Power Analysis Calculator

Figran

Get the sample size or power for your study, with a power curve and a justification to paste.

7 common study designs
Power curve as SVG or 300 DPI PNG
Sample size justification text

Design

Solve for

Enter as

Effect size (d)Small / medium / large: 0.2 / 0.5 / 0.8Significance level (α)

Tails

Target power (1 − β)Allocation ratio n₂/n₁Expected dropout (%)Effect size source (optional)

Plot

Chart titleX-axis title (optional)Y-axis title

Figure width

GrayscaleShow power as percentages

Result

64 per group · 128 total

Achieved power 80.1%

Sample size justification

An a priori power analysis for a two-sided independent-samples t-test (at significance level α = 0.05) requires 64 per group (128 total) to detect d = 0.50. The target power was 80%, and the achieved power after rounding was 80.1%. Power was calculated using the noncentral t distribution. The equivalent R calculation is: pwr::pwr.t.test(n = 64, d = 0.5, sig.level = 0.05, type = "two.sample", alternative = "two.sided").

Power analysis examples

Start with a study design, then adapt its inputs to your research.

Two-arm trial with a continuous outcome

Plan two groups using means and a shared standard deviation.

d = 0.50 · α = 0.05 · 80% power

Pre–post paired design

Estimate the number of pairs for a within-person comparison.

34 pairs · d_z = 0.50

Three-arm experiment (one-way ANOVA)

Plan an equal-sized comparison across three groups.

3 groups · f = 0.25

Binary outcome trial with dropout

Compare two event rates and account for attrition.

20% vs 30% · 10% dropout

Correlation study

Estimate the observations needed to detect a correlation.

r = 0.3 · two-sided

Animal study: what can 10 per group detect?

Find the detectable effect with a fixed sample size.

10 per group · sensitivity

How to do a power analysis

Set the assumptions for your study and take the results with you.

1Choose a design
Select the test that matches your primary analysis.
2Enter the effect size
Use a standardized effect or convert raw study values.
3Set your assumptions
Choose significance level, power and expected dropout.
4Save your results
Copy the justification and download the power curve.

Choosing an effect size for a power analysis

Use the effect that is meaningful for the question and the chosen design. To calculate Cohen's d or Hedges' g from your data, use the Effect Size Calculator.

Formulas by study design

Cohen's d

d = \frac{| μ_{1} - μ_{2} |}{σ}

μ₁ and μ₂ are group means; σ is their common standard deviation.

Paired d_z

d_{z} = \frac{| μ_{difference} |}{σ_{difference}}

μ_difference and σ_difference are the mean and standard deviation of paired differences.

Cohen's f

f = \sqrt{\frac{η^{2}}{1 - η^{2}}}

η² is the proportion of variance explained by the group factor.

Cohen's h

h = 2 | arcsin \sqrt{p_{1}} - arcsin \sqrt{p_{2}} |

p₁ and p₂ are the proportions in the two groups.

Cohen's w

w = \sqrt{\sum_{i} \frac{{(p_{1 i} - p_{0 i})}^{2}}{p_{0 i}}}

p_0i and p_1i are the null and alternative proportions for category i.

Correlation r

r = \frac{cov (X, Y)}{σ_{X} σ_{Y}}

cov(X, Y) is the covariance; σ_X and σ_Y are the standard deviations.

Cohen's f squared (f²)

f^{2} = \frac{R_{full}^{2} - R_{reduced}^{2}}{1 - R_{full}^{2}}

R²_full and R²_reduced are the explained-variance proportions for the full and reduced models.

Common reference values

Measure	Small	Medium	Large
Cohen's d	.2	.5	.8
Cohen's f	.10	.25	.40
Cohen's h	.2	.5	.8
Correlation r	.1	.3	.5
Cohen's w	.1	.3	.5
Cohen's f squared (f²)	.02	.15	.35

Reference: Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates. Lakens: Sample size justification (2022).

These conventional benchmarks are context dependent, not a substitute for a substantive justification.

Where the effect size should come from

Prefer the smallest effect that would matter in practice, supported by prior studies or a pilot. Pilot estimates can be imprecise; consider uncertainty when planning.

How sample size is calculated

A sample size calculation selects the smallest integer meeting the target power.

Power is the probability of rejecting the null hypothesis when the specified alternative is true. The calculator uses noncentral distributions for t, F and chi-square tests, and the stated approximations for correlation and proportions.

Power (1 − β)

1 - β = P (reject H_{0} | H_{1})

β is the probability of a false negative; H₀ is the null and H₁ the specified alternative.

Two independent groups (approximation)

n \approx \frac{2 {(z_{1 - α / 2} + z_{1 - β})}^{2}}{d^{2}}

n is per group; d is the standardized difference and z denotes a standard-normal quantile. This approximation assumes equal groups and a two-sided test.

Why 64, not 63?

For a two-sided independent t-test with d = 0.5 and α = 0.05, 63 per group falls short of 80% power; rounding up reaches the target.

Sample size per group	Power
63	0.7952
64	0.8015

Writing the sample size justification

Report the design, effect and its source, significance level, target and achieved power, sample size, and any allowance for dropout. The paragraph below uses the default study assumptions.

An a priori power analysis for a two-sided independent-samples t-test (at significance level α = 0.05) requires 64 per group (128 total) to detect d = 0.50. The target power was 80%, and the achieved power after rounding was 80.1%. Power was calculated using the noncentral t distribution. The equivalent R calculation is: pwr::pwr.t.test(n = 64, d = 0.5, sig.level = 0.05, type = "two.sample", alternative = "two.sided").

References: CONSORT 2010 statement

Power analysis FAQ

What is a power analysis?

A power analysis estimates the probability of detecting a specified effect if it exists. An a priori analysis uses that probability to plan sample size before collecting data.

How do I calculate the sample size for a study?

Choose a test, a significance level, a target power and an effect size. The calculator finds the smallest integer sample size that meets the target.

What power and significance level should I use?

A common starting point is 80% power and a 0.05 significance level. Some clinical studies use 90% power, and multiple comparisons can require an adjusted significance level.

What effect size should I use, and what is Cohen's d?

Use the smallest scientifically meaningful effect when possible. Cohen's d is a mean difference divided by a standard deviation; 0.2, 0.5 and 0.8 are conventional reference values, not universal rules.

What is the difference between a priori, sensitivity and post hoc power analysis?

A priori analysis finds a sample size; sensitivity analysis finds an effect detectable at a fixed sample size. Power calculated from the observed effect after a test does not add useful information about its result.

How do I adjust the sample size for dropout?

Divide the required sample by one minus the expected dropout fraction and round up. For example, 64 per group becomes 76 with 15% dropout.

Does this give the same results as G*Power?

Results depend on the selected test and approximation. For example, R pwr gives 64 per group for d = 0.5 in a two-sided t-test and 53 per group for a three-group ANOVA with f = 0.25; the correlation calculation uses a Fisher z approximation.

Is the sample size per group or in total?

Independent t-tests, two proportions and ANOVA show group and total sizes. Correlation, chi-square and regression use the total sample size; paired tests use pairs.

Is my data uploaded? What can I download?

Your inputs stay in your browser. Download SVG, 300 DPI PNG or CSV, and copy the justification or R code without signing in.

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Calculate a sample size and prepare a clear justification.

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