Power Analysis for t-tests

Statistical power is the probability of detecting a true effect. For a two-sample t-test with equal group sizes, power depends on effect size d, significance level α, tails, and sample size per group.

Parameters

• Effect size d (Cohen’s d)
• α (significance level)
• One-tailed or two-tailed test
• Sample size per group n

Formulas

• Critical Z depends on α and tails
• Noncentrality δ = d × √(n ÷ 2)
• Approximate power uses the standard normal CDF with δ and Zα
• Required n per group ≈ 2 × ((Zα + Zβ) ÷ d)²

Practical Guidance

• d ≈ 0.2 small, 0.5 medium, 0.8 large
• Common α values: 0.01, 0.05, 0.10
• Typical power target is 80% or higher
• Two-tailed tests require larger n than one-tailed for the same d and α

Related tools: Cohen’s d, t-test, statistical power.

Example

With d = 0.5, α = 0.05, two-tailed, and n = 50 per group, δ ≈ 2.5. The approximate power exceeds 80%. Increasing d or n increases power.