# What is my test’s statistical power?

Finds the statistical power of a one-sample, paired or two-sample t test or z test from the effect size (Cohen’s d), the sample size and α, with β, the noncentrality and the critical value.

- Page: https://www.acalculator.org/statistics/statistical-power-calculator
- JSON spec: https://www.acalculator.org/statistics/statistical-power-calculator.json
- Version: dc4a067df1fd

## Default answer

Example with the default inputs (Test t test: one sample or paired, Alternative Two-sided, Effect size (Cohen’s d) 0.5, Sample size (n per group) 30, Significance level (α) 0.05): The power is 75.40%, so the chance of missing a real effect of d = 0.5 is 24.60%.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| test | Test | The test you plan: a t test (σ estimated from the data) or a z test (σ known), for one mean or two. |
| tails | Alternative | Two-sided: the effect may go either way. One-sided: you test one direction only. |
| d | Effect size (Cohen’s d) | The difference you want to detect divided by the standard deviation: (μ − μ₀) ÷ σ, or (μ₁ − μ₂) ÷ σ for two groups. |
| n | Sample size (n per group) | The number of observations, or of pairs; for two samples, the number in each group. |
| a | Significance level (α) | The chance of a false positive the test allows, such as 0.05. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| power | Power (1 − β) | The chance that the test rejects H₀ when the effect is real, in percent. |
| beta | Type II error rate (β) | The chance of missing a real effect of this size: 100% − power. |
| lambda | Noncentrality (λ) | d√n for one sample, or d√(n/2) for two groups of n. |
| crit | Critical value | The z or t value the test statistic must pass: for α/2 two-sided, or α one-sided. |
| df | Degrees of freedom (ν) | n − 1, or 2(n − 1) for two groups; t tests only. |
| total | Total sample size | n, or 2n for two groups. |

## Method

λ = d√n (one sample) or d√(n/2) (two groups of n). z test: power = Φ(λ − z(1 − α/2)) + Φ(−λ − z(1 − α/2)), or Φ(λ − z(1 − α)) one-sided. t test: power = P(T′ > t(1 − α/2; ν)) + P(T′ < −t(1 − α/2; ν)) for T′ noncentral t with ν = n − 1 or 2(n − 1) and noncentrality λ; one-sided P(T′ > t(1 − α; ν)).

## Assumptions

- The data are normal (or n is large), the observations independent, and two groups have the same size and standard deviation.
- d is the true effect in the tested direction, so it is 0 or more.

## Worked examples

1. test = z1, tails = one, d = 1, n = 9, a = 0.05 gives power = 91.23%, lambda = 3, crit = 1.64485, beta = 8.77%. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2.2 Sample sizes required (N = (z(1 − α/2) + z(1 − β))² (σ/δ)² for a two-sided test and (z(1 − α) + z(1 − β))² (σ/δ)² one-sided; α = 0.05, β = 0.10 and a shift of one standard deviation give N = (1.645 + 1.282)² = 8.567 ≈ 9; with s, (1.860 + 1.397)² = 10.6 ≈ 11), https://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm (retrieved 2026-10-19).
2. test = t1, tails = one, d = 1, n = 11, a = 0.05 gives power = 92.45%, df = 10, crit = 1.81246. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2.2 Sample sizes required (N = (z(1 − α/2) + z(1 − β))² (σ/δ)² for a two-sided test and (z(1 − α) + z(1 − β))² (σ/δ)² one-sided; α = 0.05, β = 0.10 and a shift of one standard deviation give N = (1.645 + 1.282)² = 8.567 ≈ 9; with s, (1.860 + 1.397)² = 10.6 ≈ 11), https://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm (retrieved 2026-10-19); SAS/STAT, The Analyst Application, Sample Size and Power Calculations, Hypothesis Tests (Power = Prob(t > t_crit, ν, NC) one-sided, + Prob(t < t_critl, ν, NC) two-sided; ν = n − 1 and NC = δ√n for one-sample and paired t tests; ν = 2(n − 1) and NC = δ ÷ √(2/n) for two samples of n each; δ = (μ − μ₀) ÷ s), https://support.sas.com/documentation/cdl/en/anlystug/58352/HTML/default/chap12_sect7.htm (retrieved 2026-10-19).
3. test = t2, tails = two, d = 0.5, n = 64, a = 0.05 gives power = 80.15%, df = 126, total = 128, lambda = 2.82843. Source: SAS/STAT, The Analyst Application, Sample Size and Power Calculations, Hypothesis Tests (Power = Prob(t > t_crit, ν, NC) one-sided, + Prob(t < t_critl, ν, NC) two-sided; ν = n − 1 and NC = δ√n for one-sample and paired t tests; ν = 2(n − 1) and NC = δ ÷ √(2/n) for two samples of n each; δ = (μ − μ₀) ÷ s), https://support.sas.com/documentation/cdl/en/anlystug/58352/HTML/default/chap12_sect7.htm (retrieved 2026-10-19).
4. test = t1, tails = two, d = 0, n = 20, a = 0.05 gives power = 5.00%. Source: SAS/STAT, The Analyst Application, Sample Size and Power Calculations, Hypothesis Tests (Power = Prob(t > t_crit, ν, NC) one-sided, + Prob(t < t_critl, ν, NC) two-sided; ν = n − 1 and NC = δ√n for one-sample and paired t tests; ν = 2(n − 1) and NC = δ ÷ √(2/n) for two samples of n each; δ = (μ − μ₀) ÷ s), https://support.sas.com/documentation/cdl/en/anlystug/58352/HTML/default/chap12_sect7.htm (retrieved 2026-10-19).
5. test = z2, tails = two, d = 0.5, n = 64, a = 0.05 gives power = 80.74%, lambda = 2.82843, crit = 1.95996. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2.2 Sample sizes required (N = (z(1 − α/2) + z(1 − β))² (σ/δ)² for a two-sided test and (z(1 − α) + z(1 − β))² (σ/δ)² one-sided; α = 0.05, β = 0.10 and a shift of one standard deviation give N = (1.645 + 1.282)² = 8.567 ≈ 9; with s, (1.860 + 1.397)² = 10.6 ≈ 11), https://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm (retrieved 2026-10-19).
6. test = t1, tails = two, d = 0.5, n = 20, a = 0.05 gives power = 56.45%. Source: SAS/STAT, The Analyst Application, Sample Size and Power Calculations, Hypothesis Tests (Power = Prob(t > t_crit, ν, NC) one-sided, + Prob(t < t_critl, ν, NC) two-sided; ν = n − 1 and NC = δ√n for one-sample and paired t tests; ν = 2(n − 1) and NC = δ ÷ √(2/n) for two samples of n each; δ = (μ − μ₀) ÷ s), https://support.sas.com/documentation/cdl/en/anlystug/58352/HTML/default/chap12_sect7.htm (retrieved 2026-10-19).

## FAQ

### What is statistical power?

The chance that a test rejects the null hypothesis when the effect you expect is real: power = 1 − β, where β is the chance of a type II error (missing the effect). Studies usually aim for 80% or 90%.

### How do you calculate the power of a z test?

Find λ = d√n and the critical value z. One-sided: power = Φ(λ − z(1 − α)). With d = 1, n = 9 and α = 0.05: Φ(3 − 1.645) = Φ(1.355) = 91.2%, which is why the NIST e-Handbook rounds its 8.57 up to 9 for 90% power.

### How is the power of a t test calculated?

From the noncentral t distribution: power = P(T′ > t_c) (+ P(T′ < −t_c) two-sided), where T′ has ν = n − 1 degrees of freedom (2(n − 1) for two groups) and noncentrality λ = d√n (d√(n/2) for two groups). Two groups of 64 with d = 0.5 and α = 0.05 give 80.1%.

### What effect size should I use?

The smallest difference that would matter, divided by the standard deviation: d = (μ₁ − μ₂) ÷ σ. A common rule of thumb calls 0.2 small, 0.5 medium and 0.8 large. A smaller d needs a larger sample for the same power.

### Why is a t test’s power lower than a z test’s?

The t test also estimates σ from the data, which adds uncertainty: its critical value is larger and its statistic more spread out. For two groups of 64 at d = 0.5, the z test gives 80.7% and the t test 80.1%; with small samples the gap grows.

### What happens to power when the effect is 0?

It equals α: with no real effect, the test rejects only by chance. A two-sided t test at α = 0.05 with d = 0 has power exactly 5%.

## Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2.2 Sample sizes required: N = (z(1 − α/2) + z(1 − β))² (σ/δ)² (two-sided) and (z(1 − α) + z(1 − β))² (σ/δ)² (one-sided); with α = 0.05, β = 0.10 and a one standard deviation shift, N = (1.645 + 1.282)² = 8.567 ≈ 9, and with s from the data (1.860 + 1.397)² = 10.6 ≈ 11, retrieved 2026-10-19. https://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm
- SAS/STAT, The Analyst Application, Sample Size and Power Calculations, Hypothesis Tests: power = Prob(t > t_crit, ν, NC) one-sided, plus Prob(t < t_critl, ν, NC) two-sided; ν = n − 1 and NC = δ√n for one-sample and paired t tests; ν = 2(n − 1) and NC = δ ÷ √(2/n) for two samples of n each, retrieved 2026-10-19. https://support.sas.com/documentation/cdl/en/anlystug/58352/HTML/default/chap12_sect7.htm