standarddeviationcalculator.net

Updated Free · runs in your browser

Statistics

Power analysis calculator

Work out how many observations a study needs before you collect them. Choose the test, the effect size, α and the power you want, and the calculator returns the sample size, or switch to solving for the power of a sample you already have. The curve shows how power grows with n.

0.2 small, 0.5 medium, 0.8 large.

As a decimal: 0.8 for 80%.

Sample size per group 64
Achieved power80.15%
Total observations128
Type II error β0.1985
Effectd = 0.5
α0.05 (two-tailed)
MethodExact noncentral t
2040608010012014000.20.40.60.81 n = 64, power 80.1% n per group Power

━ Power of the two-sample t-test, d = 0.5

Show the working, step by step

  1. Degrees of freedom and noncentrality for n = 64 per group.

    df = 2n − 2 = 126, δ = d × √(n/2) = 0.5 × √(64/2) = 2.8284

  2. Critical t for α = 0.05 (two-tailed).

    t_crit = 1.979

  3. Power is the chance a noncentral t with that δ lands beyond the critical value.

    power = P(T′(126, 2.828) > 1.979) + P(T′ < −1.979) = 0.8015

n is the smallest whole number that reaches 80% power, so the achieved power is slightly above the target.

The four quantities

Power analysis links four numbers: the effect size, the significance level α, the power 1 − β and the sample size n. Fix any three and the fourth follows. Most often α = 0.05 and power = 0.80 are set by convention, the effect size comes from subject knowledge, and n is the unknown.

The formulas

Two-sample t: df = 2n − 2, δ = d √(n/2) One-sample / paired t: df = n − 1, δ = d √n power = P(T′(df, δ) > t_crit) + P(T′(df, δ) < −t_crit)

T′ is a noncentral t variable: the distribution the t statistic follows when the true standardised difference is d. The calculator evaluates it exactly and searches for the smallest n that reaches the target. For proportions and correlations it uses closed forms:

Proportions: n = [z_α √(2p̄(1 − p̄)) + z_β √(p₁(1 − p₁) + p₂(1 − p₂))]² ÷ (p₁ − p₂)² Correlation: n = ((z_α + z_β) ÷ atanh r)² + 3

z_α is z₀.₀₂₅ = 1.960 for a two-tailed test at 5%, and z_β is z₀.₂₀ = 0.842 for 80% power.

Worked example

A trial will compare a new teaching method with the current one. The smallest improvement worth detecting is half a standard deviation, d = 0.5. With α = 0.05 two-tailed and 80% power:

  • Try n = 64 per group: df = 126, δ = 0.5 × √32 = 2.828, and t_crit = 1.979.
  • The chance that T′(126, 2.828) exceeds 1.979 is 0.8015, so power is 80.15%.
  • At n = 63 power is 79.5%, just short. The answer is 64 per group, 128 in total.

The normal shortcut 2(1.960 + 0.842)² ÷ 0.5² = 62.8 gives 63, one fewer, because it ignores the extra uncertainty from estimating the standard deviation. For the other designs with the default settings: a one-sample or paired t-test with d = 0.5 needs 34 subjects, comparing 10% with 15% needs 686 per group, and detecting r = 0.3 needs 85 cases.

Reading the power curve

Power rises steeply at first and then flattens as it approaches 1. Moving from 80% to 90% power for d = 0.5 takes 86 per group instead of 64, about a third more. Halving the effect size to d = 0.25 roughly quadruples the sample, because n grows with 1/d².

Common mistakes

  • Using an optimistic effect size from a small pilot study, which tends to overstate effects.
  • Forgetting drop-out. If you expect 15% to leave the study, divide n by 0.85.
  • Planning with a two-sample formula for a paired design, or the reverse. Pairing usually needs far fewer subjects.
  • Computing "observed power" after a non-significant result as if it explained the result.
Power analysis calculator: the worked example on this page, with its result and chart
Power analysis calculator: the worked example above, at a glance.

Common questions

What is statistical power?

The probability that a test rejects H₀ when a real effect of the stated size exists. Power is 1 − β, where β is the chance of a false negative. A study with 50% power misses a real effect half the time, so a non-significant result from it says very little.

What effect size should I use?

The smallest effect that would matter in practice, not the effect you hope to see. Take it from earlier studies, a pilot, or a judgement of what difference is worth acting on. Cohen's labels (d = 0.2, 0.5, 0.8; r = 0.1, 0.3, 0.5) are a last resort when nothing better exists. Overestimating the effect is the most common reason studies turn out underpowered.

How accurate are the numbers?

For t-tests the calculator uses the exact noncentral t distribution (algorithm AS 243), so the results match G*Power and R's pwr package: 64 per group for d = 0.5 at 80% power. For two proportions it uses the normal approximation with the pooled standard error under H₀, and for correlations Fisher's z transformation, which can be one or two above the exact answer.

Is the sample size per group or in total?

For the two-sample t-test and two proportions the answer is per group, with equal group sizes. Double it for the total. For the one-sample or paired t-test it is the number of subjects or pairs, and for a correlation the number of cases.

Should I calculate power after the study is done?

Not with the observed effect size. "Observed power" is a direct function of the p-value and adds no information. Power is a planning tool. After the study, report the confidence interval, which shows which effect sizes the data rule out.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.