Statistical Power

Definition

Statistical power is the probability that a hypothesis test will correctly reject the null hypothesis when a true effect exists. Power equals 1 − β, where β is the Type II error rate (the probability of failing to detect a true effect). Adequate power (commonly 80% or higher) requires sufficient sample size, a meaningful effect size, and an appropriate significance level.

Why It Matters

Under-powered studies are ethically questionable because they expose participants to risk without a realistic chance of detecting the effect being studied. They also waste resources and contribute to the file-drawer problem — null results from under-powered studies are less likely to be published, distorting the evidence base. Power analysis should be conducted before data collection to ensure a study is designed to detect the expected effect.

Example

A researcher wants to test whether a new therapy reduces anxiety by a medium effect size (Cohen's d = 0.5). With α = 0.05 and a desired power of 0.80, a power analysis shows she needs 64 participants per group. Recruiting only 25 per group would yield power of approximately 0.30 — meaning there is a 70% chance of missing the effect even if it truly exists. She secures funding to recruit the full 128 participants.

Related Terms

• Sample Size
• Effect Size
• Type I Error
• Type II Error
• Meta-Analysis
• P-Value

Software Notes

• SPSS: Analyze > Power Analysis (requires the Power Analysis module). Alternatively, use G*Power (free external software) for t-tests, ANOVA, regression, and proportions.
• R: pwr.t.test(d = 0.5, sig.level = 0.05, power = 0.80) from the pwr package. For ANOVA: pwr.anova.test(k = 3, f = 0.25, sig.level = 0.05, power = 0.80). For logistic regression: pwr::pwr.chisq.test().
• Stata: power twomeans 0 0.5, sd(1) power(0.8) for two-group comparisons. power twoproportions 0.3 0.4, power(0.8) for proportions. power onemean 100 105, sd(15) power(0.8) for one-sample tests.