Statistical Power Calculator
Plan an analyzable sample size or evaluate the power of a planned study using a specific test, anticipated effect and significance level.
See the power of your design
Enter your values, or try an example.
Your results and calculation will appear here.
Show sensitivity table
| Sample configuration | Power |
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How to use this calculator
- Choose the study design first, then required sample size or prospective power.
- Specify a meaningful anticipated effect: d, paired dz, ANOVA f, or two event rates. Choose Decimal or Percent explicitly for probabilities.
- Enter α, sidedness where applicable, and the target power or planned integer sample sizes. For unequal planning, enter Group 2 / Group 1 allocation.
- Calculate. Read the sample units, achieved power, assumptions and calculation steps; compare nearby sample sizes in the curve and sensitivity table.
Worked example: two means: 80% power
This example uses sample values, not your current inputs.
For two independent means, assume Cohen’s d = 0.5 and use a two-sided pooled-variance t-test with α = 0.05. What analyzable sample size gives 80% power with equal allocation?
- Required sample size
- 64 per group · 128 total
- Achieved power
- 80.15% · Target 80%
63 per group falls short of 80% power; 64 per group reaches the target. This assumed effect is illustrative, not a universal sample-size recommendation.
Example calculation
- Two independent means; Two-sided. Find the minimum supported integer sample size.
- Inputs: α = 0.05; Cohen’s d = 0.5; target power = 80%. Allocation: n₂ = ceiling(1 × n₁).
- Evaluate 64 observations in Group 1 and 64 in Group 2 (128 total). df = n₁ + n₂ − 2 = 126; δ = d / √(1/n₁ + 1/n₂) = 0.5 / √(1/64 + 1/64) = 2.82842712.
- Noncentral t distribution. Critical t threshold under the null = 1.9789706 (both ± thresholds).
- Power = P(T > c | δ) + P(T < −c | δ) = 0.8014595579 = 80.146%.
- Predecessor: 63 observations in Group 1 and 63 in Group 2 (126 total) gives 79.5168% < 80%.
- Minimum: 64 observations in Group 1 and 64 in Group 2 (128 total); achieved power 80.146% ≥ target 80%.
Formula & assumptions
t models: δ = d√n or d/√(1/n₁ + 1/n₂). Paired model uses dz. Proportions: h = 2asin√p₂ − 2asin√p₁. ANOVA: λ = Nf², df₁ = k−1, df₂ = N−k. Power is rejection probability under the specified alternative.
- One-sample and paired t models assume normal observations or paired differences. Independent means use a pooled-variance t-test with equal population variances, not Welch’s test.
- Balanced one-way ANOVA assumes independent normal groups, equal variances and fixed effects; no repeated-measures, factorial or clustered design.
- Two-proportion power uses Cohen’s h normal approximation without continuity correction. Require at least 10 expected events and nonevents in each group. Rare-event and boundary rates need specialist consideration.
- Signed d and dz describe the anticipated difference. Positive independent effects mean Group 2 minus Group 1. One-sided direction must be prespecified.
- Supported α: 0.001–0.20; target power: 0.50–0.99. Up to 5,000 observations per group or complete pairs; ANOVA up to 500/group and 3–20 groups. Allocation ratio 0.2–5 for sample-size planning.
- Results are analyzable sample sizes, not recruitment targets. Dropout, clustering, multiple testing and survival outcomes are not modeled. Power is not the probability that a hypothesis is true.
- Use a scientifically meaningful anticipated effect. Small/medium/large conventions are rough illustrations, not justification. Observed post-hoc power is not a substitute for confidence intervals.
Sources
- R pwr: t-test power ↗
- R pwr: two-proportion power ↗
- SciPy: noncentral t distribution ↗
- SciPy: noncentral F distribution ↗
- statsmodels: independent t power ↗
- Lakens: Sample Size Justification ↗
Reviewed: October 2026
Common questions
How should I choose an effect size?
Use a smallest meaningful difference, prior reliable studies or a justified range of anticipated effects. Divide a mean difference by the appropriate population SD; paired dz uses the SD of paired differences. ANOVA f summarizes overall between-group differences. Conventional values are not study-specific evidence.
What do α, β and power mean?
α is the false-positive rejection probability under the null model. β is the chance of failing to reject under the specified alternative; power = 1 − β. Power does not give the probability that a hypothesis is true.
Why are per-group and total sample sizes different?
Independent two-group designs need participants in both groups; total N = n₁ + n₂. Balanced ANOVA needs k groups, so total N = k × n per group. A paired design counts complete pairs or participants, not two independent groups.
Is this a survey sample-size calculator?
No. It plans power for specified statistical tests, not population surveys, margin of error or generic confidence-level sampling.
Can I enter the observed effect after a study?
This tool is intended for prospective planning with an anticipated effect. Observed post-hoc power based on an observed effect or p-value can be misleading; report effect uncertainty and confidence intervals instead.
Why might another calculator give a different sample size?
Compare test family, effect definition, sidedness, allocation, count restrictions and rounding. The proportions model is a specific arcsine-normal approximation, not an exact binomial or pooled/unpooled z-test model.
Use these tools for mathematical information, not as a substitute for checking assumptions. Read our methodology or disclaimer.