Compute the sample size needed for your study, or solve for power or detectable effect size. Supports independent and paired t-tests, one-way ANOVA, chi-square tests, and Pearson correlations with preset effect size conventions.
Compute the required sample size per group for an independent-samples t-test, or solve for power or minimum detectable effect size.
n = ((zα/2 + zβ)² × (1 + 1/ratio)) / d²
Choose the statistical test you plan to use: independent t-test, paired t-test, one-way ANOVA, chi-square, or correlation.
Select whether to solve for sample size (most common), statistical power, or minimum detectable effect size given a fixed sample.
Provide the effect size (or use a preset), alpha level, and power. Use the preset buttons for Cohen’s conventional small, medium, and large benchmarks.
Review the required sample size, achieved power, and critical values. Copy all results to your clipboard for your protocol, grant application, or manuscript.
Most research guidelines accept power = 0.80 as the minimum threshold. This means you accept a 20% chance of failing to detect a true effect. For confirmatory trials, power = 0.90 or higher is often recommended to reduce false negative risk.
The standard 5% significance level is widely used but should be justified in context. Fields with high false positive costs (e.g., clinical trials with safety implications) may use stricter thresholds like 0.01 or 0.005, which increases the required sample size.
Cohen’s small/medium/large benchmarks (d = 0.2/0.5/0.8) are useful defaults, but your power analysis should ideally use effect sizes from pilot data, meta-analyses, or the smallest clinically meaningful difference. Generic conventions may lead to over- or under-powered designs.
State the test type, expected effect size (and its source), alpha level, desired power, and the resulting sample size. Many journals and CONSORT/STROBE guidelines require this. Include any adjustments for anticipated dropout, multiple comparisons, or unequal allocation ratios.
A power analysis calculator operationalizes the relationship between four interdependent parameters that govern the sensitivity of any statistical test: sample size, significance level (alpha), effect size, and statistical power (1 − beta). Cohen (1988) established the foundational framework for these calculations, defining power as the probability that a test correctly rejects the null hypothesis when the alternative hypothesis is true. An a priori power analysis — conducted before data collection — determines the minimum number of participants needed to detect a specified effect with a given probability, and the CONSORT statement (Schulz et al., 2010) explicitly requires that trial protocols include a sample size justification grounded in such analysis. Underpowered studies waste resources and expose participants to experimental conditions without a reasonable chance of yielding informative results, making power analysis an ethical obligation as much as a statistical one. G*Power (Faul et al., 2007) is the most widely used reference software for validating power calculations across test families, and results from this calculator can be cross-checked against its exact non-central distribution solutions. Notably, post-hoc (observed) power analysis—computing power from the observed effect size after data collection—is widely discouraged because it is a monotonic function of the p-value and adds no new information (Hoenig & Heisey, 2001).
The sample size calculator in this tool supports five common test families: independent t-tests, paired t-tests, one-way ANOVA, chi-square tests of association, and Pearson correlations. Faul et al. (2007, 2009) demonstrated that normal- approximation formulas produce sample size estimates within 1% of exact non-central distribution solutions for most practical scenarios. Each test requires an anticipated effect size as input — Cohen (1992) proposed conventional benchmarks (small, medium, and large) as starting points, but the Cochrane Handbook (Higgins et al., 2023) recommends deriving effect size estimates from pilot data, prior meta-analyses, or the smallest clinically meaningful difference whenever possible. Using the minimum clinically important difference (MCID) as the target effect size—rather than Cohen's generic small, medium, or large conventions—ensures the study is powered to detect effects that patients and clinicians would consider meaningful. Our effect size calculator converts between Cohen's d, Hedges' g, odds ratios, and correlations, making it straightforward to derive the input your power calculation requires. For studies with categorical outcomes, our chi-square test calculator helps you explore Cramér's V from existing data, which can then be converted to Cohen's w for use in the chi-square power tab above.
The statistical power calculator also serves an important role in the planning of systematic reviews and meta- analyses. Valentine et al. (2010) showed that meta-analytic power depends on the number of included studies, the within-study sample sizes, and the between-study heterogeneity — factors that reviewers can estimate during protocol development. Bayesian sample size determination offers a modern alternative that incorporates prior information about the effect size distribution, while adaptive and sequential trial designs allow researchers to re-estimate sample size at pre-planned interim analyses without inflating the type I error rate. Our sample size and heterogeneity calculator complements this tool by computing I², Cochran's Q, and the minimum number of studies required to detect a meaningful pooled effect. For assessing the reliability of measurement instruments used across studies, our intraclass correlation coefficient calculator quantifies inter-rater and test-retest agreement, which informs the precision of the data feeding into your power estimates.
Transparent reporting of power analyses strengthens the credibility of both primary research and evidence syntheses. CONSORT (Schulz et al., 2010) requires authors to state the test type, expected effect size and its source, alpha level, desired power, and the resulting sample size, along with any adjustments for anticipated dropout or multiple comparisons. Cohen (1988) cautioned against treating his conventional benchmarks as universal standards, urging researchers to justify their effect size assumptions in the context of their specific domain. When your study is complete and you are ready to present pooled results, our forest plot generator creates publication-ready visualizations showing each study's weight, confidence interval, and the diamond summary estimate — the visual counterpart to the statistical precision that an adequately powered design ensures.
A power analysis determines the minimum sample size needed to detect a specified effect size with a given probability (statistical power). It balances four parameters: sample size (N), significance level (alpha), effect size, and power (1 – beta). By fixing any three, you can solve for the fourth. Power analyses are essential for ethical research design because underpowered studies waste resources and participant time, while overpowered studies may expose unnecessary participants to experimental conditions.
You should perform a power analysis during the study planning phase, before collecting any data. This is called an a priori power analysis. It helps justify your sample size to ethics committees, funding agencies, and peer reviewers. Post hoc (observed) power analyses after data collection are widely discouraged because they add no information beyond the p-value itself. Some journals and reporting guidelines (CONSORT, STROBE) explicitly require a priori sample size justifications.
The most widely accepted conventions come from Cohen (1988): alpha = 0.05 (two-tailed) and power = 0.80. These values mean you accept a 5% false positive rate and a 20% false negative rate. For confirmatory or high-stakes research, stricter thresholds such as alpha = 0.01 or power = 0.90 are recommended. The choice should be justified based on the costs of Type I vs. Type II errors in your specific context, not simply adopted by default.
This calculator uses the same normal-approximation formulas that underlie many sample size calculators, including parts of G*Power. For t-tests, chi-square tests, and correlations, the results should closely match G*Power. For ANOVA, this tool uses an approximation rather than the exact non-central F distribution, so results may differ slightly. For complex designs (repeated measures, multivariate tests, mixed models), we recommend using G*Power or simulation-based methods for more precise estimates.
No. The required sample size depends entirely on the expected effect size, the desired power, the alpha level, and the statistical test being used. Small effects require much larger samples: detecting d = 0.2 at 80% power requires about 394 participants per group, while d = 0.8 requires only about 26 per group. Rules of thumb like 'n = 30 per group' or 'N = 200 total' have no statistical basis and should be avoided. Always base your sample size on a formal power analysis with a justified effect size estimate.
There is no universal minimum — the required sample size depends on the effect size, significance level (α), desired power (1–β), and the statistical test used. A power analysis calculates this for your specific scenario. For example, detecting a medium effect (d = 0.5) with 80% power at α = 0.05 requires about 64 participants per group in a two-sample t-test.
No. Post-hoc (observed or retrospective) power analysis — calculating power using the observed effect size after a study is complete — is widely discouraged. Hoenig and Heisey (2001) demonstrated that observed power is a monotonic function of the p-value: low p-values always yield high observed power. It adds no information beyond the p-value itself. Only a priori power analysis (before data collection) is scientifically meaningful.
Ideally, use the smallest clinically meaningful difference (MCID) for your outcome measure, informed by clinical judgment and prior literature. If no MCID exists, use estimates from pilot studies, published meta-analyses, or systematic reviews. Cohen’s conventions (d = 0.2/0.5/0.8) are a last resort — they were intended as generic benchmarks, not substitutes for domain-specific reasoning.
Our biostatisticians can run power analyses for complex designs, calculate sample sizes for multi-arm trials, and provide publication-ready statistical analysis plans for your systematic review or primary study.