Statistical Power Calculator

What Is Statistical Power?

Statistical power is the probability of detecting an effect when one truly exists. If an intervention really works, what's the chance your study will find it?

$\text{Power} = 1 - \beta$

Where $\beta$ is the probability of a Type II error (false negative—missing a real effect).

A study with 80% power has an 80% chance of detecting a true effect and a 20% chance of missing it.

The Four Connected Quantities

Power analysis involves four interrelated quantities—specify any three to find the fourth:

1. Sample size (n): Number of subjects
2. Effect size (d): Magnitude of the difference
3. Significance level (α): False positive rate
4. Power (1 - β): Detection probability

This calculator finds power given the other three.

The Formula Concept

For a two-sample t-test:

$\text{Power} = P(\text{reject } H_0 \mid H_1 \text{ is true}) = P(|t| > t_{\text{critical}} \mid \text{true effect} = \delta)$

This probability comes from the non-central t-distribution, shifted by the true effect.

Interpreting Power

The 80% convention: Most fields consider 80% power adequate. This means accepting a 20% chance of missing a real effect.

Practical Examples

Evaluating a Planned Study

"I can recruit 50 subjects per group. What's my power to detect d = 0.5?"

Given: n = 50, d = 0.5, α = 0.05 Power ≈ 0.70

You have a 70% chance of detecting a medium effect—borderline acceptable.

Assessing Feasibility

"Can I detect a small effect (d = 0.2) with available resources?"

Given: n = 100 per group, d = 0.2, α = 0.05 Power ≈ 0.29

Only 29% power—likely to miss even a real effect. Need n ≈ 400 per group for 80% power.

Retrospective Evaluation

"The study found no significant effect. Was it adequately powered?"

This informs interpretation but note: post-hoc power based on observed effect sizes is problematic (see below).

Factors Affecting Power

Sample Size

Power increases with n:

Doubling n doesn't double power—there are diminishing returns.

Effect Size

Power increases with effect size:

Large effects are easy to detect; small effects are hard.

Significance Level

More stringent α reduces power:

Higher bars for significance require larger samples.

Variability

Higher variance (larger σ) reduces power. Reducing measurement error increases power without adding subjects.

The Power Curve

A power curve plots power vs. effect size for fixed n and α:

• Power = α when effect = 0 (Type I error rate)
• Power approaches 1 as effect increases
• Steeper curves indicate more sensitive tests

One-Tailed vs. Two-Tailed

One-tailed tests have higher power for the same n:

But one-tailed tests should only be used with strong prior justification.

Post-Hoc Power: A Caution

Calculating power after seeing results, using the observed effect size, is problematic:

• It's mathematically determined by the p-value
• High p-value → low post-hoc power (circular logic)
• Doesn't answer "Was the study adequately powered?"

For retrospective assessment, calculate power for a meaningful effect size, not the observed one.

Historical Context

Jerzy Neyman and Egon Pearson (1930s) developed the theory of Type I and Type II errors, establishing power as a fundamental concept.

Jacob Cohen (1962, 1988) advocated for power analysis, noting that most psychological research was underpowered.

Despite decades of awareness, power remains a problem. Reviews consistently find:

• Median power in psychology: 35-50%
• Many published findings likely false positives from underpowered studies

Relationship to Confidence Intervals

A 95% CI that doesn't exclude zero implies a non-significant result at α = 0.05.

Wide CIs indicate low precision—often due to small samples (low power for detecting meaningful effects).

Narrow CIs indicate high precision—the study would detect small effects.

Meta-Analysis and Power

Individual studies may be underpowered, but combining studies in meta-analysis increases effective sample size and power.

Fixed-effects meta-analysis power:

• Approximately: √(Σnᵢ) replaces √n
• More studies = more power

Related Calculations

• Sample Size for Two Means — Find n for desired power
• Sample Size for Two Proportions — For binary outcomes
• Minimum Detectable Effect Calculator — What effect can you detect?
• Cohen's d Calculator — Estimate effect size
• T-Distribution Calculator — P-values from t-statistics

Summary

Statistical power—the probability of detecting a true effect—depends on sample size, effect size, and significance level. The conventional 80% power threshold accepts a 20% miss rate for real effects. Underpowered studies waste resources, produce inconclusive results, and contribute to publication bias. Always calculate power before conducting a study, use realistic effect sizes, and ensure your design has adequate power to answer your research question.