T-Test and Z-Test Power Calculator

Calculates the statistical power of a one-sample or two-sample t-test or z-test: the probability that the test will detect the effect you specify. Shows β, the required sample size, an interactive power analysis chart and a power curve.

Test

Distribution:

Sample:

Variances:

Tails:

Significance level (α):

Effect to detect

Effect:

Effect type:

Effect size (Cohen's d)

Sample

Sample size (n)

Sample size (n₂)

Standard deviation (S₁)

Standard deviation (S₂)

Statistical power

Power is the probability that a test rejects the null hypothesis when a real effect of the size you specified exists. Its complement, β = 1 - power, is the probability of a Type II error — missing a real effect. The conventional target is a power of 0.8, meaning the study would miss such an effect one time in five.

Power is decided by four things, and fixing any three determines the fourth: the significance level (α), the effect size, the sample size, and the number of tails. This page holds the first three and reports the power; it also solves the other direction and reports the sample size needed to reach your target power.

How to use this calculator

Choose the test. Use T when the standard deviation is estimated from your data — that is almost always the case. Use Normal only when the population σ is genuinely known.
Choose one or two samples. For a paired design, run a one-sample test on the within-pair differences: the power is the same calculation.
Set the effect to detect. Either pick Small / Medium / Large, which fills in Cohen's d (0.2 / 0.5 / 0.8), or type the effect yourself. Choosing Unstandardized lets you enter a difference in the original units, and then also asks for the standard deviation so it can be converted to d.
Enter the sample size you plan to collect.
Press Calculate. Open More options to change the target power or the rounding.

Reading the power analysis chart

The dashed curve is the distribution of the test statistic when H₀ is true; the solid violet curve is its distribution when the effect you specified is real. The dashed vertical line(s) are the critical values. The four shaded areas are the four possible outcomes:

α — H₀ is true but the statistic lands in the rejection region: a false positive.
1 - α — H₀ is true and correctly not rejected.
β — the effect is real but the statistic lands in the acceptance region: a false negative.
Power — the effect is real and the test detects it.

Moving the two curves apart — a larger effect, a larger sample, or less noise — moves area out of β and into power. The power curve below the chart plots exactly that, as the sample size grows.

Two-tailed power is the strict interpretation

For a two-tailed test this page counts a rejection in either tail, including the tail opposite the true effect. That matches R's strict = TRUE. The difference is negligible for a well-powered test and becomes visible only when the power is low.

Assumptions

The observations are independent, and normally distributed (or the sample is large enough for the central limit theorem).
For the z-test, the population standard deviation is known.
The effect size you enter is the effect you want to be able to detect — not an effect already observed in the data. Computing power from the observed effect ("post hoc power") adds no information beyond the p-value.