Power in Statistics: Formula and Worked Example

Statistical power is the probability that a hypothesis test rejects the null hypothesis when the null hypothesis is false. It equals 1 − β, where β is the probability of a Type II error. This guide is for researchers who need to plan a sample size or judge whether a study could have detected the effect it was looking for. After reading it you can compute power for a two-group comparison, interpret the result, and report it.

What is power in statistics? (definition)

Power is “the probability that a test will reject the null hypothesis” (gerontology effect-size and power review, PMC6736231), or, in the words of another review, “the probability of rejecting a false null hypothesis”, calculated as 1 − β (PMC7745163). A study with 80% power has an 80% chance of returning a statistically significant result if the true effect is exactly the size the study was designed to detect. The remaining 20% is the Type II error rate, β. The related Type I error rate, α, is the chance of rejecting a true null; the two are compared on Type I and Type II errors.

Power depends on the effect size, the sample size, the significance level α, and the variability of the data. Both small samples and small effect sizes reduce power (PMC7745163). Power is “conventionally set at .80” (PMC6736231).

Power formula and sample-size formula

Power itself is 1 − β. For a two-sided, two-group comparison of means with equal group sizes, the normal-approximation sample size per group is:

n = 2 (Z_1−α/2 + Z_1−β)² / d²

n: participants per group.
Z_1−α/2: the standard normal quantile for a two-sided test. It is 1.96 when α = 0.05.
Z_1−β: the standard normal quantile for the target power. It is 0.84 when power = 0.80.
d: the standardised effect size, the difference between the two means divided by the (pooled) standard deviation.

The values 1.96 and 0.84 are given in PMC7745163; writing the formula with a standardised d expresses the same calculation in units of standard deviations. Software that uses the exact t distribution returns slightly larger values, as the worked example shows.

Worked example: how many participants for 80% power?

Suppose you plan a two-group comparison and the smallest difference you care about is a medium effect, d = 0.5, with α = 0.05 (two-sided) and target power 0.80.

Z_1−α/2 + Z_1−β = 1.96 + 0.84 = 2.80.
(2.80)² = 7.84, and 2 × 7.84 = 15.68.
n = 15.68 / 0.5² = 15.68 / 0.25 = 62.7, which rounds up to 63 per group.
The exact t-test calculation gives 63.77, so 64 per group (128 in total), which matches the n = 64 per group cited in PMC6736231 for a medium effect.

Changing one input at a time shows the trade-offs (two-sided, power 0.80; exact values computed with statsmodels):

Effect size d	α	Normal approximation, n per group	Exact t, n per group
0.5	0.05	63	64
0.5	0.01	94	96
0.2	0.05	392	394
0.8	0.05	25	26

Run in the other direction, 30 per group at d = 0.5 and α = 0.05 gives a power of only about 0.48, while 64 per group gives 0.80.

How to interpret power

Power	Reading
0.80 or higher	Meets the conventional target (PMC6736231). If the true effect is at least the planned size, about 4 in 5 such studies would reach significance.
About 0.50	Roughly a coin flip for the planned effect; the 30-per-group example above sits here.
Below 0.50	More likely than not to miss the planned effect, so a non-significant result says little about whether the effect exists.

How to increase power

Increase the sample size. Small samples reduce power (PMC7745163).
Target a larger effect or reduce noise. Low effect sizes reduce power (PMC7745163); more precise measurement lowers the standard deviation and so raises d.
Raise α. Lowering α lowers power at a fixed sample size (PMC7745163), so raising it does the opposite, at the price of more false positives.

Why is 30 a good sample size? (and 0.01 vs 0.05)

It is not a power calculation. PMC7745163 notes that “some statisticians arbitrarily regard 30 as a critical sample size.” The worked example shows why it is no guarantee: 30 per group reaches about 48% power for a medium effect. Choose n from the power formula, not from the rule of thumb.

Neither 0.01 nor 0.05 is better in general. The most common α is 0.05, a lower α such as 0.001 is chosen when false positives are costly, and such conventions are “generally arbitrary” (PMC7745163). The table above shows the price of the stricter level: at d = 0.5, moving from α = 0.05 to 0.01 raises the required sample from 64 to 96 per group.

Assumptions and common mistakes

The calculation assumes a specific test, a specific effect size and equal group sizes; unequal groups need the ratio entered (statsmodels takes a ratio argument).
Take the effect size from prior work, a pilot study or the smallest effect of practical interest, and say which.
Round sample sizes up, never down.
Plan power before collecting data. It is a design property, not a verdict on a non-significant p-value.
Different tests need different formulas; this page covers the two-group comparison of means. For software that handles many tests, see the G*Power guide.

How to report power (APA-style sentence)

An a priori power analysis is usually reported with the test, α, target power, effect size and the resulting n. A template: “An a priori power analysis for an independent-samples t test (two-tailed, α = .05, power = .80, d = 0.50) indicated that 64 participants per group were required.” Replace the numbers with your own inputs and name the software and version used.

How to calculate power in Excel, R or Python

Excel (normal approximation)

=ROUNDUP(2*(NORM.S.INV(1-0.05/2)+NORM.S.INV(0.80))^2/0.5^2,0)

Returns 63.

R (pwr package, exact t)

library(pwr)
pwr.t.test(d = 0.5, sig.level = 0.05, power = 0.80, type = "two.sample")

Reports n = 63.77 per group; round up to 64.

Python (statsmodels)

from statsmodels.stats.power import TTestIndPower
n = TTestIndPower().solve_power(effect_size=0.5, alpha=0.05, power=0.80, ratio=1.0, alternative='two-sided')
print(n)  # 63.77, round up to 64

In solve_power exactly one of effect_size, nobs1, alpha and power is left as None and the method solves for it; effect_size is the difference between the two means divided by the standard deviation (statsmodels documentation). Related background: sampling methods and sampling variance.

FAQ

What do you mean by statistical power?

It is the probability of rejecting the null hypothesis when it is false, equal to 1 − β. Higher power means a lower chance of a Type II error.

What is the formula for calculating power?

Power = 1 − β. To find the sample size for a two-group mean comparison use n = 2(Z_1−α/2 + Z_1−β)² / d² per group, or solve for power with software such as the R pwr package or statsmodels.

Why is 30 a good sample size?

It is a convention, not the result of a power analysis. See the section above.

Which is better, 0.01 or 0.05 significance level?

Neither is universally better; 0.01 cuts false positives but needs a larger sample for the same power.

Sources

statsmodels documentation, TTestIndPower.solve_power, statsmodels.org, read 2026-11-07.
Effect size and statistical power review, PMC6736231, read 2026-11-07.
Sample size, power and effect size revisited, PMC7745163, read 2026-11-07.

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