# Power Analysis Calculator (Sample Size)

URL: https://speedycalculator.com/calculators/statistics/power-analysis/
Last updated: 2026-10-30

> The power analysis calculator finds the minimum sample size for a study comparing two groups or one group against a known value, with a yes/no (incidence) or continuous (mean) outcome. Set α, power and the enrollment ratio to get the size of each group, the effect size and a power curve.

## Inputs

- Study design
- Primary endpoint
- Enter group 2 as
- Incidence in group 1 (or known population) (%)
- Incidence in group 2 (or study group) (%)
- Mean of group 1 (or known population mean)
- Mean of group 2 (or study group mean)
- Change of group 2 relative to group 1 (%)
- Standard deviation (group 1 or population)
- Standard deviation of group 2 (blank = same)
- Enrollment ratio (group 2 ÷ group 1)
- Significance level (α)
- Test sides
- Power (1 − β) (%)

## Results

- Total sample size
- Group 1 sample size
- Group 2 sample size
- Power achieved with these sizes
- Standardised effect size
- Effect size measure
- Unrounded size of group 1
- z for α (z₁₋α/₂ or z₁₋α)
- z for power (z₁₋β)
- Power curve

## Formula

```
Two groups, proportions: n₁ = [z₁₋α/₂ √(p̄q̄(1 + 1/k)) + z₁₋β √(p₁q₁ + p₂q₂/k)]² ÷ Δ², p̄ = (p₁ + k·p₂) ÷ (1 + k), n₂ = k·n₁ · Two groups, means: n₁ = (σ₁² + σ₂²/k)(z₁₋α/₂ + z₁₋β)² ÷ Δ² · One group: N = p₀q₀[z₁₋α/₂ + z₁₋β √(p₁q₁/(p₀q₀))]² ÷ Δ² or N = [σ(z₁₋α/₂ + z₁₋β) ÷ Δ]²
```

## Explanation

Statistical power is the probability that a study detects an effect that really exists. A study with too few participants may miss a genuine treatment benefit, waste resources and expose people to research without a useful answer. Power analysis works backwards: you decide how big an effect matters, how much false-positive risk (α) you accept and how much power you want, and it tells you how many participants you need.

Four ingredients drive the answer. The effect size – the difference in incidence or means you want to detect – has the biggest influence: halving it roughly quadruples the sample. Variability matters for continuous outcomes, so a noisier measurement needs more people. Lower α and higher power both increase n. Finally, unequal group sizes (an enrollment ratio other than 1) are less efficient than a balanced design.

The formulas here use the normal approximation that most clinical-trial calculators use. They give the sample size to analyse; in practice add extra participants to allow for dropouts and missing data. The chart shows how power rises with the total sample size.

### How to use the Power Analysis Calculator

- **Study design**
- **Primary endpoint**
- **Enter group 2 as**
- **Incidence in group 1 (or known population)**
- **Incidence in group 2 (or study group)**
- **Mean of group 1 (or known population mean)**
- **Mean of group 2 (or study group mean)**
- **Change of group 2 relative to group 1**
- **Standard deviation (group 1 or population)**
- **Standard deviation of group 2 (blank = same)**
- **Enrollment ratio (group 2 ÷ group 1)**
- **Significance level (α)**
- **Test sides**
- **Power (1 − β)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Choose the design and the type of outcome, then enter the expected incidences or means for each group (or a relative change), the enrollment ratio, α and the desired power. The calculator applies the normal-approximation formula, rounds each group up to a whole number and reports the power actually achieved.

### Worked example

A trial expects 20% of control patients and 12% of treated patients to have a complication. With α = 0.05 (two-sided), 80% power and equal groups: p̄ = 0.16, z₁₋α/₂ = 1.960 and z₁₋β = 0.842.

n₁ = [1.960 × √(0.16 × 0.84 × 2) + 0.842 × √(0.2 × 0.8 + 0.12 × 0.88)]² ÷ 0.08² = 328.5, so 329 patients per group and 658 in total.

For a continuous outcome – blood pressure 120 vs 128 mmHg with SD 15 – n₁ = (15² + 15²)(1.960 + 0.842)² ÷ 8² = 55.2, so 56 per group.

### What changes the sample size

ChangeEffect on nHalve the detectable differenceAbout × 4Power 80% → 90%About × 1.34α 0.05 → 0.01 (two-sided)About × 1.49Enrollment 1 : 1 → 2 : 1Total about × 1.13Double the SD (means)× 4

## Worked example

Inputs — Study design: Two independent groups (e.g. treatment vs control); Primary endpoint: Dichotomous – yes/no incidence; Enter group 2 as: Its own value; Incidence in group 1 (or known population): 20 %; Incidence in group 2 (or study group): 12 %; Enrollment ratio (group 2 ÷ group 1): 1; Significance level (α): 0.05; Test sides: Two-sided; Power (1 − β): 80 %.

Results — **Total sample size = 658**; Group 1 sample size = 329; Group 2 sample size = 329; Power achieved with these sizes = 80.06 %; Standardised effect size = 0.2198; Effect size measure = Cohen's h.

## Key terms

- **Power (1 − β)**: Probability of rejecting H₀ when the assumed effect is real.
- **Type II error (β)**: Failing to detect a real effect.
- **Effect size**: The smallest difference worth detecting; standardised as Cohen's d for means or Cohen's h for proportions.
- **Enrollment ratio (k)**: Size of group 2 divided by size of group 1.

## FAQ

**What power should a study have?**
80% is the usual minimum and 90% is common in clinical trials. Power of 80% means a 20% chance (β) of missing a real effect of the assumed size.

**How many participants do I need to detect a medium effect?**
For Cohen's d = 0.5, two-sided α = 0.05 and 80% power, about 63 per group (126 in total) by the normal approximation; t-based software gives 64.

**What sample size detects a drop in incidence from 20% to 12%?**
About 329 per group (658 in total) at α = 0.05 two-sided and 80% power. A larger drop, 30% to 15%, needs only 121 per group.

**Why does a smaller effect need so many more people?**
Sample size scales with 1 ÷ Δ². Halving the difference you want to detect roughly quadruples n.

**What is the enrollment ratio?**
The number in group 2 for every person in group 1. A 2 : 1 design (k = 2) needs more people in total than 1 : 1 for the same power, but can make recruitment easier.

**Should I use a one-sided test to reduce the sample size?**
Only if an effect in the opposite direction would be irrelevant and this is stated in advance. Regulators generally expect two-sided tests.

**How do I allow for dropouts?**
Divide the calculated size by (1 − expected dropout rate). With 10% dropout, 658 becomes 732.


## Sources

- [Wikipedia – Power of a test](https://en.wikipedia.org/wiki/Power_of_a_test)
- [Penn State STAT 500 – Applied Statistics](https://online.stat.psu.edu/stat500/)
- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)