Statistical Power Calculator

Compute statistical power, required sample size, and visualize power curves for two-sample t-tests.Understand the interplay between sample size, effect size, significance level (α), and power (1−β).

Two-sample t-test is the most common design.
Two-tailed is more conservative; one-tailed has higher power for directional predictions.
Small=0.2, Medium=0.5, Large=0.8 (Cohen, 1988).
Probability of Type I error (false positive).
Total N = 2n for two-sample t-test.
0.80
The calculator will estimate the sample size needed to achieve this power.
? Small effect (d=0.2, n=50)
? Medium effect (d=0.5, n=30)
? Large effect (d=0.8, n=20)
⚠️ Underpowered (d=0.2, n=20)
✅ Well-powered (d=0.5, n=80)
Privacy first: All computations are performed locally in your browser. No data is sent to any server.

What Is Statistical Power and Why Does It Matter?

Statistical power is the probability that a hypothesis test correctly rejects the null hypothesis when a true effect exists. Formally, power = 1 − β, where β is the probability of a Type II error (failing to detect a real effect). A study with low power is a waste of resources and, more troubling, may lead to false negative conclusions that stall scientific progress.

Power = P( reject H₀  |  H₁ is true )

where H₀ is the null hypothesis and H₁ is the alternative hypothesis.

The four key pillars of power analysis are: effect size (magnitude of the phenomenon), sample size (number of observations), significance level α (tolerance for false positives), and power 1−β (sensitivity to detect effects). Change any one, and the others adjust accordingly.

How This Calculator Works

This tool computes power for the two-sample independent t-test, one-sample t-test, and paired t-test using the non-central t-distribution. Unlike normal approximations, the non-central t-distribution exactly models the behavior of the test statistic under the alternative hypothesis. The non-centrality parameter (NCP) is:

δ = d · √(n/2)    (for two-sample t-test, equal n per group)

where d is Cohen's d and n is the sample size per group. Power is then computed as:

Power = P( Tν,δ > tcrit ) + P( Tν,δ < −tcrit )

with ν = 2n − 2 degrees of freedom and tcrit the critical value from the central t-distribution at level α. The calculator uses numerical integration (Gauss–Legendre quadrature) to evaluate the non-central t CDF, ensuring high accuracy across all sample sizes.

For sample size estimation, we perform a bisection search to find the smallest n that achieves the target power (default 0.80). The power curve is rendered on an interactive canvas, showing how power increases with sample size and marking your current design.

Practical Applications Across Disciplines

  • Clinical Trials: Determine the number of patients needed to detect a clinically meaningful treatment effect.
  • Social Sciences: Plan survey-based studies with adequate sensitivity for small-to-moderate effects.
  • A/B Testing: Calculate sample sizes for online experiments to reliably detect conversion rate differences.
  • Quality Control: Ensure that process changes are detectable with high probability in manufacturing.
  • Ecology & Biology: Design field studies to detect population shifts or treatment responses.

Interpreting Your Results

A power of 0.80 is conventionally considered adequate in many fields (Cohen, 1988). However, some domains (e.g., genomics, physics) demand higher power (0.90–0.99) due to multiple testing or safety requirements. Our badge system gives you a quick visual cue:

  • High (≥0.80) – Your study is well-powered.
  • Moderate (0.60–0.79) – Consider increasing sample size or effect size.
  • Low (<0.60) – Your study is underpowered; results may be inconclusive.
Case Study: A Psychology Experiment

A researcher hypothesizes that a new teaching method improves test scores. Based on prior literature, the expected effect size is Cohen's d = 0.35 (small-to-moderate). With α = 0.05 and n = 40 per group, the calculator returns power ≈ 0.53. This means the study has only a 53% chance of detecting the effect—nearly a coin flip. The researcher increases n to 80 per group, boosting power to 0.83. The additional cost is justified by the much higher probability of obtaining meaningful results.

Common Misconceptions About Power

  • “Power of 0.80 means I have an 80% chance of being correct.” – No, power is the probability of rejecting H₀ when H₁ is true. It does not reflect the probability that your conclusion is correct.
  • “A significant result implies the study was well-powered.” – Not necessarily. A significant result can occur even with low power, but low power increases the risk of false negatives and may indicate publication bias.
  • “Power analysis is only for grant proposals.” – Power analysis is essential at every stage: planning, interpreting null results, and meta-analysis.
  • “Larger sample size always solves everything.” – While larger samples increase power, they also increase cost and may detect trivial effects. Always consider the practical significance alongside statistical significance.

The Relationship Between α, β, and Power

  • α (Type I error): Probability of rejecting H₀ when it is true. Commonly set to 0.05.
  • β (Type II error): Probability of failing to reject H₀ when H₁ is true. Power = 1 − β.
  • Decreasing α (more stringent) reduces power unless sample size is increased.
  • Increasing sample size increases power for any fixed α and effect size.

Reference Table: Cohen's d Effect Size Benchmarks

Effect Size Cohen's d Typical Use Case Power (n=30, α=0.05, two-tailed)
Very Small 0.10 Subtle behavioral differences 0.08
Small 0.20 Educational interventions, small effects 0.17
Medium 0.50 Typical psychological / social effects 0.70
Large 0.80 Strong medical / biological effects 0.95
Very Large 1.20 Obvious, easily detectable effects 0.999

Power Analysis in the Era of Replication

With the growing emphasis on replication in science, power analysis has become more critical than ever. Underpowered studies contribute to the “replication crisis” because they produce unreliable estimates and inconsistent results. By using this calculator, you can design studies that are adequately powered, increasing the likelihood that your findings are reproducible. We encourage researchers to report power analyses in their methods sections and to consider sensitivity power analyses when sample size is fixed.

Frequently Asked Questions

A power of 0.80 (80%) is widely accepted as adequate. Some fields require 0.90 or 0.99, especially when false negatives are costly. The target power should be chosen based on the context and consequences of Type II errors.

Use prior literature, pilot studies, or meta-analyses to estimate the expected effect size. If no information is available, use Cohen's benchmarks (0.2 small, 0.5 medium, 0.8 large) as a starting point, but be aware that these are conventions, not universal truths.

Currently, this tool focuses on t-tests. For ANOVA, use effect size measures like η² or f and consult our ANOVA Power Calculator (separate tool). For regression, consider using G*Power or similar specialized software.

A one-tailed test has higher power for detecting an effect in a specified direction, but it is only appropriate when you have a strong directional hypothesis. A two-tailed test is more conservative and is standard in most exploratory research.

This occurs when the effect size is extremely small (e.g., d < 0.05) or the target power is very high (e.g., 0.99). In such cases, an impossibly large sample size would be needed. Consider whether such a tiny effect is practically meaningful.

Start with Cohen (1988) Statistical Power Analysis for the Behavioral Sciences. For online resources, see Statistics How To and Wikipedia: Power (statistics). Our blog also features in-depth tutorials on experimental design.
References: Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates. Faul, F., Erdfelder, E., Lang, A.-G., & Buchner, A. (2007). G*Power 3: A flexible statistical power analysis program. Button et al. (2013). Power failure: why small sample size undermines the reliability of neuroscience.