# Statistical Power Calculator

> Approximate the power of a test of means from effect size d, sample size and α, and the sample size needed for a target power, one- or two-sided.

Category: Statistics & probability · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/statistical-power-approximation-calculator

Power is the chance a study detects an effect that really exists. Enter Cohen’s d, the sample size and α to see the power, or the sample needed to reach, say, 80%.

Also called: Power analysis calculator.

## Inputs
- Design (options: Two independent groups, One sample or paired; default two)
- Effect size (Cohen’s d) (default 0.5)
- Sample size (per group, or pairs) (default 64)
- Significance level α (unit %; default 5)
- Test (options: Two-sided, One-sided; default two)
- Target power (unit %; default 80)

## Results
- Power at this sample size
- n for target power (t-test, Guenther-adjusted)
- n for target power (z approximation)
- Total participants (t-test)
- Noncentrality d√(n/k)
- Critical z

## Formula

```
δ = d√(n/k), k = 2 for two groups, 1 for one sample;  two-sided power = Φ(δ − z₁₋α/₂) + Φ(−δ − z₁₋α/₂);  n = k(z_α + z_β)² ÷ d²;  t-adjusted n = n + z_α² ÷ (2k)
```

Where:
- d: standardised effect size
- n: per-group size (or number of pairs)
- z_α: critical value (z₁₋α/₂ two-sided, z₁₋α one-sided)
- z_β: normal quantile of the target power

## Worked example
1. δ = 0.5 × √(64 ÷ 2) = 2.8284; power ≈ Φ(2.8284 − 1.96) = Φ(0.8685) = 80.7%.
2. n = 2 × (1.96 + 0.8416)² ÷ 0.25 = 62.8 → 63 per group (z).
3. Guenther adjustment: 62.8 + 1.96² ÷ 4 = 63.75 → 64 per group, matching Cohen’s table for a t-test.

**Result:** Detecting d = 0.5 with 80% power at α = 0.05 (two-sided) needs about 64 people per group, 128 in total.

## What the result means

Plan for the smallest effect that would matter in practice, not the effect you hope for. Halving d roughly quadruples the sample needed.

## Assumptions and limits
- Normal approximation to the t-test; the exact noncentral-t power is slightly lower for small samples (0.8015 rather than 0.8074 in the example).
- Equal group sizes and a common standard deviation.
- Not for proportions (use the A/B test sample size calculator), ANOVA, regression or clustered designs.
- Power analysis is only as good as the assumed effect size.

## Questions

### Why 80% power?

It is a common convention balancing cost against a 20% risk of missing a real effect. Many trials now target 90%.

### Should I use a one-sided test to save participants?

Only if an effect in the other direction would be treated exactly like no effect, and you decided before collecting data.

## Related calculators
- [Cohen’s d Effect Size Calculator](https://calculators.geekmidst.com/calculators/cohen-d-effect-size-calculator)
- [A/B Test Sample Size Calculator](https://calculators.geekmidst.com/calculators/ab-test-sample-size-calculator)
- [Sample Size Calculator](https://calculators.geekmidst.com/calculators/sample-size-calculator)
- [P-value Calculator](https://calculators.geekmidst.com/calculators/p-value-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/statistical-power-approximation-calculator