Sample Size Estimation for Two-Group Comparison (t-test / Proportion Test)

Per-group sample size for comparing two means:

n = 2 × (z₁₋α∕₂ + z₁₋β)² / d²

where d = |μ₁ − μ₂| / σ is the standardized effect size (Cohen’s d). At α = 0.05, z₁₋α∕₂ = 1.960; at 80% power, z₁₋β = 0.842, giving the commonly cited

n ≈ 15.7 / d²

What this formula tells you: Sample size scales inversely with the square of the effect size. Halving the effect size quadruples the required sample size. This is why detecting a small difference is so costly.

Before you use this:

• Where does σ come from? It must come from a pilot experiment or comparable published data. A σ pulled from thin air produces a meaningless n.
• This is a normal approximation. It slightly underestimates at small sample sizes (n < 20); the rigorous approach is to iterate with the t-distribution. This tool will flag results when the estimate is small.
• The result is per group, not the total. In animal studies, add an allowance for expected attrition on top of this.
• Sample size cannot be determined after the fact. Running the experiment, finding a non-significant result, and then calculating how many subjects were “needed” is post-hoc power analysis and is statistically meaningless. Sample size must be fixed before the experiment begins.