Power Analysis Calculator

Q: What is statistical power?

Statistical power (1 − β) is the probability of correctly rejecting the null hypothesis when it is false - i.e., detecting a true effect. A study with 80% power has an 80% chance of finding a significant result if the hypothesised effect size is real. Conversely, β = 20% is the probability of a Type II error (missing a real effect). Power depends on three things: effect size, sample size, and significance level α.

Q: What is Cohen's d?

Cohen's d is a standardised effect size for mean-based tests: d = (μ₁ − μ₀) / σ. It expresses how many standard deviations the population mean is from the null value. Jacob Cohen (1988) proposed conventional benchmarks: d = 0.2 (small), d = 0.5 (medium), d = 0.8 (large). For power analysis, you estimate d from pilot data, from the literature, or by specifying the minimum effect you care about detecting.

Q: What sample size do I need for 80% power?

For a two-tailed Z-test at α = 0.05 with 80% power, the required n depends on effect size: d = 0.2 (small): n ≈ 197; d = 0.5 (medium): n ≈ 32; d = 0.8 (large): n ≈ 13. These are approximate for one-sample tests; two-sample tests need roughly twice as many per group. This calculator finds the exact n for any d and α combination.

Q: What is the relationship between α, β, n, and effect size?

These four quantities are mathematically related - fixing any three determines the fourth. (1) Decreasing α (more strict) increases β (less power) for fixed n and d. (2) Increasing n increases power. (3) Larger d (bigger effect) means more power. (4) For fixed power, a smaller d requires a larger n. Power analysis exploits this relationship to answer: 'How large does n need to be to detect an effect of size d with power 1−β?'

Q: What is a priori vs post-hoc power analysis?

A priori power analysis is done before data collection to determine the required sample size. This is the correct use of power analysis. Post-hoc power analysis computes power after seeing the data, using the observed effect size - this is widely criticised because the observed effect size is noisy, and 'observed power' is directly determined by the p-value and carries no additional information. Always plan sample size before the study.

Q: Why is 80% the conventional power threshold?

Jacob Cohen (1988) suggested 80% power as a reasonable standard for social science research, based on a trade-off between the cost of larger samples and the acceptable Type II error rate. He argued that a 4:1 ratio of β to α (80% power with α = 0.05, giving β = 0.20) was reasonable. Medical and clinical research often requires 90% power (β = 0.10) due to higher stakes. These are conventions, not mathematical necessities.

Q: How does the Z-test power formula work?

For a two-tailed one-sample Z-test with effect size d and sample size n: Power = Φ(d√n − z_α/2) + Φ(−d√n − z_α/2), where Φ is the standard normal CDF and z_α/2 is the upper α/2 critical value (e.g., 1.96 for α = 0.05). The term d√n is called the non-centrality parameter - it represents how many standard errors the true mean is from H₀. Larger n or larger d shifts the distribution further from H₀, increasing power.

Q: What is the difference between power for Z-tests and t-tests?

For Z-tests (σ known), power is computed exactly using the normal distribution. For t-tests (σ estimated), power depends on the non-central t-distribution. This calculator uses the normal approximation to non-central t, which is accurate for practical purposes. Exact t-test power requires the non-central t-CDF, which converges to the Z-based formula for large n. The approximation gives conservative (slightly lower) power estimates for small n.

Find statistical power and required sample size for your study design - enter effect size, n, and α.

📖 What is Statistical Power?

Statistical power is the probability that a hypothesis test will correctly detect a true effect - that is, reject the null hypothesis H₀ when it should be rejected. Power = 1 − β, where β is the probability of a Type II error (failing to detect a real effect). If a study has 80% power, it has an 80% chance of finding statistical significance when the hypothesised effect is real. Low-powered studies miss real effects and waste resources.

Power analysis is most valuable before data collection, when designing a study. Given a desired power level (typically 80% or 90%), a significance level α (typically 0.05), and an expected effect size, power analysis tells you the minimum sample size needed. Alternatively, given a fixed n, it tells you what power you have - and whether the study is worth running.

The key input is the effect size (Cohen's d) - a standardised measure of how large the true difference is relative to variability. For a one-sample test, d = (μ − μ₀) / σ. Larger effects require smaller samples to detect. Jacob Cohen's benchmarks - d = 0.2 (small), 0.5 (medium), 0.8 (large) - are widely used when the true effect is unknown, though it is always better to use domain-specific knowledge or prior studies to estimate d.

This calculator supports both Z-tests (when σ is known) and t-tests (σ estimated from sample), with both one-tailed and two-tailed options. It computes power for your current n and also finds the required n to achieve 80% and 90% power.

📐 Formulas

Power (two-tailed Z) = Φ(d√n − z_α/2) + Φ(−d√n − z_α/2)

One-tailed Z-test power: Power = Φ(d√n − z_α)

where Φ = standard normal CDF, d = Cohen's d effect size, n = sample size, z_α/2 = critical Z (e.g., 1.96 for α = 0.05 two-tailed)

Non-centrality parameter: δ = d√n - the expected Z-score under H₁. Larger δ = more power.

t-test power (approximation): Replace z_α/2 with t_crit(df = n−1) in the formula above. This normal approximation to non-central t is accurate for practical purposes.

Required n for target power: Solve n = (z_α/2 + z_β)² / d² for Z-tests. For example, at α = 0.05 (two-tailed), 80% power (z_β = 0.842), d = 0.5: n = (1.96 + 0.842)² / 0.5² ≈ 32.

Type I error: α (probability of rejecting H₀ when it is true, set by the researcher)

Type II error: β = 1 − Power (probability of failing to reject H₀ when H₁ is true)

📖 How to Use This Calculator

Select test type (Z-test or t-test) and tail type (two-tailed for most research). Two-tailed tests require larger n for the same power.

Enter Cohen's d: estimate from pilot data, literature, or use d = 0.2 (small), 0.5 (medium), 0.8 (large) as a starting point. Also enter your planned sample size n.

Choose your significance level α (typically 0.05) and click Calculate Power.

Read the results: statistical power, Type II error β, and required n for 80% and 90% power. If power < 80%, increase n to the recommended value.

📝 Example Calculations

Example 1 - Clinical Trial Design

A clinical trial plans to test a new drug. Based on prior studies, the expected effect is d = 0.5 (medium). The trial uses α = 0.05 two-tailed. How many patients are needed for 80% power?

Power formula: n = (z_0.025 + z_0.20)² / d² = (1.960 + 0.842)² / 0.25 = 7.849 / 0.25 ≈ 32 patients

With n = 32 and d = 0.5: Power ≈ 80%. To achieve 90% power, n ≈ 44 patients are needed.

Result = Power ≈ 80% with n = 32

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Example 2 - Survey Planning (Small Effect)

A market researcher expects a small effect (d = 0.2) and uses a two-tailed Z-test at α = 0.05. What sample size achieves 80% power?

n = (1.96 + 0.842)² / 0.2² = 7.849 / 0.04 ≈ 197 respondents

Small effects require large samples. With n = 100 and d = 0.2, power ≈ only 29% - the study would likely miss the effect.

Result = n ≈ 197 for 80% power

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Example 3 - Psychology Experiment (Current Power Check)

A psychology study has n = 25 participants and expects d = 0.6. Test is two-tailed at α = 0.05. What is the power?

δ = d√n = 0.6 × √25 = 3.0; Power ≈ Φ(3.0 − 1.96) + Φ(−3.0 − 1.96) = Φ(1.04) + Φ(−4.96)

Power ≈ 0.8508 + 0.0000 ≈ 85%. The study has adequate power.

Result = Power ≈ 85%

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Example 4 - Underpowered Study Warning

A pilot study is planned with n = 15 and d = 0.4 at α = 0.05, two-tailed. Is it adequately powered?

δ = 0.4 × √15 = 1.549; Power ≈ Φ(1.549 − 1.96) = Φ(−0.411) ≈ 34%

Power is only 34% - far below 80%. The study needs n ≈ 50 for 80% power with d = 0.4. Running with n = 15 risks a Type II error probability of 66%.

Result = Power ≈ 34% (underpowered)

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❓ Frequently Asked Questions

What is statistical power?+

Statistical power (1 − β) is the probability of correctly rejecting the null hypothesis when it is false - i.e., detecting a true effect. A study with 80% power has an 80% chance of finding a significant result if the hypothesised effect size is real. Conversely, β = 20% is the probability of a Type II error (missing a real effect). Power depends on three things: effect size, sample size, and significance level α.

What is Cohen's d?+

Cohen's d is a standardised effect size for mean-based tests: d = (μ₁ − μ₀) / σ. It expresses how many standard deviations the population mean is from the null value. Jacob Cohen (1988) proposed conventional benchmarks: d = 0.2 (small), d = 0.5 (medium), d = 0.8 (large). For power analysis, you estimate d from pilot data, from the literature, or by specifying the minimum effect you care about detecting.

What sample size do I need for 80% power?+

For a two-tailed Z-test at α = 0.05 with 80% power, the required n depends on effect size: d = 0.2 (small): n ≈ 197; d = 0.5 (medium): n ≈ 32; d = 0.8 (large): n ≈ 13. These are approximate for one-sample tests; two-sample tests need roughly twice as many per group. This calculator finds the exact n for any d and α combination.

What is the relationship between α, β, n, and effect size?+

These four quantities are mathematically related - fixing any three determines the fourth. (1) Decreasing α (more strict) increases β (less power) for fixed n and d. (2) Increasing n increases power. (3) Larger d (bigger effect) means more power. (4) For fixed power, a smaller d requires a larger n. Power analysis exploits this relationship to answer: 'How large does n need to be to detect an effect of size d with power 1−β?'

What is a priori vs post-hoc power analysis?+

A priori power analysis is done before data collection to determine the required sample size. This is the correct use of power analysis. Post-hoc power analysis computes power after seeing the data, using the observed effect size - this is widely criticised because the observed effect size is noisy, and 'observed power' is directly determined by the p-value and carries no additional information. Always plan sample size before the study.

Why is 80% the conventional power threshold?+

Jacob Cohen (1988) suggested 80% power as a reasonable standard for social science research, based on a trade-off between the cost of larger samples and the acceptable Type II error rate. He argued that a 4:1 ratio of β to α (80% power with α = 0.05, giving β = 0.20) was reasonable. Medical and clinical research often requires 90% power (β = 0.10) due to higher stakes. These are conventions, not mathematical necessities.

How does the Z-test power formula work?+

For a two-tailed one-sample Z-test with effect size d and sample size n: Power = Φ(d√n − z_α/2) + Φ(−d√n − z_α/2), where Φ is the standard normal CDF and z_α/2 is the upper α/2 critical value (e.g., 1.96 for α = 0.05). The term d√n is called the non-centrality parameter - it represents how many standard errors the true mean is from H₀. Larger n or larger d shifts the distribution further from H₀, increasing power.

What is the difference between power for Z-tests and t-tests?+

For Z-tests (σ known), power is computed exactly using the normal distribution. For t-tests (σ estimated), power depends on the non-central t-distribution. This calculator uses the normal approximation to non-central t, which is accurate for practical purposes. Exact t-test power requires the non-central t-CDF, which converges to the Z-based formula for large n. The approximation gives conservative (slightly lower) power estimates for small n.

Power Analysis Calculator

📖 What is Statistical Power?

📐 Formulas

📖 How to Use This Calculator

📝 Example Calculations

❓ Frequently Asked Questions

🔗 Related Calculators

📌 Quick Tips