Specialty › Power (hypothesis tests)

t-test

A priori or post-hoc power analysis for one- and two-sample t-tests. Solve for any three of {n, Cohen's d, α, power}; the fourth is computed.

What is t-test?

A priori power analysis is the standard sample-size step before a study: pick the smallest effect you'd care to detect (d), fix α (usually 0.05), pick a power target (usually 0.80), solve for n. The resulting n is what you need in your protocol.

Post-hoc power (computing power from the observed effect after the study) is widely regarded as useless or misleading — it's a deterministic function of the p-value (Hoenig & Heisey 2001). Reach for it only when a journal demands it, and pair it with the more informative confidence interval on the effect.

Cohen's d effect-size anchors: 0.2 small, 0.5 medium, 0.8 large. Field-specific norms usually exist and override these.

When should I use t-test?

Planning sample size before a study.
Justifying an n in a protocol or grant.
Avoid for post-hoc power; report the CI on the observed effect instead.

What data does it need?

Three of {n, d, α, power} + design (one-sample / two-sample / paired) + alternative.

What does it report?

Fourth quantity, with the solved row highlighted.

Formula

For two-sample: d = (μ₁ − μ₂) / σ;  required n per group ≈ 2(z_{1−α/2} + z_{power})² / d²

How do I interpret the result?

Always sanity-check the d you assume against pilot data or published effects in the same field. Powering for d = 0.8 when the realistic effect is d = 0.3 produces a study that's badly underpowered for the realistic alternative.

References

Cohen (1988). Statistical Power Analysis for the Behavioral Sciences, 2nd ed.
Hoenig & Heisey (2001). The abuse of power. American Statistician 55(1).