Compute the exact per-group (`n_1`, `n_2`) and total (`N`) sample size required to detect a difference between two independent proportions (`p_1` vs. `p_2`), including pooled normal approximation, Fleiss continuity correction, and Cohen's `h` effect size.
Enter the inputs, then select Calculate Required Sample Size.
Result will appear here
Comparing two independent binomial proportions requires accounting for both the null pooled variance p_bar * (1 - p_bar) and the alternative unpooled variance p_1(1 - p_1) + p_2(1 - p_2)/k.
p_bar = (p_1 + k * p_2) / (1 + k); n_1 = [z_{alpha} * sqrt((1 + 1/k) * p_bar * (1 - p_bar)) + z_{beta} * sqrt(p_1(1 - p_1) + p_2(1 - p_2)/k)]^2 / (p_2 - p_1)^2.
For continuous outcomes (means and standard deviations), switch to our Sample Size Calculator for Means (/tools/sample-size-calculator-for-means).
Required sample size per group (`n_1 = n_2`) and total `N` at 80% and 90% power across common clinical and A/B testing scenarios.
| Baseline `p_1` vs. Target `p_2` | Absolute / Relative Lift | 80% Power (`n` per group / Total `N`) | 80% Power with Fleiss CC (`n_cc` / `N_cc`) |
|---|---|---|---|
| `0.10` vs. `0.15` (Default) | `+5.0%` abs (`+50%` rel, `h = 0.152`) | `686` per arm (`1,372` total) | `725` per arm (`1,450` total) |
| `0.05` vs. `0.10` (Low-rate conversion) | `+5.0%` abs (`+100%` rel, `h = 0.193`) | `435` per arm (`870` total) | `474` per arm (`948` total) |
| `0.20` vs. `0.30` (10-point lift) | `+10.0%` abs (`+50%` rel, `h = 0.232`) | `294` per arm (`588` total) | `313` per arm (`626` total) |
| `0.50` vs. `0.60` (Max-variance region) | `+10.0%` abs (`+20%` rel, `h = 0.201`) | `388` per arm (`776` total) | `408` per arm (`816` total) |
Size your clinical trial or conversion experiment before collecting data.
Input your control group rate p1 (e.g., 0.10 for 10%) and the smallest target rate p2 worth detecting (e.g., 0.15 for 15%).
Choose alpha = 0.05 (two-sided) and 80% or 90% power, with k = 1 for an equal 50/50 split between control and treatment.
Use the unadjusted N when analyzing with a standard two-proportion Z-test, or use the Fleiss continuity-corrected N_cc when analyzing with Fisher's exact test or Yates's chi-square.
See how baseline rates, minimum detectable effects, and unequal allocation change required N.
Free sample size calculator for comparing two independent proportions (p1 vs. p2). Compute unadjusted and Fleiss continuity-corrected n1, n2, total N, and Cohen's h.
p_bar = (p_1 + k * p_2) / (1 + k); n_1 = [z_{alpha} * sqrt((1 + 1/k) * p_bar * (1 - p_bar)) + z_{beta} * sqrt(p_1(1 - p_1) + p_2(1 - p_2)/k)]^2 / (p_2 - p_1)^2.
For continuous outcomes (means and standard deviations), switch to our Sample Size Calculator for Means (/tools/sample-size-calculator-for-means).
Keep these three statistical rules in mind when budgeting traffic or patient recruitment.
First, sample size scales inversely with the square of the absolute difference (p_2 - p_1)^2. Halving your minimum detectable lift from 5 percentage points to 2.5 percentage points quadruples the required sample size.
Second, always inflate your final recruitment target by 1 / (1 - dropout_rate) if you expect subject attrition or incomplete follow-up.
Verify independence and test alignment before locking your protocol.
1. Independent Observations: Both formulas assume independent Bernoulli trials; cluster-randomized or repeated-measures designs require multiplying N by the Design Effect DEFF = 1 + (m - 1) * ICC.
2. Match Analysis to Sizing: If you size your study using the unadjusted normal formula, analyze it with the unadjusted two-proportion Z-test or Pearson chi-square test.
Calculate sample size for comparing two independent means, testing a single mean, or estimating a mean within a target margin of error. Includes Z-approximation, ceiling rounding, and allocation ratio k = n2/n1.
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Open toolAnswers about using Sample Size Calculator for Two Proportions (A/B Test & Fleiss CC) and understanding its results.
The unadjusted normal formula aligns with the uncorrected Pearson chi-square or two-proportion Z-test. The Fleiss (1981) continuity correction inflates n1 and n2 slightly so your study achieves full nominal power when analyzed with Yates's continuity-corrected chi-square or Fisher's exact test.
The binomial variance p(1 - p) reaches its maximum at p = 0.50 (0.50 * 0.50 = 0.25) and shrinks near 0 or 1 (at p = 0.05, 0.05 * 0.95 = 0.0475). Higher variance around 50% requires a larger denominator to separate signal from noise.
Cohen's h = 2 * arcsin(sqrt(p_2)) - 2 * arcsin(sqrt(p_1)) stabilizes binomial variance across the [0, 1] scale. Values around |h| = 0.20, 0.50, and 0.80 represent small, medium, and large proportion differences.
Use this calculator when your primary outcome is binary (yes/no, converted/not converted, cured/relapsed). When your outcome is a continuous measurement with a mean and standard deviation, use our Sample Size Calculator for Means (/tools/sample-size-calculator-for-means).
Choose another tool for your next calculation, conversion, or text task.
