Power Analysis Calculator

About the Power Analysis Calculator

Statistical power is the probability that a test correctly rejects the null hypothesis when it's false — in other words, the probability of detecting a real effect. Power analysis is the essential planning step before any study, answering the question: "How many subjects do I need to have a good chance of finding the effect I'm looking for?"

This calculator helps you determine the required sample size for a desired power level, or compute the achieved power for a given sample size. It supports one-sample and two-sample z-tests, t-tests, paired t-tests, and chi-square tests. Enter your expected effect size (Cohen's d), significance level, and desired power to get the minimum sample size.

Power analysis is critical in clinical trials (ethical requirement to not waste patient resources), psychology experiments, A/B tests, market research, and any study where you need to justify your sample size to reviewers, grant agencies, or ethics boards.

When This Page Helps

An underpowered study wastes resources by being unlikely to detect real effects, while an overpowered study wastes resources by enrolling far more subjects than necessary. This calculator finds the sweet spot. It also provides a power curve showing how power changes with sample size, and a reference table of Cohen's d benchmarks to help you choose a realistic effect size.

How to Use the Inputs

1. Select the type of statistical test you plan to use.
2. Enter the expected effect size (Cohen's d). Use 0.2 for small, 0.5 for medium, 0.8 for large if unsure.
3. Set the significance level alpha (typically 0.05).
4. Set your desired power (typically 0.80 or 0.90).
5. Review the required sample size — this is the minimum per group.
6. Optionally enter a specific sample size to see its achieved power.
7. Use the power curve and sensitivity table to explore trade-offs.

Example Calculation

Tips & Best Practices

• Always perform power analysis BEFORE collecting data. Post-hoc power analysis (after the study) is widely discouraged as misleading.
• Use the smallest effect size that would be clinically or practically meaningful, not the largest effect you hope for.
• Power of 0.80 (80%) is conventional; 0.90 is preferred for important studies. Below 0.60 is generally considered unacceptable.
• For two-sample tests, the required n is per group — you need twice as many subjects total.
• If the required sample size is impractical, consider whether a one-tailed test or a larger effect size is justified.
• Unequal group sizes reduce power. Keep groups balanced when possible.

The Four Components of Power Analysis

Power analysis involves four quantities, and knowing any three determines the fourth: (1) significance level α, (2) effect size d, (3) sample size n, and (4) power 1−β. Most commonly, you fix α and d, then solve for n given a desired power. Alternatively, you can compute what power a given n achieves, or what effect size is detectable with your n and power.

Choosing an Appropriate Effect Size

The most critical (and difficult) decision in power analysis is selecting a realistic effect size. Overly optimistic effect sizes lead to underpowered studies. Sources include: pilot studies, published literature meta-analyses, subject-matter judgment about the minimum clinically important difference (MCID), and standardized benchmarks. When in doubt, use a smaller effect size for a more conservative (larger) sample size estimate.

Power in Different Study Designs

Paired designs (e.g., before-after) typically have more power than independent-groups designs because they control for individual differences. Unequal group sizes always reduce power relative to balanced designs with the same total n. For ANOVA and regression, power depends on the number of groups/predictors and the expected effect size (Cohen's f or f²). Multi-site studies pool power across locations but introduce complexity in the analysis.

Frequently Asked Questions

• What is Cohen's d?

  Cohen's d is a standardized effect size measuring the difference between two means divided by the pooled standard deviation. d = 0.2 is small (hard to see), 0.5 is medium (visible to careful observer), and 0.8 is large (obvious). It's the most common effect size for power analysis.

• What power level should I target?

  80% (0.80) is standard in most fields, meaning you have an 80% chance of detecting a real effect. For clinical trials or high-stakes research, 90% is preferred. Power below 50% means you're more likely to miss the effect than detect it.

• What if I don't know the expected effect size?

  Use pilot study data, published meta-analyses, or Cohen's benchmarks (0.2, 0.5, 0.8). Some researchers compute sample sizes for a range of effect sizes. A sensitivity analysis showing what effect sizes your planned n can detect is also informative.

• Why is post-hoc power analysis discouraged?

  Because after a study, post-hoc power is a direct function of the p-value obtained — it adds no new information. A non-significant result with "adequate post-hoc power" is a contradiction. Design-stage power analysis is the correct approach.

• How does alpha affect sample size?

  A stricter alpha (e.g., 0.01 instead of 0.05) requires larger samples to maintain the same power. This is because the rejection region shrinks, making it harder to reject H₀ and requiring more data to compensate.

• What is the relationship between power and Type II error?

  Power = 1 − β, where β is the Type II error rate (probability of failing to detect a real effect). If power is 0.80, β is 0.20 — there's a 20% chance of a false negative.