Sample Size Calculators

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Sample size (n) is the number of observations, individuals, or units included in a study. It is one of the most critical decisions in experimental and survey design. The required sample size depends on: the desired statistical power (1 − β, typically 0.80 or 0.90), the significance level (α, typically 0.05), and the effect size (the magnitude of the effect you want to detect). Larger sample sizes are needed to detect smaller effects, achieve higher power, or use more stringent significance criteria. Underpowered studies are a major source of unreliable research findings — they frequently produce false negatives and may also yield inflated effect size estimates.

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Sample Size Formula for Two-Means t-test

n per group = 2(z_α/2 + z_β)² × σ² / δ²

z_α/2 = 1.96 (α = 0.05, two-tailed); z_β = 0.842 (power = 0.80) or 1.282 (power = 0.90). σ = standard deviation; δ = minimum detectable difference. Simplified: n ≈ 16 × σ² / δ² per group (for 80% power, α = 0.05).

Effect Size (Cohen's d)

d = (μ₁ − μ₂) / σ = δ/σ. Small effect: d = 0.2. Medium: d = 0.5. Large: d = 0.8. For t-test: n ≈ 16/d² (per group) for 80% power, α = 0.05. Large effect (d=0.8): n ≈ 26/group. Medium (d=0.5): n ≈ 64/group. Small (d=0.2): n ≈ 400/group.

Sample Size for Proportions

n = (z_α/2 + z_β)² × [p₁(1−p₁) + p₂(1−p₂)] / (p₁ − p₂)². Or the pooled formula using p̄ = (p₁ + p₂)/2.

Software Tools

G*Power (free); R: pwr package; PASS (commercial); R: power.t.test(), power.prop.test(). Always perform a priori power analysis before collecting data — post-hoc power analysis is generally not informative.

Glossary

Sample Size (n)
The number of observations in a study; determined by power, effect size, and significance level; n ≈ 16/d² per group for two-means t-test at 80% power, α = 0.05.
Statistical Power
1 − β; probability of detecting a true effect; target ≥ 0.80; increases with larger n, larger effect, and higher α; underpowered studies generate unreliable, non-replicable findings.
Effect Size (Cohen's d)
d = (μ₁−μ₂)/σ; standardized measure of the difference between two means; small d = 0.2; medium d = 0.5; large d = 0.8; required for sample size calculation; smaller d → larger n needed.

Frequently Asked Questions

Sample size is critical for two main reasons: (1) Statistical power: larger samples have more power to detect true effects (avoid Type II errors / false negatives). Underpowered studies miss real effects and produce unreliable results. (2) Precision: larger samples give narrower confidence intervals — estimates are more precise. Consequences of inadequate sample size: studies fail to detect real effects (false negatives); estimates are imprecise and highly variable; if effects are detected, they may be overestimated ('winner's curse') because only the most extreme random samples reach significance. The replication crisis in science is partly explained by underpowered studies being published based on chance-significant results.

For comparing two group means (two-sample t-test): n per group = 2(z_α/2 + z_β)² × σ² / δ². z_α/2 = 1.96 (α = 0.05, two-tailed); z_β = 0.842 (power = 0.80); σ = population SD; δ = minimum detectable difference. For effect size d = δ/σ: n ≈ 16/d² per group (approximate formula). Examples: d = 0.5 (medium effect): n = 16/0.25 = 64 per group; d = 0.8 (large): n = 16/0.64 = 25; d = 0.2 (small): n = 16/0.04 = 400. In practice: use G*Power software or R pwr::pwr.t.test() to get exact values. Specify: n=NULL; d=effect size; sig.level=0.05; power=0.80; type='two.sample'.

Statistical power = 1 − β = probability of rejecting H₀ when it is truly false (detecting a real effect). β = Type II error rate (false negative probability). Power depends on: sample size (larger n → more power); effect size (larger effect → more power); significance level α (relaxing α increases power but also Type I error); variability (less variance → more power). 80% power as target: this convention (Cohen, 1988) reflects a practical balance — 80% power means a 20% chance of missing a real effect (β = 0.20). Higher power (90%, 95%) requires substantially larger samples and expense. In clinical trials: 90% power is often required. In exploratory basic research: 80% is commonly accepted.

Sample size ∝ 1/d² (for t-tests), where d = effect size (Cohen's d). This inverse square relationship means: quadrupling the sample size halves the minimum detectable effect size. Practical implications: to detect a small effect (d = 0.2), you need 16× more participants than to detect a large effect (d = 0.8): n(d=0.2) ≈ 400 vs. n(d=0.8) ≈ 25. Effect size estimation: use pilot data, prior literature, or expert judgment. Overestimating expected effect size → underpowered study. Common mistake: estimating effect size from a small pilot study (the pilot effect size is usually too large due to sampling noise) → underpowered main study. Better: use published meta-analytic effect size estimates when available.