Sample Size Power Calculator

Sample Size & Power

Statistical Test Family

Hypothesis Direction

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

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

Study Design & Sample Planning

What Is Sample Size & Power Analysis?

Power analysis mathematically links four core study parameters: Sample Size ($N$), Effect Size ($\delta$), Significance Level ($\alpha$), and Statistical Power ($1 - \beta$).

Specifying any three parameters allows you to solve directly for the fourth. Calculating the required sample size a priori ensures your study is adequately powered ($\ge 80\%$) to detect meaningful experimental effects without wasting resources on an excessively large cohort.

When to Use It

Dissertation & Grant Planning: Determining the minimum required sample size to justify research methodology to committees.
Clinical Trials & A/B Testing: Establishing cohort sizes needed to detect clinically or commercially meaningful differences.
Post-Hoc Sensitivity Testing: Evaluating the statistical power achieved by an already-completed empirical experiment.
Mitigating Type II Errors ($\beta$): Preventing underpowered studies from failing to reject false null hypotheses.

The Mathematical Formulas

$$\text{Two-Sample t: } n = 2 \left( \frac{z_{1-\alpha/2} + z_{1-\beta}}{d} \right)^2, \quad N = 2n$$ $$\text{Proportions: } n = \frac{\left( z_{\alpha/2}\sqrt{2\bar{p}\bar{q}} + z_\beta\sqrt{p_1 q_1 + p_2 q_2} \right)^2}{(p_1 - p_2)^2}$$ $$\text{Correlation: } N = \left( \frac{z_{1-\alpha/2} + z_{1-\beta}}{\frac{1}{2}\ln\left(\frac{1+r}{1-r}\right)} \right)^2 + 3$$

Where $z_{1-\alpha/2}$ is the standard normal critical value for Type I error, $z_{1-\beta}$ is the quantile for statistical power, $d$ is Cohen's $d$, and $r$ is the Pearson correlation coefficient.

Worked Example

Scenario: You plan an independent two-sample t-test with a medium effect size ($d = 0.50$), targeting $80\%$ power at $\alpha = 0.05$ (two-tailed).

Step 1: Obtain Standard Normal Quantiles:
$z_{1 - \alpha/2} = z_{0.975} = 1.9600, \quad z_{1 - \beta} = z_{0.80} = 0.8416$.

Step 2: Substitute into Sample Size Formula:
$n = 2 \left( \frac{1.9600 + 0.8416}{0.50} \right)^2 = 2 \left( \frac{2.8016}{0.50} \right)^2 = 2(5.6032)^2 = 62.79$.

Step 3: Degrees-of-Freedom $t$-Distribution Adjustment:
Iterating using exact Student's $t$ ($df = 126$) yields $n = 64$ per group ($N = 128$ total).

Conclusion: You must recruit **64 participants per group** (**128 total**) to achieve **80.2% statistical power**.

Analysis Complete

Power & Sample Size Results

Required Sample Size

N = 128 Total (64 per group)

Achieved Power: 80.15% (Meets ≥ 80% Scientific Standard)

Total Sample (N) 128

Group Size (n) 64

Power (1 - β) 80.15%

Type II Error (β) 19.85%

Non-Centrality (δ) 2.8284

Critical Value 1.9790

Sample Size Power Calculator

Sample Size & Power

Power & Sample Size Results

Step-by-Step Calculation

Interpretation

Sample Size vs. Statistical Power Curve

Perfect for students, researchers, and professionals looking to build real statistical skills.