Power Analysis: How to Calculate and Justify Sample Size

Power analysis is the calculation that links four quantities: significance level (α), statistical power (1 − β), effect size, and sample size (n). If you fix any three, the fourth is determined. Researchers most often use it a priori, before collecting data, to find the sample size needed to detect a meaningful effect with high probability.

Ethics committees, grant reviewers, and journals increasingly expect a written sample size justification. This guide explains the four ingredients, works through the most common example (an independent-samples t test), and gives wording you can adapt for your methods section.

The four ingredients

1. Alpha (α): the false-positive rate

Alpha is the probability of rejecting a true null hypothesis (a Type I error). The convention is α = .05, two-tailed. Lowering α (for example, to .01, or after a Bonferroni correction for multiple outcomes) requires a larger sample.

2. Power (1 − β): the true-positive rate

Beta (β) is the probability of missing a real effect (a Type II error). Power is 1 − β: the probability that your study detects an effect of the assumed size if it exists. The common minimum is .80, following Cohen (1988); .90 is common in clinical trials and confirmatory studies. Higher power requires a larger sample.

3. Effect size: how big a difference you want to detect

The effect size is the size of the difference or relationship in standardized units. For comparing two means, the usual measure is Cohen's d:

d = (mean 1 − mean 2) / pooled SD

Cohen's (1988) benchmarks for d are 0.2 (small), 0.5 (medium), and 0.8 (large). Smaller effects need far larger samples. The effect size calculator can compute d from means and SDs reported in earlier studies.

4. Sample size (n)

The output of an a priori analysis. Note whether it is per group or total; confusing the two is a common error.

Types of power analysis

• A priori: given α, power, and effect size, compute the required n. This is the one to use when planning a study.
• Sensitivity: given α, power, and your actual n, compute the smallest effect you could reliably detect. Useful when the sample size is fixed by circumstances (for example, a whole class or all available patient records).
• Post hoc (observed) power: computing power from the effect observed in your own data. This is widely discouraged, because observed power is a direct function of the p value and adds no new information. If a study was non-significant, report a sensitivity analysis or confidence intervals instead.

Worked example: a priori power for an independent t test

A researcher plans to compare a new study-skills intervention with a control group on exam scores, using an independent-samples t test.

• Effect size: d = 0.5 (a medium effect; see below on how to justify this)
• α = .05, two-tailed
• Power = .80
• Allocation: equal group sizes

Result: 64 participants per group, 128 in total.

How this number is obtained

The power of a two-sided independent t test comes from the noncentral t distribution with df = 2n − 2 and noncentrality parameter δ = d × √(n / 2). Software increases n until power reaches the target:

At 63 per group, power falls just short of .80; at 64, it reaches .80. So 64 per group is the minimum.

A quick check with the normal approximation

A hand formula gives a close answer:

n per group ≈ 2 × (z for 1 − α/2 + z for power)² / d²

With z = 1.96 (for α = .05, two-tailed) and z = 0.84 (for power = .80): 2 × (1.96 + 0.84)² / 0.25 = 2 × 7.84 / 0.25 ≈ 62.7, which rounds up to 63. The exact t-based calculation adds about one participant per group because the t distribution has heavier tails than the normal. Use software for the final number.

How the inputs change the answer

Exact values for a two-sided independent t test, α = .05:

Halving the effect size roughly quadruples the required sample. This is why the effect size assumption deserves more thought than any other input.

Adjusting for attrition

If you expect 15% of participants to drop out or provide unusable data, inflate the target: 64 / 0.85 = 75.3, so recruit 76 per group (152 total). State the attrition assumption and where it came from.

Choosing and justifying the effect size

"We assumed a medium effect" is a weak justification on its own. Better options, roughly from strongest to weakest:

1. Smallest effect size of interest. The smallest difference that would matter in practice, such as a minimal clinically important difference on a validated scale, converted to d.
2. Prior studies or meta-analyses. Use a pooled estimate from similar studies. Published effects tend to be inflated by publication bias, so a somewhat smaller value is a reasonable, conservative choice.
3. Pilot data. Pilot effect sizes are very imprecise; treat them with caution.
4. Conventional benchmarks. Cohen's small/medium/large labels, used only when nothing better exists, and stated as such.

Tools for power analysis

G*Power is a free, widely used program for Windows and macOS that covers t tests, ANOVA, regression, chi-square, correlations, and more. For the example above, choose "t tests", then "Means: Difference between two independent means (two groups)", type of power analysis "A priori", and enter tails = two, effect size d = 0.5, α = .05, power = .80, and allocation ratio = 1. It returns 64 per group.

R users can use the pwr package (for example, pwr.t.test(d = 0.5, sig.level = 0.05, power = 0.80, type = "two.sample")). For complex designs such as multilevel models or mediation, simulation-based power analysis is often more realistic than closed-form formulas.

Once you have collected data, the t-test calculator runs the planned comparison, and you should report the observed effect size with its confidence interval.

Other common designs

The same logic applies to other tests; only the effect size measure changes:

• One-way ANOVA: Cohen's f (0.10 small, 0.25 medium, 0.40 large). See the one-way ANOVA guide.
• Correlation: r (0.10, 0.30, 0.50).
• Multiple regression: f² (0.02, 0.15, 0.35).
• Chi-square tests: w (0.10, 0.30, 0.50).
• Paired t test: d based on the SD of the difference scores, which usually requires fewer participants than an independent design.

These benchmarks come from Cohen (1988) and should be treated as rough defaults, not targets.

How to report a sample size justification (APA 7)

Report the test, effect size and its source, α, tails, power, the software, the resulting n, and any adjustment for attrition. A template:

If your sample size was fixed, report a sensitivity analysis instead:

(For two groups of 30, the minimum detectable effect at 80% power is about d = 0.74.)

Replace "a meta-analysis of similar study-skills interventions" with your actual source and cite it; do not cite a meta-analysis you have not read.

Common mistakes

• Confusing per-group and total n. G*Power reports both; check which one you copy.
• Using an effect size without a source. Reviewers will ask where d = 0.5 came from.
• Reporting post hoc observed power to explain a non-significant result.
• Ignoring multiple outcomes or comparisons. If you will adjust α, use the adjusted α in the power analysis.
• Forgetting attrition, missing data, or clustering. Students nested in classrooms or patients in clinics need a design effect adjustment.
• Mismatching the analysis. The power analysis should match the test you will actually run (for example, ANCOVA needs fewer participants than a t test when the covariate is strongly correlated with the outcome).

FAQ

What is a good statistical power?

At least .80 is the common minimum (Cohen, 1988). Many confirmatory and clinical studies aim for .90.

Why do I need 64 participants per group for a medium effect?

With d = 0.5, α = .05 (two-tailed), and power of .80, the noncentral t calculation shows that 63 per group gives just under 80% power and 64 gives just over. That is the smallest n meeting the target.

Is G*Power free?

Yes. G*Power is free to download and is one of the most commonly cited tools for power analysis in the social and health sciences.

Can I do a power analysis after collecting data?

You can do a sensitivity analysis to show what effect sizes your sample could detect. Avoid "observed power" based on your own effect estimate, which does not add information beyond the p value.

What if I cannot reach the required sample size?

Be transparent: report a sensitivity analysis, label the study as exploratory or a pilot if appropriate, emphasize effect sizes and confidence intervals, and discuss the limitation.

After the data are in

Power analysis belongs in the methods section, but it only pays off once the analysis and write-up are done. If you have the full dataset, Data2Paper's Generate Paper produces a draft paper with methods, results, tables, figures, and references. The first half is free to preview, and the full draft costs a one-time $19.90. Review every statistic and citation before submission.