Statistical Power Calculator

Statistical Test Family

Hypothesis Direction

Significance Level (α) Default threshold is 0.05 (5% Type I error risk)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Study Design & Sample Planning

What Is Statistical Power?

Statistical Power ($1 - \beta$) is the probability that a statistical test will correctly reject a false null hypothesis—that is, the probability of detecting a genuine true effect when one actually exists.

A study with low statistical power risks committing a Type II error ($\beta$) (a "false negative"), potentially discarding a truly effective clinical treatment or meaningful empirical finding. In scientific research, power of **80% ($\ge 0.80$)** or **90%** is the universally recognized benchmark.

When to Use It

Prospective Study Planning: Computing the power of a planned sample size before collecting dissertation or trial data.
Post-Hoc / Sensitivity Evaluation: Determining the minimum detectable effect size for an already-completed study.
IRB & Grant Proposals: Demonstrating to ethics boards and funding agencies that your study is adequately powered.
Controlling False Negative Risk: Balancing Type I error ($\alpha$) and Type II error ($\beta$) tolerances.

The Mathematical Formulas

$$\text{Two-Sample t-Test: } \delta = d \sqrt{\frac{n_1 n_2}{n_1 + n_2}}, \quad \text{Power} = 1 - \Phi(z_{\text{crit}} - \delta)$$ $$\text{ANOVA: } \lambda = N f^2, \quad \text{Power} = 1 - F_{\text{central}}\left(\frac{F_{\text{crit}}}{1 + \lambda/df_1}; df_1, df_2\right)$$ $$\text{Correlation: } z_\beta = \text{arctanh}(|r|)\sqrt{n - 3} - z_{\text{crit}}, \quad \text{Power} = \Phi(z_\beta)$$

Where $\delta$ and $\lambda$ represent non-centrality parameters, $d$ is Cohen's $d$, $f$ is Cohen's $f$, and $z_{\text{crit}}$ is the critical standard normal quantile corresponding to chosen significance level $\alpha$.

Worked Example

Scenario: Independent two-sample t-test with medium effect size $d = 0.50$, $n_1 = 64, n_2 = 64$ ($N = 128$), and $\alpha = 0.05$ (two-tailed).

Step 1: Compute Non-Centrality Parameter ($\delta$):
$\delta = 0.50 \times \sqrt{\frac{64 \times 64}{64 + 64}} = 0.50 \times \sqrt{32} = 0.50 \times 5.6569 = 2.8284$.

Step 2: Determine Critical Value ($df = 126$):
$t_{\text{crit}} = t_{0.975, 126} \approx 1.9790 \quad (z_{\text{crit}} \approx 1.9600)$.

Step 3: Calculate Power ($1 - \beta$):
$\text{Power} \approx 1 - \Phi(1.9790 - 2.8284) = 1 - \Phi(-0.8494) = \Phi(0.8494) \approx 0.8015 \text{ (80.2\%)}$.

Conclusion: The design achieves **80.2% statistical power**, successfully meeting the standard $\ge 80\%$ benchmark with a 19.8% Type II error risk.

Computation Complete

Statistical Power Results

Computed Statistical Power (1 - β)

80.15%

Adequately Powered Study Design (Power ≥ 80%)

Power (1 - β) 0.8015

Type II Error (β) 0.1985

Non-Centrality (δ / λ) 2.8284

Critical Threshold 1.9790

Total Sample Size (N) 128

Significance (α) 0.05

Step-by-Step Calculation

Interpretation

Null (H₀) vs Alternative (H₁) Hypothesis Power Curve

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Statistical Power Calculator