Proportion Z-Test Calculator (One and Two Proportions)

One-proportion and two-proportion z-tests

• One proportion: is the proportion of “successes” in a sample different from a claimed value p₀? For example, do 56 out of 100 people prefer a product, when the claim is 50%?
• Two proportions: are the success rates in two groups different? For example, 45 of 100 in group 1 against 30 of 90 in group 2.

Worked examples

• One proportion: p̂ = 56 ÷ 100 = 0.56, p₀ = 0.5. z = (0.56 − 0.5) ÷ √(0.5 × 0.5 ÷ 100) = 1.20, two-tailed p = 0.230. Not significant at 0.05.
• Two proportions: p̂₁ = 0.50 and p̂₂ = 0.333. The pooled proportion is 0.395, so z = 1.64, two-tailed p = 0.100. Not significant at 0.05, but close.

Formulas

• One proportion: z = (p̂ − p₀) ÷ √(p₀(1 − p₀) ÷ n).
• Two proportions: z = (p̂₁ − p̂₂) ÷ √(p̂(1 − p̂)(1/n₁ + 1/n₂)), where p̂ is the pooled proportion (x₁ + x₂) ÷ (n₁ + n₂).
• The confidence interval for the difference uses the unpooled standard error √(p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂).

When it is valid

The z-test uses a normal approximation. A common rule is that you should expect at least 10 successes and 10 failures in each group (n × p ≥ 10 and n × (1 − p) ≥ 10). For smaller samples, use an exact binomial test or Fisher’s exact test. For a 2 × 2 table of counts, the chi-square test of independence gives the same answer as the two-proportion z-test.

How to cite this page

CalcPulse. (2026). Proportion Z-Test Calculator. https://calcpulse.com/proportion-z-test-calculator/