Statistical Power Calculator

StatisticsLast updated: August 22, 2026

Statistical power is the probability that your test detects an effect that really exists: 1 − β. An underpowered study misses that effect most of the time and ends with "no significant difference found" — a result that is evidence of an inadequate study, not of an absent effect. Power depends on three quantities — sample size, effect size, and significance level — and once all three are known it can be computed as a single number.

This calculator returns the power and the Type II error probability (β) for comparisons of two independent groups as well as for one-sample and paired designs; it uses the normal (z) approximation instead of the noncentral t distribution and says so explicitly in the results. It produces a power curve showing how power grows with the sample, a second curve showing power against effect size, and a power table, then judges the result against the 80% benchmark used in the literature.

In a two-group design the effective sample enters as n/2; in a one-sample design it enters as n.
In a two-group design, the number of observations PER GROUP; in a one-sample or paired design, the total number of observations (pairs).
d = (μ₁ − μ₂) / σ. Rough benchmark: 0.20 small · 0.50 medium · 0.80 large.
Sets the false positive risk. In the social sciences 5% is the common choice.
A one-tailed test gives higher power for the same sample; the direction must be fixed before seeing the data.

Statistical Power (Normal Approximation)

Two independent groups:  δ = d · √(n / 2)      (n = per group)
One sample / paired:     δ = d · √n
Two-tailed power:  1 − β = Φ(δ − z₍α/₂₎) + Φ(−δ − z₍α/₂₎)
One-tailed power:  1 − β = Φ(δ − z₍α₎)
Type II error:     β = 1 − (1 − β)
Inverse solution:  n = 2 · (z₍α/₂₎ + z₍β₎)² / d²   (two groups, per group)
Effect size:       d = (μ₁ − μ₂) / σ

Power is computed with the standard normal (z) approximation rather than the noncentral t distribution. This is a deliberate simplification: the approximation overstates power by a few points in small samples, and the difference vanishes as n grows. For publications that need exact decimals, verifying the result with G*Power or the pwr package in R is recommended.

How to Calculate

  1. Choose your study design: two independent groups, or a one-sample / paired measurement?
  2. Enter the sample size — in a two-group design write the number of observations PER GROUP, not the total.
  3. Enter the effect size (d) you want to detect; d = the difference considered meaningful / the standard deviation.
  4. Set the significance level (α) and the test direction; a two-tailed test is the standard choice.
  5. Read the power (1 − β) and the Type II error probability (β); if the power falls below the 80% benchmark, take the warning seriously.
  6. Use the power curve to see how enlarging the sample raises power, and pick the n needed for 80% power from the table.

Worked Examples

A typical underpowered study (30 per group)

In the very common design of 30 people per group, a medium effect (d = 0.50) gives δ = 0.50 · √15 = 1.9365 and a power of only 49.07%. In other words, if a real medium effect exists, the study misses it roughly as often as a coin toss (β = 0.5093); reaching 80% power would require 63 people per group.

Statistical power (1 − β): 49.07% · Type II error probability (β): 0.5093 · Noncentrality parameter (δ): 1.9365

Cohen's classic scenario (64 per group)

In the design of 64 people per group, treated as the standard for a medium effect in the literature, δ = 0.50 · √32 = 2.8284 and the power is 80.74%; the Type II error drops to β = 0.1926. Under the normal approximation the 80% threshold is already crossed at 63 people per group, so 64 is the rounded, safe-side design.

Statistical power (1 − β): 80.74% · Type II error probability (β): 0.1926 · Noncentrality parameter (δ): 2.8284

Paired measurement, 25 pairs

In a before-after design with 25 pairs and d = 0.60, δ = 0.60 · √25 = 3.0000 and the power is 85.08%, with a Type II error of β = 0.1492. For the same effect, a paired design reaches adequate power with far fewer observations than two independent groups: 22 pairs are enough for 80% power.

Statistical power (1 − β): 85.08% · Type II error probability (β): 0.1492 · Noncentrality parameter (δ): 3.0000

Frequently Asked Questions

What exactly does statistical power measure?
Power is the probability of rejecting the null hypothesis when it is genuinely false — that is, of catching an effect that exists: 1 − β. A power of 80% means a real effect will be detected in four studies out of five and missed in one. Power is determined by three quantities: sample size, effect size, and α; once those three are known, the fourth (power) follows as a single number.
Why use the normal approximation instead of the noncentral t?
The exact power calculation for an independent samples t-test requires the cumulative probability of the noncentral t distribution, which needs numerical integration in the browser. This calculator applies the widely used normal (z) approximation instead and states so with every result. The difference is a few points in small samples and negligible from about n ≥ 30 upward.
How can I increase power?
The most direct route is a larger sample; power grows with the square root of the sample size. The second is to shrink σ by reducing measurement error or choosing a more homogeneous sample, which enlarges d. The third is to change the design: paired measurement, analysis of covariance (ANCOVA), or a block design gives markedly higher power with the same number of participants. Raising α also increases power, but it is not preferred because it raises the Type I error risk.
Is it correct to compute power after the study is finished?
Post-hoc power computed from the observed effect size is a direct transformation of the p-value: the larger the p, the smaller the power. That makes the sentence "it was not significant because power was low" circular reasoning that adds no information. The right way to interpret a non-significant result is to look at the confidence interval or to compute the minimum detectable effect (MDE) of the study.
Where does the 80% power benchmark come from, and is it mandatory?
Cohen (1988) proposed β = 4α on the assumption that a Type I error is about four times as serious as a Type II error; for α = 0.05 that means β = 0.20, that is, 80% power. This is a convention, not a law of nature. Irreversible clinical decisions call for 90–95% power, while exploratory pilot studies may accept lower power as long as the study is reported as a pilot.
What will I see if I compare the results with G*Power?
Selecting "t tests → Means: Difference between two independent means" and "Post hoc: Compute achieved power" in G*Power gives a very close power value for the same inputs. For 64 per group, d = 0.50, and α = 0.05 two-tailed, for instance, this calculator gives 80.74% and G*Power gives 80.15%. The difference comes from the use of the noncentral t distribution and shrinks as the sample grows.
Does this power apply to nonparametric tests as well?
Not directly, but it gives a good upper bound. If the data is normally distributed, the power of the Mann-Whitney U test is about 95% of that of the t-test (asymptotic relative efficiency 3/π ≈ 0.955). When the distribution is markedly skewed or heavy-tailed, the nonparametric test can be more powerful. For an exact answer, a simulation-based power analysis is recommended.