could be described as statistically significant. And there are times when a lack of significance is positive.

Imagine if your main competitor ran a multi-million dollar ad campaign in a major city and a huge pre-post study to detect campaign effects, only to discover that there were no statistically significant differences in brand awareness . This may be terrible news for your competitor, but it would be great news for you.

With Stats iQ™ you can analyze your research results and conduct significance testing

As you determine your sample size, you should consider the real-world constraints to your research.

Factors revolving around timings, budget and target population are among the most common constraints, impacting virtually every study. But by understanding and acknowledging them, you can definitely navigate the practical constraints of your research when pulling together your sample.

Timeline constraints

Gathering a larger sample size naturally requires more time. This is particularly true for elusive audiences, those hard-to-reach groups that require special effort to engage. Your timeline could become an obstacle if it is particularly tight, causing you to rethink your sample size to meet your deadline.

Budgetary constraints

Every sample, whether large or small, inexpensive or costly, signifies a portion of your budget. Samples could be like an open market; some are inexpensive, others are pricey, but all have a price tag attached to them.

Population constraints

Sometimes the individuals or groups you’re interested in are difficult to reach; other times, they’re a part of an extremely small population. These factors can limit your sample size even further.

What’s a good sample size?

A good sample size really depends on the context and goals of the research. In general, a good sample size is one that accurately represents the population and allows for reliable statistical analysis.

Larger sample sizes are typically better because they reduce the likelihood of sampling errors and provide a more accurate representation of the population. However, larger sample sizes often increase the impact of practical considerations, like time, budget and the availability of your audience. Ultimately, you should be aiming for a sample size that provides a balance between statistical validity and practical feasibility.

4 tips for choosing the right sample size

Choosing the right sample size is an intricate balancing act, but following these four tips can take away a lot of the complexity.

1) Start with your goal

The foundation of your research is a clearly defined goal. You need to determine what you’re trying to understand or discover, and use your goal to guide your research methods – including your sample size.

If your aim is to get a broad overview of a topic, a larger, more diverse sample may be appropriate. However, if your goal is to explore a niche aspect of your subject, a smaller, more targeted sample might serve you better. You should always align your sample size with the objectives of your research.

2) Know that you can’t predict everything

Research is a journey into the unknown. While you may have hypotheses and predictions, it’s important to remember that you can’t foresee every outcome – and this uncertainty should be considered when choosing your sample size.

A larger sample size can help to mitigate some of the risks of unpredictability, providing a more diverse range of data and potentially more accurate results. However, you shouldn’t let the fear of the unknown push you into choosing an impractically large sample size.

3) Plan for a sample that meets your needs and considers your real-life constraints

Every research project operates within certain boundaries – commonly budget, timeline and the nature of the sample itself. When deciding on your sample size, these factors need to be taken into consideration.

Be realistic about what you can achieve with your available resources and time, and always tailor your sample size to fit your constraints – not the other way around.

4) Use best practice guidelines to calculate sample size

There are many established guidelines and formulas that can help you in determining the right sample size.

The easiest way to define your sample size is using a sample size calculator , or you can use a manual sample size calculation if you want to test your math skills. Cochran’s formula is perhaps the most well known equation for calculating sample size, and widely used when the population is large or unknown.

Beyond the formula, it’s vital to consider the confidence interval, which plays a significant role in determining the appropriate sample size – especially when working with a random sample – and the sample proportion. This represents the expected ratio of the target population that has the characteristic or response you’re interested in, and therefore has a big impact on your correct sample size.

If your population is small, or its variance is unknown, there are steps you can still take to determine the right sample size. Common approaches here include conducting a small pilot study to gain initial estimates of the population variance, and taking a conservative approach by assuming a larger variance to ensure a more representative sample size.

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Equip yourself with the tools for success with our essential eBook, “The ultimate guide to conducting market research” .

With this front-to-back guide, you’ll discover the latest strategies and best practices that are defining effective market research. Learn about practical insights and real-world applications that are demonstrating the value of research in driving business growth and innovation.

Related resources

Selection bias 11 min read, systematic random sampling 15 min read, convenience sampling 18 min read, probability sampling 8 min read, non-probability sampling 17 min read, stratified random sampling 12 min read, simple random sampling 9 min read, request demo.

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8   Sample Size Justification

You can listen to an audio recording of this chapter here . You can download a translation of this chapter in Chinese here

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. In this chapter, which is largely identical to Lakens ( 2022 ) , we will explore in detail how to justify your sample size.

8.1 Six Approaches to Justify Sample Sizes

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  8.1 ). This is not an exhaustive overview, but it includes the most common and applicable approaches for single studies. The topic of power analysis for meta-analyses is outside the scope of this chapter, but see Hedges & Pigott ( 2001 ) and Valentine et al. ( 2010 ) . 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.

8.2 Six Ways to Evaluate Which Effect Sizes are Interesting

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  8.2 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 Lakens ( 2022 ) 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.

8.3 The Value of Information

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.

8.4 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.

8.5 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 & 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 & 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  8.3 ). 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 chapter can be found at https://shiny.ieis.tue.nl/sample_size_justification/.

8.6 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 \(\alpha\) level. A test yields a significant result, even though the null hypothesis is true.
• A false negative (or Type II error), determined by \(\beta\) , or 1 - power. A test yields a non-significant result, even though the alternative hypothesis is true.
• A true negative, determined by 1- \(\alpha\) . A test yields a non-significant result when the null hypothesis is true.
• A true positive, determined by 1- \(\beta\) . 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. 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. Figure  8.1 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, 2021 ) .

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  8.2 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. These figures can be reproduced and adapted in an online shiny app .

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 (that is, 136 participants in each condition) 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.

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, 2021 ; 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  8.5 (if the effect size is based on a meta-analysis) or Table  8.6 (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, et al. ( 2018 ) ). For an overview of all aspects that should be reported when describing an a-priori power analysis, see Table  8.4 .

8.7 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:

\[N = \left(\frac{z \cdot sd}{error}\right)^2\]

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. 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 (see also the section on which confidence interval to use in Bayesian tests of range predictions ).

8.8 Heuristics

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 that 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 & 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  8.6 ).

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. That said, 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.

8.9 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.

8.10 What is Your Inferential Goal?

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 effects that is likely to be observed in a specific research area.

8.11 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. The 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 the 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 et al. ( 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 the 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). Put differently, although the 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 smaller 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), once researchers determine the smallest effect size of interest, 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 ) .

8.12 The Minimal Statistically Detectable Effect

The minimal statistically detectable effect is the smallest effect size that, if observed, would yield a statistically significant p -value ( Cook et al., 2014 ) . In Figure  8.4 , 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  8.2 , 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.

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  8.3 ).

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.

8.13 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.

8.14 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  8.5 ). 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 there is substantial heterogeinity in the meta-analytic effect sizes, 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).

8.15 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  8.5 shows the distribution for \(\eta_p^2\) for a study with three conditions with 25 participants in each condition when the null hypothesis is true (dotted grey curve) and when there is a ‘medium’ true effect of \(\eta_p^2\) = 0.0588 [solid black curve; Richardson ( 2011 ) ]. As in Figure  8.4 the critical effect size is indicated, which shows observed effects smaller than \(\eta_p^2\) = 0.08 will not be significant with the given sample size. If the null hypothesis is true, effects larger than \(\eta_p^2\) = 0.08 will be a Type I error (the dark grey area), and when the alternative hypothesis is true effects smaller than \(\eta_p^2\) = 0.08 will be a Type II error (light grey area). It is clear all significant effects are larger than the true effect size ( \(\eta_p^2\) = 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  8.5 shows it is quite likely to observe an effect of \(\eta_p^2\) = 0.01 in a small pilot study, even when the true effect size is 0.0588. Entering an effect size estimate of \(\eta_p^2\) = 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  8.5 . The only studies that would be accessible to researchers would come from the part of the distribution where \(\eta_p^2\) > 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, \(\eta_p^2\) = 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 ( Anderson et al., 2017 ) to perform a power analysis that attempts to correct for bias. In the example above, 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 instead. 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 et al. ( 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  8.6 ). 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.

8.16 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.

8.17 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 - \(\alpha\) 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 = -0.716 to d = 0.716. Confidence intervals around effect sizes can be computed using the MOTE Shiny app: https://www.aggieerin.com/shiny-server/. The margin of error is half the width of the 95% CI, 0.716. 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 = 0.748). 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  8.6 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 to 1.484, 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.

8.18 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  8.7 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. Fixing the sample 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  8.8 . 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  8.8 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.

8.19 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, 2021 ) . 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., 2021 ) . 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.

8.20 Additional Considerations When Designing an Informative 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 & 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.

8.21 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 the smallest effect size of interest of d = 0.2 the researcher is 8.3 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. This default use of 80% power (or a Type II error rate/ \(\beta\) of 0.20) 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 \(\beta\) 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 \(\alpha\) of .05. The \(\beta\) 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 et al. ( 1996 ) created the G*Power software in part to give researchers a tool to perform compromise power analysis.

Figure  8.9 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 (that is, when the \(\beta/\alpha\) ratio = 1), 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 \(\alpha\) = .05 (in the case of small samples or effect sizes) or less than \(\alpha\) = .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. Alternatively, using compromise power analysis, we can choose to weigh both types of error equally ( \(\beta/\alpha\) ratio = 1) by setting both the alpha level and Type II error rate to 0.149. Doing so, we would have 85.10% power to detect the expected population effect size of d = 0.5 instead. The choice of \(\alpha\) and \(\beta\) 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 < \(\alpha\) ) 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  8.7 lists three aspects that should be discussed alongside a reported compromise power analysis.

8.22 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  8.6 ), 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  8.10 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. Alternatively, if no smallest effect size of interest was specified when designing the study, researchers can report a sensitivity power analysis.

8.23 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 or gsDesign . Shiny apps are available for both rpact and gsDesign .

8.24 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 (one-sided) tests where relevant, instead of two-sided tests. 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 et al. ( 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:

\[NW = \frac{NB (1-\rho)}{2}\]

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- \(\rho\) ) 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.

In Figure  8.11 we see two normally distributed scores with a mean difference of 6, where the standard deviation of each mean is 15, and the correlation between the measurements is 0. The standard deviation of the difference score is \(\sqrt{2}\) times as large as the standard deviation in each measurement, and indeed, 15× \(\sqrt{2}\) = 21.21, which is rounded to 21. This situation where the correlation between measurements is zero equals the situation in an independent t -test, where the correlation between measurements is not taken into account.

In Figure  8.12 we can see what happens when the two variables are correlated, for example with r = 0.7. Nothing has changed when we plot the means. The correlation between measurements is now strongly positive, and the important difference is in the standard deviation of the difference scores, which is 11 instead of 21 in the uncorrelated example. Because the standardized effect size is the difference divided by the standard deviation, the effect size (Cohen’s \(d_z\) in within designs) is larger in this test than in the uncorrelated test.

The correlation between dependent variables is an important aspect of within designs. I recommend explicitly reporting the correlation between dependent variables in within designs (e.g., participants responded significantly slower ( M = 390, SD = 44) when they used their feet than when they used their hands ( M = 371, SD = 44, r = .953), t (17) = 5.98, p < 0.001, Hedges’ g = 0.43, \(M_{diff}\) = 19, 95% CI [12; 26]). Since most dependent variables in within designs in psychology are positively correlated, within designs will increase the power you can achieve given the sample size you have available. Use within-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-subject design ( Maxwell et al., 2017 ) .

You can use this Shiny app to play around with different means, standard deviations, and correlations, and see the effect of the distribution of the difference scores.

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.

8.25 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, 2021 ; 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 \(\sigma^2\) is provided in the following formula, and the confidence for the standard deviation is the square root of these limits:

\[(N - 1)s^2/\chi^2_{N-1:\alpha/2},(N - 1)s^2/\chi^2_{N-1:1-\alpha/2}\]

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 ) .

8.26 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 et al. ( 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.

8.27 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.

8.28 Discussion

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.

8.29 Test Yourself

Q1 : A student has at most 2 months to collect data. They need to pay participants for their participation, and their budget is limited to 250 euro. They decide to collect all the participants they can in the amount of time, and with the money they have available. What type of sample size justification is this?

Q2 : What is the goal of an a-priori power analysis?

Q3 : A researcher already knows the sample size they will be able to collect. Given this sample size, they choose to compute equal Type 1 and Type 2 error rates for an effect size of interest. This is known as:

Q4 : Looking at the formula in the section ‘Increasing Power Without Increasing the Sample Size’. which two factors contribute to the fact that within subject designs can have much more power, with the same number of participants, than between subject designs?

Q5 : Which factors determine the minimal statistically detectable effect?

Q6 : All else equal, if you want to perform a study that has the highest possible informational value, which approach to specifying the effect size of interest is the best choice?

Q7 : In an a-priori power analysis based on an empirical estimate of the literature, which 2 issues are important to consider, both when using an estimate from a meta-analysis, as from a single study?

Q8 : Imagine a researcher did not justify their sample size before performing the study, and had no justification for the sample size they choose. After submitting their scientific article to a journal reviewers ask for a justification of the sample size. Of course, honesty requires the authors to write down there was no justification, but how can they still evaluate the informational value of the study for effect sizes of interest?

Q9 : Why can it be useful to consider the effect size distribution of findings in a specific research area when evaluating the informational value of the study you are planning?

Q10 : Why is it nonsensical to ask researchers to perform a post-hoc or retrospective power analysis, where the observed effect size and the collected sample size is used to calculate the statistical power of a test, when a non-significant finding is observed?

Q11 : Researchers should not perform a post-hoc power analysis. There are two solutions, one that can be implemented when designing a study, and one when interpreting a non-significant result after the data is in. Which solution can be implemented when the data is in?

Q12 : What is a way/are ways to increase the statistical power of a test, without increasing the sample size?

8.29.1 Open Questions

Why are resource constraints, if not the primary justification, always a secondary justification (if it is not possible to collect data from the entire population)?

What is the goal of an a-priori power analysis, and why is the goal not to achieve a desired Type 2 error rate for the true effect size?

Which factors determine the Minimal Statistically Detectable Effect, and why can it be useful to compute it for a study you are planning to perform?

What is a benefit of planning for precision, given that the effect size is typically unknown (and might even be 0). Which aspect of the decisions that need to be made when planning for precision is most difficult to justify?

What is a problem of using heuristics as the basis of a sample size justification?

It seems counter-intuitive to have a ‘no justification’ category in a chapter on sample size justification, but why is it important to explicitly state there was no justification?

From all effect sizes that might be related to the inferential goal in a study, which of the 6 categories in Table  8.2 is the best approach (if it can be specified)?

Why can’t you simply take an effect size estimate from a meta-analysis as the basis of an a-priori power analysis for a related study?

Why can’t you simply take an effect size estimate from a single study as the basis of an a-priori power analysis for a related study?

What is the goal in a compromise power analysis?

Why is ‘post-hoc’ or ‘retrospective’ power not a useful way to draw inferences about non-significant results?

When would you perform a sensitivity power analysis?

How can the statistical power of a study be increased, without increasing the sample size?

Why can it be beneficial to use a within-design compared to a between-design (where possible)?

Sample Size Calculator

Sample size calculation formula, how to calculate sample size: an example, other useful tools beyond the sample size calculator.

If you're conducting research and wonder how many measurements you need so that it is statistically significant , this sample size calculator is here to help you. All you need to do is ask yourself these three questions before you use it:

• How accurate should your result be? (margin of error)
• What level of confidence do you need? (confidence level)
• What is your initial estimate ? (proportion estimate)

Read on to learn how to calculate the sample size using this tool, and what do all the variables in the sample size calculation formula mean.

The equation that our sample size calculator uses is:

• Z Z Z — The z-score associated with the confidence level you chose. Our statistical significance calculator calculates this value automatically (shown in Advanced mode ), but if you want to learn how to calculate it by hand, take a look at the instructions of our confidence interval calculator .
• M E \mathrm{ME} ME — Margin of error, also known as the confidence interval . It tells you that you can be sure (with a probability of confidence level, for example, 95%), that the real value doesn't differ from the one that you obtained by more than this percentage. You can learn more about it at our margin of error calculator .
• p p p — Your initial proportion estimate. For example, if you are conducting a survey among students trying to find out how many of them read more than 5 books last year, you may know a result of a previous survey — 40%. If you have no such estimate, use the conservative value of 50%.
• n 1 n_1 n 1 ​ — Required sample size.

If your population is finite — for example, you are conducting a survey among students of only one faculty — you need to include a correction in the following form:

• N N N — Total population size.
• n 2 n_2 n 2 ​ — Size of the sample taken from the whole population that will make your research statistically significant.

We will analyze a survey case step-by-step, so you can get a clear picture of how to use our sample size calculator . You are planning to conduct a survey to find out what is the proportion of students on your campus who regularly eat their lunch at the campus canteen.

Decide how accurate you want your result to be. Let's say that it is important for the canteen to know the result, with a margin of error of 2 % 2\% 2% maximum.

Decide on your confidence level . We can assume you want to be 99 % 99\% 99% sure that your result is correct.

Do you have an initial proportion guess ? Let's say you accessed a similar survey from 10 years ago and the proportion was equal to 30 % 30\% 30% . You can assume it as your initial estimate.

Is the total population of students so high that you can assume it's infinite ? Probably not. You need to find the current data for the number of students on the campus — let's assume it is 25 , 000 25,000 25 , 000 .

All you need to do now is to input all this data into our sample size calculator. It finds the sample size required for the result to be statistically significant is 3 , 051 3,051 3 , 051 . You need to ask that many students the same question… Are you sure you can't settle for a 95 % 95\% 95% confidence level? 😀

Now that you know how to calculate sample size, you can go beyond and use it to calculate other statistics of interest in your research:

Sampling error calculator : sample size is the most influential feature when predicting the sampling error. Use it to calculate the error of your sample.

Normal probability calculator for sampling distributions : use your sample size, along with the population mean and standard deviation, to find the probability that your sample mean falls within a specific range.

Sampling distribution of the sample proportion calculator : use your sample size and the population proportion to find the probability that your sample proportion falls within a specific range.

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Sample Size Calculator

Find out the sample size.

This calculator computes the minimum number of necessary samples to meet the desired statistical constraints.

Find Out the Margin of Error

This calculator gives out the margin of error or confidence interval of observation or survey.

Related Standard Deviation Calculator | Probability Calculator

In statistics, information is often inferred about a population by studying a finite number of individuals from that population, i.e. the population is sampled, and it is assumed that characteristics of the sample are representative of the overall population. For the following, it is assumed that there is a population of individuals where some proportion, p , of the population is distinguishable from the other 1-p in some way; e.g., p may be the proportion of individuals who have brown hair, while the remaining 1-p have black, blond, red, etc. Thus, to estimate p in the population, a sample of n individuals could be taken from the population, and the sample proportion, p̂ , calculated for sampled individuals who have brown hair. Unfortunately, unless the full population is sampled, the estimate p̂ most likely won't equal the true value p , since p̂ suffers from sampling noise, i.e. it depends on the particular individuals that were sampled. However, sampling statistics can be used to calculate what are called confidence intervals, which are an indication of how close the estimate p̂ is to the true value p .

Statistics of a Random Sample

The uncertainty in a given random sample (namely that is expected that the proportion estimate, p̂ , is a good, but not perfect, approximation for the true proportion p ) can be summarized by saying that the estimate p̂ is normally distributed with mean p and variance p(1-p)/n . For an explanation of why the sample estimate is normally distributed, study the Central Limit Theorem . As defined below, confidence level, confidence intervals, and sample sizes are all calculated with respect to this sampling distribution. In short, the confidence interval gives an interval around p in which an estimate p̂ is "likely" to be. The confidence level gives just how "likely" this is – e.g., a 95% confidence level indicates that it is expected that an estimate p̂ lies in the confidence interval for 95% of the random samples that could be taken. The confidence interval depends on the sample size, n (the variance of the sample distribution is inversely proportional to n , meaning that the estimate gets closer to the true proportion as n increases); thus, an acceptable error rate in the estimate can also be set, called the margin of error, ε , and solved for the sample size required for the chosen confidence interval to be smaller than e ; a calculation known as "sample size calculation."

Confidence Level

The confidence level is a measure of certainty regarding how accurately a sample reflects the population being studied within a chosen confidence interval. The most commonly used confidence levels are 90%, 95%, and 99%, which each have their own corresponding z-scores (which can be found using an equation or widely available tables like the one provided below) based on the chosen confidence level. Note that using z-scores assumes that the sampling distribution is normally distributed, as described above in "Statistics of a Random Sample." Given that an experiment or survey is repeated many times, the confidence level essentially indicates the percentage of the time that the resulting interval found from repeated tests will contain the true result.

Confidence Interval

In statistics, a confidence interval is an estimated range of likely values for a population parameter, for example, 40 ± 2 or 40 ± 5%. Taking the commonly used 95% confidence level as an example, if the same population were sampled multiple times, and interval estimates made on each occasion, in approximately 95% of the cases, the true population parameter would be contained within the interval. Note that the 95% probability refers to the reliability of the estimation procedure and not to a specific interval. Once an interval is calculated, it either contains or does not contain the population parameter of interest. Some factors that affect the width of a confidence interval include: size of the sample, confidence level, and variability within the sample.

There are different equations that can be used to calculate confidence intervals depending on factors such as whether the standard deviation is known or smaller samples (n<30) are involved, among others. The calculator provided on this page calculates the confidence interval for a proportion and uses the following equations:

Within statistics, a population is a set of events or elements that have some relevance regarding a given question or experiment. It can refer to an existing group of objects, systems, or even a hypothetical group of objects. Most commonly, however, population is used to refer to a group of people, whether they are the number of employees in a company, number of people within a certain age group of some geographic area, or number of students in a university's library at any given time.

It is important to note that the equation needs to be adjusted when considering a finite population, as shown above. The (N-n)/(N-1) term in the finite population equation is referred to as the finite population correction factor, and is necessary because it cannot be assumed that all individuals in a sample are independent. For example, if the study population involves 10 people in a room with ages ranging from 1 to 100, and one of those chosen has an age of 100, the next person chosen is more likely to have a lower age. The finite population correction factor accounts for factors such as these. Refer below for an example of calculating a confidence interval with an unlimited population.

EX: Given that 120 people work at Company Q, 85 of which drink coffee daily, find the 99% confidence interval of the true proportion of people who drink coffee at Company Q on a daily basis.

Sample Size Calculation

Sample size is a statistical concept that involves determining the number of observations or replicates (the repetition of an experimental condition used to estimate the variability of a phenomenon) that should be included in a statistical sample. It is an important aspect of any empirical study requiring that inferences be made about a population based on a sample. Essentially, sample sizes are used to represent parts of a population chosen for any given survey or experiment. To carry out this calculation, set the margin of error, ε , or the maximum distance desired for the sample estimate to deviate from the true value. To do this, use the confidence interval equation above, but set the term to the right of the ± sign equal to the margin of error, and solve for the resulting equation for sample size, n . The equation for calculating sample size is shown below.

EX: Determine the sample size necessary to estimate the proportion of people shopping at a supermarket in the U.S. that identify as vegan with 95% confidence, and a margin of error of 5%. Assume a population proportion of 0.5, and unlimited population size. Remember that z for a 95% confidence level is 1.96. Refer to the table provided in the confidence level section for z scores of a range of confidence levels.

Thus, for the case above, a sample size of at least 385 people would be necessary. In the above example, some studies estimate that approximately 6% of the U.S. population identify as vegan, so rather than assuming 0.5 for p̂ , 0.06 would be used. If it was known that 40 out of 500 people that entered a particular supermarket on a given day were vegan, p̂ would then be 0.08.

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Sample Size Determination: Definition, Formula, and Example

Are you ready to survey your research target? Research surveys help you gain insights from your target audience. The data you collect gives you insights to meet customer needs, leading to increased sales and customer loyalty. Sample size calculation and determination are imperative to the researcher to determine the right number of respondents, keeping in mind the research study’s quality.

So, how should you do the sample size determination? How do you know who should get your survey? How do you decide on the number of the target audience?

Sending out too many surveys can be expensive without giving you a definitive advantage over a smaller sample. But if you send out too few, you won’t have enough data to draw accurate conclusions. 

Knowing how to calculate and determine the appropriate sample size accurately can give you an edge over your competitors. Let’s take a look at what a good sample includes. Also, let’s look at the sample size calculation formula so you can determine the perfect sample size for your next survey.

What is Sample Size?

‘Sample size’ is a market research term used for defining the number of individuals included in conducting research. Researchers choose their sample based on demographics, such as age, gender questions , or physical location. It can be vague or specific. 

For example, you may want to know what people within the 18-25 age range think of your product. Or, you may only require your sample to live in the United States, giving you a wide population range. The total number of individuals in a particular sample is the sample size.

What is sample size determination?

Sample size determination is the process of choosing the right number of observations or people from a larger group to use in a sample. The goal of figuring out the sample size is to ensure that the sample is big enough to give statistically valid results and accurate estimates of population parameters but small enough to be manageable and cost-effective.

In many research studies, getting information from every member of the population of interest is not possible or useful. Instead, researchers choose a sample of people or events that is representative of the whole to study. How accurate and precise the results are can depend a lot on the size of the sample.

Choosing the statistically significant sample size depends on a number of things, such as the size of the population, how precise you want your estimates to be, how confident you want to be in the results, how different the population is likely to be, and how much money and time you have for the study. Statistics are often used to figure out how big a sample should be for a certain type of study and research question.

Figuring out the sample size is important in ensuring that research findings and conclusions are valid and reliable.

Why do you need to determine the sample size?

Let’s say you are a market researcher in the US and want to send out a survey or questionnaire . The survey aims to understand your audience’s feelings toward a new cell phone you are about to launch. You want to know what people in the US think about the new product to predict the phone’s success or failure before launch.

Hypothetically, you choose the population of New York, which is 8.49 million. You use a sample size determination formula to select a sample of 500 individuals that fit into the consumer panel requirement. You can use the responses to help you determine how your audience will react to the new product.

However, determining a sample size requires more than just throwing your survey at as many people as possible. If your estimated sample sizes are too big, it could waste resources, time, and money. A sample size that’s too small doesn’t allow you to gain maximum insights, leading to inconclusive results.

LEARN ABOUT: Survey Sample Sizes

What are the terms used around the sample size?

Before we jump into sample size determination, let’s take a look at the terms you should know:

1. Population size: 

Population size is how many people fit your demographic. For example, you want to get information on doctors residing in North America. Your population size is the total number of doctors in North America. 

Don’t worry! Your population size doesn’t always have to be that big. Smaller population sizes can still give you accurate results as long as you know who you’re trying to represent.

2. Confidence level: 

The confidence level tells you how sure you can be that your data is accurate. It is expressed as a percentage and aligned to the confidence interval. For example, if your confidence level is 90%, your results will most likely be 90% accurate.

3. The margin of error (confidence interval): 

There’s no way to be 100% accurate when it comes to surveys. Confidence intervals tell you how far off from the population means you’re willing to allow your data to fall. 

A margin of error describes how close you can reasonably expect a survey result to fall relative to the real population value. Remember, if you need help with this information, use our margin of error calculator .

4. Standard deviation: 

Standard deviation is the measure of the dispersion of a data set from its mean. It measures the absolute variability of a distribution. The higher the dispersion or variability, the greater the standard deviation and the greater the magnitude of the deviation. 

For example, you have already sent out your survey. How much variance do you expect in your responses? That variation in response is the standard deviation.

Sample size calculation formula – sample size determination

With all the necessary terms defined, it’s time to learn how to determine sample size using a sample calculation formula.

Your confidence level corresponds to a Z-score. This is a constant value needed for this equation. 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 choose a different confidence level, various online tools can help you find your score.

Necessary Sample Size = (Z-score)2 * StdDev*(1-StdDev) / (margin of error)2

Here is an example of how the math works, assuming you chose a 90% confidence level, .6 standard deviation, and a margin of error (confidence interval) of +/- 4%.

((1.64)2 x .6(.6)) / (.04)2

( 2.68x .0.36) / .0016

.9648 / .0016

603 respondents are needed, and that becomes your sample size.

Free Sample Size Calculator

How is a sample size determined?

Determining the right sample size for your survey is one of the most common questions researchers ask when they begin a market research study. Luckily, sample size determination isn’t as hard to calculate as you might remember from an old high school statistics class.

Before calculating your sample size, ensure you have these things in place:

Goals and objectives: 

What do you hope to do with the survey? Are you planning on projecting the results onto a whole demographic or population? Do you want to see what a specific group thinks? Are you trying to make a big decision or just setting a direction? 

Calculating sample size is critical if you’re projecting your survey results on a larger population. You’ll want to make sure that it’s balanced and reflects the community as a whole. The sample size isn’t as critical if you’re trying to get a feel for preferences. 

For example, you’re surveying homeowners across the US on the cost of cooling their homes in the summer. A homeowner in the South probably spends much more money cooling their home in the humid heat than someone in Denver, where the climate is dry and cool. 

For the most accurate results, you’ll need to get responses from people in all US areas and environments. If you only collect responses from one extreme, such as the warm South, your results will be skewed.

Precision level: 

How close do you want the survey results to mimic the true value if everyone responded? Again, if this survey determines how you’re going to spend millions of dollars, then your sample size determination should be exact. 

The more accurate you need to be, the larger the sample you want to have, and the more your sample will have to represent the overall population. If your population is small, say, 200 people, you may want to survey the entire population rather than cut it down with a sample.

Confidence level: 

Think of confidence from the perspective of risk. How much risk are you willing to take on? This is where your Confidence Interval numbers become important. How confident do you want to be — 98% confident, 95% confident? 

Understand that the confidence percentage you choose greatly impacts the number of completions you’ll need for accuracy. This can increase the survey’s length and how many responses you need, which means increased costs for your survey. 

Knowing the actual numbers and amounts behind percentages can help make more sense of your correct sample size needs vs. survey costs. 

For example, you want to be 99% confident. After using the sample size determination formula, you find you need to collect an additional 1000 respondents. 

This, in turn, means you’ll be paying for samples or keeping your survey running for an extra week or two. You have to determine if the increased accuracy is more important than the cost.

Population variability: 

What variability exists in your population? In other words, how similar or different is the population?

If you are surveying consumers on a broad topic, you may have lots of variations. You’ll need a larger sample size to get the most accurate picture of the population. 

However, if you’re surveying a population with similar characteristics, your variability will be less, and you can sample fewer people. More variability equals more samples, and less variability equals fewer samples. If you’re not sure, you can start with 50% variability.

Response rate: 

You want everyone to respond to your survey. Unfortunately, every survey comes with targeted respondents who either never open the study or drop out halfway. Your response rate will depend on your population’s engagement with your product, service organization, or brand. 

The higher the response rate, the higher your population’s engagement level. Your base sample size is the number of responses you must get for a successful survey.

Consider your audience: 

Besides the variability within your population, you need to ensure your sample doesn’t include people who won’t benefit from the results. One of the biggest mistakes you can make in sample size determination is forgetting to consider your actual audience. 

For example, you don’t want to send a survey asking about the quality of local apartment amenities to a group of homeowners.

Select your respondents

Focus on your survey’s objectives: 

You may start with general demographics and characteristics, but can you narrow those characteristics down even more? Narrowing down your audience makes getting a more accurate result from a small sample size easier. 

For example, you want to know how people will react to new automobile technology. Your current population includes anyone who owns a car in a particular market. 

However, you know your target audience is people who drive cars that are less than five years old. You can remove anyone with an older vehicle from your sample because they’re unlikely to purchase your product.

Once you know what you hope to gain from your survey and what variables exist within your population, you can decide how to calculate sample size. Using the formula for determining sample size is a great starting point to get accurate results. 

After calculating the sample size, you’ll want to find reliable customer survey software to help you accurately collect survey responses and turn them into analyzed reports.

LEARN MORE: Population vs Sample

In sample size determination, statistical analysis plan needs careful consideration of the level of significance, effect size, and sample size. 

Researchers must reconcile statistical significance with practical and ethical factors like practicality and cost. A well-designed study with a sufficient sample size can improve the odds of obtaining statistically significant results.

To meet the goal of your survey, you may have to try a few methods to increase the response rate, such as:

• Increase the list of people who receive the survey.
• To reach a wider audience, use multiple distribution channels, such as SMS, website, and email surveys.
• Send reminders to survey participants to complete the survey.
• Offer incentives for completing the survey, such as an entry into a prize drawing or a discount on the respondent’s next order.
• Consider your survey structure and find ways to simplify your questions. The less work someone has to do to complete the survey, the more likely they will finish it. 
• Longer surveys tend to have lower response rates due to the length of time it takes to complete the survey. In this case, you can reduce the number of questions in your survey to increase responses.  

QuestionPro’s sample size calculator makes it easy to find the right sample size for your research based on your desired level of confidence, your margin of error, and the size of the population.

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Frequently Asked Questions (FAQ)

The four ways to determine sample size are: 1. Power analysis 2. Convenience sampling, 3. Random sampling , 4. Stratified sampling

The three factors that determine sample size are: 1. Effect size, 2. Level of significance 3. Power

Using statistical techniques like power analysis, the minimal detectable effect size, or the sample size formula while taking into account the study’s goals and practical limitations is the best way to calculate the sample size.

The sample size is important because it affects how precise and accurate the results of a study are and how well researchers can spot real effects or relationships between variables.

The sample size is the number of observations or study participants chosen to be representative of a larger group

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