Question 1

What is statistical power and why does 0.80 matter?

Accepted Answer

Statistical power is the probability of detecting a true effect. Power = 1 − β, where β is the Type II error rate (probability of a false negative). The 0.80 convention (Jacob Cohen, 1988) was chosen as a pragmatic balance: it means a 20% false-negative rate. Many fields now require 0.90 power for pre-registered studies. Power below 0.50 means you are more likely to miss a real effect than detect it.

Question 2

What inputs do I need for a power analysis?

Accepted Answer

Four quantities are related — knowing any three allows you to solve for the fourth: (1) Effect size (d): the size of the effect you want to detect. (2) Sample size (n): observations per group. (3) Significance level (α): typically 0.05. (4) Power (1−β): probability of detection. Most commonly, you know d and α, set your target power (0.80 or 0.90), and solve for required n.

Question 3

What Cohen's d should I use?

Accepted Answer

Use the smallest effect size of practical importance — not necessarily 'medium'. If prior research exists, use the effect size from previous studies or a meta-analysis. Cohen's rough benchmarks: d=0.2 (small), d=0.5 (medium), d=0.8 (large). Avoid 'medium' as a default — it is arbitrary and may not reflect your specific context.

Question 4

What happens to power as sample size increases?

Accepted Answer

Power increases with n via √n: doubling n increases the effective effect size (d√n) by √2 ≈ 1.41. For a fixed d=0.5 and α=0.05: n=25 gives power≈0.41; n=50 gives power≈0.70; n=85 gives power≈0.80; n=134 gives power≈0.90. Power approaches 1.0 as n → ∞ — with a large enough sample, even a trivial effect will be detected.

Question 5

Is the power formula exact or approximate?

Accepted Answer

This calculator uses the normal approximation to the non-central t-distribution: Power ≈ Φ(d√n − z_{α/2}). This is accurate for n ≥ 30 and is standard for planning purposes. For small samples (n < 30) with t-tests, exact power uses the non-central t-distribution, which gives slightly different results. Software like G*Power uses the exact non-central t.

Question 6

What is a priori vs post hoc power analysis?

Accepted Answer

A priori power analysis is done before data collection to determine required sample size — this is the correct and primary use. Post hoc power analysis (after seeing results) is controversial: if you have p > 0.05 and compute power using the observed d, you get "observed power" which is mathematically equivalent to a transformation of the p-value and adds no new information. A priori power analysis is what journals and IRBs require.

Symbol	Meaning
Cohen's d	(μ₁ − μ₂) / σ — standardized mean difference (effect size)
n	Sample size per group
α	Significance level (Type I error rate) — typically 0.05
β	Type II error rate = 1 − Power (probability of missing a real effect)
Power (1−β)	Probability of correctly rejecting H₀ when the effect is real

d	Label	n for 80% power	n for 90% power
0.20	Small	197	265
0.35	Small-medium	66	89
0.50	Medium	34	46
0.80	Large	14	19
1.20	Very large	7	9

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