sample size calculation for two proportions - Aaron Graves, PhDude Replica

Two-Proportion Sample Size Calculator

Use this tool to estimate the required sample size for comparing two independent proportions (e.g., conversion rate, response rate, event rate).

Group 1 proportion (p₁)

Example: 0.10 = 10%

Group 2 proportion (p₂)

Expected proportion in treatment or comparison group

Significance level (α)

Common choice: 0.05

Power (1 - β)

Common choices: 0.80 or 0.90

Allocation ratio (n₂ / n₁)

1 = equal group sizes

Test type

Two-sided is usually preferred

Expected dropout / non-response (%)

Optional inflation for attrition

Enter your assumptions and click Calculate Sample Size.

Why sample size matters for two proportions

When your outcome is binary (yes/no, success/failure, converted/not converted), comparing two proportions is one of the most common study designs in clinical trials, public health, social science, and A/B testing. A sample size that is too small can miss a true difference. A sample size that is too large can waste time, money, and participant effort.

The goal is to choose a sample size that gives a high probability (power) of detecting a meaningful difference at a chosen false-positive rate (alpha).

Inputs used by this calculator

1) Proportion in each group (p₁ and p₂)

These are your expected event rates. For example, if you expect 10% conversion in control and 14% in treatment, then: p1 = 0.10 and p2 = 0.14.

2) Significance level (α)

Alpha is the probability of a Type I error (finding a difference when none exists). A common setting is 0.05.

3) Power (1 - β)

Power is the probability of detecting the specified difference if it truly exists. Common values are 0.80 or 0.90.

4) Allocation ratio (n₂/n₁)

Equal allocation (ratio = 1) is often most efficient. Unequal allocation is used when one group is easier to recruit, less expensive, or ethically favored.

5) One-sided vs two-sided test

A two-sided test checks for any difference. A one-sided test checks only one direction and usually yields a smaller required sample size, but it must be justified before data collection.

6) Dropout inflation

If you expect attrition, inflate your sample size. For example, with 10% dropout, divide by 0.90 (or multiply by 1.111...).

Formula overview

This page uses the standard normal approximation for two independent proportions with optional unequal allocation ratio. The tool reports required participants in each group and total sample size after rounding up.

Effect size: |p1 - p2|
Critical values: z(1-α/2) for two-sided or z(1-α) for one-sided, and z(power)
Outputs: required n for group 1, group 2, and total

Practical tips before finalizing your design

Use realistic baseline rates: pull from pilot data, historical studies, or registry data.
Define a meaningful difference: statistical significance is not the same as practical significance.
Stress test assumptions: run best-case and worst-case scenarios for p₁, p₂, and dropout.
Document choices: include your assumptions in a protocol or analysis plan.
Consult a statistician for confirmatory work: especially for cluster designs, repeated measures, non-inferiority, or interim analyses.

Common mistakes

Using an optimistic effect size that is unlikely in practice.
Ignoring dropout/non-response in recruitment targets.
Switching from two-sided to one-sided after seeing data.
Confusing percentage points with relative changes (e.g., 10% to 12% is +2 points, not +2%).

Bottom line

A good sample size calculation is where scientific intent meets operational reality. Use this calculator to get a solid starting estimate for two-proportion comparisons, then refine with domain knowledge and study-specific constraints.