{"id":1009,"date":"2026-10-13T18:16:00","date_gmt":"2026-10-13T18:16:00","guid":{"rendered":"https:\/\/freestatisticaltools.com\/blog\/?p=1009"},"modified":"2026-10-13T18:16:00","modified_gmt":"2026-10-13T18:16:00","slug":"sample-size-calculation","status":"publish","type":"post","link":"https:\/\/freestatisticaltools.com\/blog\/sample-size-calculation\/","title":{"rendered":"How to Calculate Sample Size: Surveys, Experiments and Power Analysis"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">&quot;How many participants do I need?&quot; is one of the first questions in any research proposal, and one ethics committees and supervisors will ask you to justify. The answer depends on <strong>what you want to do with the data<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If you want to <strong>estimate<\/strong> something, such as the percentage of customers who are satisfied, you need a sample size based on the <strong>margin of error<\/strong>.<\/li>\n\n<li>If you want to <strong>test a hypothesis<\/strong>, such as whether two groups differ, you need a sample size based on <strong>statistical power<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This guide covers both, with formulas, worked examples and quick-reference tables.<\/p>\n\n\n<div class=\"fst-tool\"><span class=\"ico\"><svg viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"1.8\" stroke-linecap=\"round\" stroke-linejoin=\"round\" aria-hidden=\"true\"><circle cx=\"12\" cy=\"12\" r=\"9\"\/><path d=\"M12 7v5l3 3\"\/><\/svg><\/span><span><span class=\"t-eyebrow\">Free tool \u00b7 runs in your browser<\/span><a class=\"t-name\" href=\"https:\/\/freestatisticaltools.com\/power-analysis-sample-size-calculator\/\" style=\"text-decoration:none;color:inherit\">Power Analysis and Sample Size Calculator<\/a><span class=\"t-desc\">Free a priori power analysis for t-tests, ANOVA, correlation, proportions, chi-square and regression. Required sample size and power curve, like G*Power.<\/span><\/span><a class=\"btn accent\" href=\"https:\/\/freestatisticaltools.com\/power-analysis-sample-size-calculator\/\" data-tool=\"power-analysis-sample-size-calculator\">Calculate sample size \u2192<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Part 1: Sample size for surveys (margin of error)<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">The formula<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">To estimate a proportion with a given margin of error:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><code>n\u2080 = z\u00b2 \u00d7 p \u00d7 (1 \u2212 p) \u00f7 e\u00b2<\/code><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>z<\/strong> is the z-value for your confidence level: 1.96 for 95%, 2.576 for 99%.<\/li>\n\n<li><strong>p<\/strong> is the expected proportion. If you don&#39;t know it, use 0.5, which gives the largest (safest) sample size.<\/li>\n\n<li><strong>e<\/strong> is the margin of error as a decimal (0.05 for \u00b15%).<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example:<\/strong> 95% confidence, \u00b15% margin, p = 0.5:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">n\u2080 = 1.96\u00b2 \u00d7 0.5 \u00d7 0.5 \u00f7 0.05\u00b2 = 384.2 \u2192 <strong>385 respondents<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Finite population correction<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If your population is small (for example, 2,000 employees in one company), you need fewer respondents:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><code>n = n\u2080 \u00f7 (1 + (n\u2080 \u2212 1) \u00f7 N)<\/code><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For N = 2,000: n = 384.2 \u00f7 (1 + 383.2 \u00f7 2,000) = <strong>323 respondents<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Adjust for response rate<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If you expect only 30% of invited people to respond, invite 323 \u00f7 0.30 \u2248 <strong>1,077 people<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Quick-reference table (95% confidence, p = 0.5)<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Population size<\/th><th>\u00b13%<\/th><th>\u00b15%<\/th><th>\u00b110%<\/th><\/tr><\/thead><tbody><tr><td>500<\/td><td>341<\/td><td>218<\/td><td>81<\/td><\/tr><tr><td>1,000<\/td><td>517<\/td><td>278<\/td><td>88<\/td><\/tr><tr><td>5,000<\/td><td>880<\/td><td>357<\/td><td>95<\/td><\/tr><tr><td>10,000<\/td><td>965<\/td><td>370<\/td><td>96<\/td><\/tr><tr><td>100,000+<\/td><td>1,056<\/td><td>383<\/td><td>96<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Beyond about 100,000, population size hardly matters: a national survey and a city survey need roughly the same sample for the same precision. The free <a href=\"\/survey-sample-size-calculator\/\">survey sample size calculator<\/a> does these calculations, including the response-rate adjustment.<\/p>\n\n\n\n<div class=\"fst-note warning\"><b>Big samples don&#039;t fix biased samples<\/b><p>The margin-of-error formula assumes a <strong>random sample<\/strong>. A convenience sample of 2,000 people recruited on social media is not more representative than a random sample of 400; it is just more precise about the wrong population. Describe your sampling method honestly.<\/p><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Part 2: Sample size for hypothesis tests (power analysis)<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">The four ingredients<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A power analysis links four quantities. Fix any three and you can solve for the fourth:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Effect size:<\/strong> how big a difference or relationship you want to be able to detect (Cohen&#39;s d, f, r, f\u00b2 or w).<\/li>\n\n<li><strong>Significance level (\u03b1):<\/strong> usually .05.<\/li>\n\n<li><strong>Power (1 \u2212 \u03b2):<\/strong> the probability of detecting the effect if it really exists. 80% is the conventional minimum; 90% is better for confirmatory studies.<\/li>\n\n<li><strong>Sample size (n).<\/strong><\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">An <strong>a priori<\/strong> power analysis, done <em>before<\/em> data collection, solves for n. This is what proposals and ethics applications need.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Choosing the effect size<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">This is the hardest part, and the most important. In order of preference:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>The smallest effect that would matter in practice<\/strong>, for example a 5-point improvement on a clinical scale.<\/li>\n\n<li><strong>Effect sizes from previous studies or a meta-analysis<\/strong> on the same topic. Published effects are often inflated, so consider using a somewhat smaller value.<\/li>\n\n<li><strong>Cohen&#39;s conventions<\/strong> (small, medium, large) as a last resort.<\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Test<\/th><th>Effect size<\/th><th>Small<\/th><th>Medium<\/th><th>Large<\/th><\/tr><\/thead><tbody><tr><td>t-test<\/td><td>d<\/td><td>0.20<\/td><td>0.50<\/td><td>0.80<\/td><\/tr><tr><td>ANOVA<\/td><td>f<\/td><td>0.10<\/td><td>0.25<\/td><td>0.40<\/td><\/tr><tr><td>Correlation<\/td><td>r<\/td><td>0.10<\/td><td>0.30<\/td><td>0.50<\/td><\/tr><tr><td>Multiple regression<\/td><td>f\u00b2<\/td><td>0.02<\/td><td>0.15<\/td><td>0.35<\/td><\/tr><tr><td>Chi-square<\/td><td>w<\/td><td>0.10<\/td><td>0.30<\/td><td>0.50<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Quick-reference table (\u03b1 = .05 two-tailed, power = .80, medium effect)<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Analysis<\/th><th>Effect size<\/th><th>Total sample needed<\/th><\/tr><\/thead><tbody><tr><td>Independent t-test<\/td><td>d = 0.50<\/td><td>128 (64 per group)<\/td><\/tr><tr><td>Paired t-test<\/td><td>dz = 0.50<\/td><td>34<\/td><\/tr><tr><td>One-way ANOVA, 3 groups<\/td><td>f = 0.25<\/td><td>159 (53 per group)<\/td><\/tr><tr><td>Pearson correlation<\/td><td>r = 0.30<\/td><td>about 85<\/td><\/tr><tr><td>Multiple regression, 3 predictors<\/td><td>f\u00b2 = 0.15<\/td><td>77<\/td><\/tr><tr><td>Multiple regression, 5 predictors<\/td><td>f\u00b2 = 0.15<\/td><td>92<\/td><\/tr><tr><td>Chi-square, df = 1<\/td><td>w = 0.30<\/td><td>88<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">For a <strong>small<\/strong> effect the numbers grow fast: an independent t-test with d = 0.20 needs about 394 per group. For a <strong>large<\/strong> effect (d = 0.80), about 26 per group is enough.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Worked example<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">You plan to compare anxiety scores between a mindfulness group and a control group. A meta-analysis suggests d \u2248 0.45. With \u03b1 = .05, two-tailed, and 90% power, the <a href=\"\/power-analysis-sample-size-calculator\/\">power analysis calculator<\/a> gives <strong>105 participants per group<\/strong>. Allowing for 15% dropout, you recruit 105 \u00f7 0.85 \u2248 124 per group.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Rules of thumb for complex models<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Power analysis for complex models is harder, so researchers often fall back on rules of thumb. Use them cautiously, and say that you did:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Multiple regression:<\/strong> at least 10\u201315 cases per predictor, or N \u2265 50 + 8k for testing the overall model (Green, 1991).<\/li>\n\n<li><strong>Logistic regression:<\/strong> at least 10 events (cases in the smaller outcome group) per predictor.<\/li>\n\n<li><strong>Exploratory factor analysis:<\/strong> at least 5\u201310 respondents per item, and 200+ overall when communalities are modest.<\/li>\n\n<li><strong>CB-SEM:<\/strong> about 200 as a common minimum for moderately complex models with good loadings; more for complex models or non-normal data. Monte Carlo simulation gives a model-specific answer.<\/li>\n\n<li><strong>PLS-SEM:<\/strong> the inverse square root method (Kock &amp; Hadaya, 2018) or a power analysis for the most complex regression in the model.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">How to report your sample size justification<\/h2>\n\n\n\n<div class=\"fst-report\"><b>Example justification<button type=\"button\" data-copy-report>Copy<\/button><\/b><div><p>An a priori power analysis indicated that 105 participants per group would provide 90% power to detect an effect of <i>d<\/i> = 0.45 (based on a recent meta-analysis) in an independent-samples <i>t<\/i>-test with \u03b1 = .05 (two-tailed). Allowing for 15% attrition, we aimed to recruit 124 participants per group.<\/p><\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Frequently asked questions<\/h2>\n\n\n\n<div class=\"fst-faq\"><details><summary>What sample size do I need for a survey?<\/summary><div class=\"a\"><p>For a large population, 95% confidence and a \u00b15% margin of error, you need about 385 respondents. Smaller populations need fewer once you apply the finite population correction, and you should invite more people to allow for non-response.<\/p><\/div><\/details><details><summary>What is statistical power?<\/summary><div class=\"a\"><p>Power is the probability that a study will detect an effect of a given size if it really exists. 80% is the usual minimum, which means a 20% chance of missing a real effect.<\/p><\/div><\/details><details><summary>Why is 80% power the standard?<\/summary><div class=\"a\"><p>It is a convention proposed by Jacob Cohen, balancing the cost of a larger sample against the risk of missing a real effect. Many funders now prefer 90% power for confirmatory studies.<\/p><\/div><\/details><details><summary>Should I do a post hoc power analysis?<\/summary><div class=\"a\"><p>Generally no. Power calculated from the observed effect after the study adds no information beyond the p-value. Report confidence intervals for your effects instead.<\/p><\/div><\/details><details><summary>Is there a free alternative to G*Power?<\/summary><div class=\"a\"><p>Yes. The free power analysis calculator on FreeStatisticalTools.com runs a priori power analyses for t-tests, ANOVA, correlation, proportions, chi-square and regression in your browser, with a power curve.<\/p><\/div><\/details><\/div>\n\n\n<div class=\"fst-report\"><b>Cite the tool (APA 7)<button type=\"button\" data-copy-report>Copy<\/button><\/b><div>Xavier, B. (2026). <i>Free Statistical Tools<\/i> [Computer software]. Zenodo. https:\/\/doi.org\/10.5281\/zenodo.23372149 (web app: https:\/\/freestatisticaltools.com\/power-analysis-sample-size-calculator\/)<\/div><small>Name the tool in your methods: \u201cThe analysis was carried out with the Power Analysis and Sample Size Calculator in Free Statistical Tools.\u201d<\/small><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Work out how many participants you need, whether you are estimating a percentage in a survey or testing a hypothesis, with formulas, quick-reference tables and free calculators.<\/p>\n","protected":false},"author":1,"featured_media":1010,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_fst_seo_title":"How to Calculate Sample Size for a Survey or Study (Free)","_fst_meta_desc":"Calculate the sample size you need for a survey (margin of error) or an experiment (power analysis), with formulas, examples, tables and free calculators.","_fst_primary_tool":"power-analysis-sample-size-calculator","_fst_canonical":"","_fst_noindex":false,"_fst_no_toc":false,"_fst_no_autolink":false,"footnotes":""},"categories":[9],"tags":[51,15,50,47],"class_list":["post-1009","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-sample-size-power-calculators","tag-power-analysis","tag-research-methods","tag-sample-size","tag-thesis"],"_links":{"self":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/posts\/1009","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/comments?post=1009"}],"version-history":[{"count":0,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/posts\/1009\/revisions"}],"wp:attachment":[{"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/media?parent=1009"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/categories?post=1009"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/freestatisticaltools.com\/blog\/wp-json\/wp\/v2\/tags?post=1009"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}