Power Analysis Calculator | Sample Size, Effect Size & Statistical Power
Determine the required sample size for a study given desired statistical power (1 − β), significance level α, and effect size. Supports Cohen's d for t-tests and z-tests, Cohen's h for proportion tests, and Cohen's w for chi-square tests. Also computes achieved power for a given sample size.
TEST TYPE
DIRECTION
MODE
PRESETS
Required n (per group)
63
Power = 80%
Critical value z_α
1.960
two-sided
Type I error α
5.0%
False positive rate
Type II error β
20.0%
False negative rate
POWER CURVE (n vs POWER)
| n | Power | Bar |
|---|---|---|
| 10 | 20.0% | |
| 20 | 35.2% | |
| 30 | 49.1% | |
| 40 | 60.9% | |
| 50 | 70.5% | |
| 60 | 78.2% | |
| 80 | 88.5% | |
| 100 | 94.2% | |
| 120 | 97.2% | |
| 150 | 99.1% | |
| 200 | 99.9% |
What Is the Power Analysis Calculator | Sample Size, Effect Size & Statistical Power?
Determine the required sample size given a desired power (1 − β), significance level α, and effect size. Supports Cohen's d for t/z-tests, Cohen's h for proportion tests, and Cohen's w for chi-square tests. Also calculates achieved power for a given n.
Formula
n = ((z_{α/2} + z_β) / d)² — one-sample; n = 2×((z_{α/2} + z_β) / d)² — two-sample; power = Φ(|d|√(n/2) − z_{α/2})How to Use
- 1
Choose the test type: One-Sample Z, Two-Sample T, Proportion Z (Cohen's h), or Chi-Square.
- 2
Select direction: Two-Sided (H₁: μ ≠ μ₀) or One-Sided (H₁: μ > μ₀).
- 3
Choose mode: Find n (enter desired power) or Find Power (enter known n).
- 4
Click Small / Medium / Large to fill in Cohen's effect size convention, or type a custom value.
- 5
Enter α (significance level, commonly 0.05) and the desired power (commonly 0.80 or 0.90).
- 6
Read the required sample size, achieved power, critical value, and β error from the results.
- 7
Review the power curve table to see how power grows with increasing n for your effect size.
Select the test type, direction, and mode (find n or find power), then enter the effect size, α, and power or n.
Example Calculation
Medium effect, two-sample t-test: d = 0.5, α = 0.05, desired power = 0.80. n per group = 2 × ((1.96 + 0.8416) / 0.5)² = 2 × (5.603)² ≈ 2 × 31.4 ≈ 63 total, 32 per group. For 95% power with the same effect: n per group ≈ 53.
Understanding Power Analysis | Sample Size, Effect Size & Statistical Power
Cohen's Effect Size Conventions
Jacob Cohen established widely used benchmarks for effect size magnitudes. These are guidelines, not rules — context determines what constitutes a meaningful effect in your field.
| Statistic | Test | Small | Medium | Large |
|---|---|---|---|---|
| Cohen's d | t-test / z-test | 0.2 | 0.5 | 0.8 |
| Cohen's h | Two proportions | 0.2 | 0.5 | 0.8 |
| Cohen's w | Chi-square / goodness of fit | 0.1 | 0.3 | 0.5 |
| Cohen's f | ANOVA / F-test | 0.1 | 0.25 | 0.4 |
| Pearson r | Correlation | 0.1 | 0.3 | 0.5 |
Sample Size Quick Reference (Two-Sample t-test, α = 0.05, Two-Sided)
Required n per group for two-sample t-test with equal groups. Values computed from n = 2 × ((z_α/2 + z_β) / d)².
| Effect size d | Power 70% | Power 80% | Power 90% | Power 95% |
|---|---|---|---|---|
| 0.2 (small) | 265 | 394 | 527 | 651 |
| 0.3 | 118 | 175 | 234 | 290 |
| 0.5 (medium) | 43 | 64 | 85 | 105 |
| 0.8 (large) | 17 | 25 | 34 | 42 |
| 1.0 | 11 | 17 | 22 | 28 |
| 1.5 | 5 | 8 | 10 | 13 |
Type I vs Type II Errors
- ›Type I error (α, false positive): rejecting H₀ when it is true. Controlled directly by setting α = 0.05 or 0.01.
- ›Type II error (β, false negative): failing to reject H₀ when it is false. Power = 1 − β. Industry standard: β ≤ 0.20 (power ≥ 80%).
- ›Effect size d = (μ₁ − μ₀)/σ measures how far apart the null and alternative means are in standard deviation units.
- ›Cohen's h for proportions: h = 2 arcsin(√p₁) − 2 arcsin(√p₀). The arcsin transformation stabilises the variance.
- ›Increasing sample size n reduces β (raises power) without affecting α, the most direct lever for improving study sensitivity.
- ›One-sided tests require smaller n than two-sided for the same power, but are only valid when direction is pre-specified.
- ›Pilot studies are often used to estimate effect size; if the pilot d is inflated by sampling error, the main study will be underpowered.
- ›For clinical trials, regulatory agencies typically require power ≥ 80% at α = 0.05 (two-sided) for primary endpoints.
Frequently Asked Questions
What is statistical power and why does it matter?
Power (1 − β) is the probability of correctly detecting a real effect. A study with 80% power will miss 20% of true effects. Low-powered studies waste resources and produce unreliable results — if they do find an effect, it may be inflated; if they do not, the null result is inconclusive.
What is Cohen's d?
Cohen's d = (μ₁ − μ₀) / σ, the difference between means divided by the pooled standard deviation. It is a dimensionless measure of effect size. d = 0.2 is small (like a 2-point IQ difference), d = 0.5 is medium, and d = 0.8 is large.
Why do small effects require such large samples?
Sample size scales as 1/d². A small effect d = 0.2 requires 25× more participants than a large effect d = 1.0 for the same power. Detecting a 1% improvement in a process requires thousands of observations; detecting a 20% improvement may need only tens.
What is the difference between one-sided and two-sided tests?
A two-sided test asks "Is μ ≠ μ₀?" and splits α between both tails. A one-sided test asks "Is μ > μ₀?" and puts all α in one tail. One-sided tests are more powerful but only valid when direction is pre-specified before seeing the data.
Can I use this calculator for clinical trial planning?
Yes, the z-test approximation is standard for clinical trial sample size estimation. For superiority trials use d and the two-sample formula. Regulatory submissions (FDA, EMA) typically require power ≥ 80% at α = 0.05 two-sided. Consult a biostatistician for complex designs.
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