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Two-Proportion Z-Test Calculator

Q: Why does the test use a pooled proportion but the interval does not?

Under H0 both groups share one proportion, so the test pools them for the best estimate of the null standard error; the interval describes the actual difference and therefore uses each group's own variance.

Q: Is this the same as an A/B test significance calculator?

Yes for a two-sided test of conversion rates; z² equals the chi-square statistic of the corresponding 2×2 table without continuity correction.

Tests whether two independent proportions differ using the pooled two-proportion z-test and reports the z statistic, two- and one-sided p-values and a Wald confidence interval for the difference.

When to use

You have successes and trials for two independent groups (conversion rates, response rates, defect rates) and want to know whether the rates differ significantly.

Do not use when: Counts are very small (n·p or n·(1 − p) below 5 in a group: use Fisher's exact test), the groups are paired (McNemar's test), or there are more than two groups or categories (use chi-square-test).

Formula

p_a = x_a/n_a, p_b = x_b/n_b, p̂ = (x_a + x_b)/(n_a + n_b); z = (p_a − p_b) / √(p̂(1 − p̂)(1/n_a + 1/n_b)); p = 2(1 − Φ(|z|)); CI = (p_a − p_b) ± z(1 − alpha/2) × √(p_a(1 − p_a)/n_a + p_b(1 − p_b)/n_b)

Normal approximation to the binomial without continuity correction (equivalent to the 2×2 chi-square test without Yates' correction: χ² = z²). The confidence interval uses the unpooled standard error, as is standard.

Inputs

Parameter	Type	Required	Description
`successes_a`	integer	yes	Number of successes (events) in group A. Range: ≥ 0
`trials_a`	integer	yes	Number of observations in group A. Range: ≥ 1
`successes_b`	integer	yes	Number of successes in group B. Range: ≥ 0
`trials_b`	integer	yes	Number of observations in group B. Range: ≥ 1
`alpha`	number	default 0.05	Significance level as a probability (0.05 = 5 %), not a percentage. Confidence intervals and critical values use level 1 − alpha. Range: > 0

Outputs

Output	Type	Description
`proportion_a`	number	successes_a / trials_a.
`proportion_b`	number	successes_b / trials_b.
`difference`	number	proportion_a − proportion_b.
`pooled_proportion`	number	(successes_a + successes_b) / (trials_a + trials_b), used for the test's standard error.
`standard_error`	number	√(p̂ (1 − p̂) (1/n_a + 1/n_b)).
`z_statistic`	number	difference / standard_error.
`p_value_two_sided`	number	2 × (1 − Φ(\|z\|)).
`p_value_one_sided`	number	1 − Φ(\|z\|): for the alternative in the observed direction.
`ci_lower`	number	Lower limit of the 100·(1 − alpha) % Wald interval for the difference (unpooled standard error).
`ci_upper`	number	Upper limit of the same interval.
`significant`	boolean	true when p_value_two_sided < alpha.
`decision`	string	Plain-language conclusion of the two-sided test.

Example

60/200 (30 %) vs 40/200 (20 %): {"successes_a":60,"trials_a":200,"successes_b":40,"trials_b":200} → {"proportion_a":0.3,"proportion_b":0.2,"difference":0.1,"pooled_proportion":0.25,"standard_error":0.0433,"z_statistic":2.3094,"p_value_two_sided":0.020921,"p_value_one_sided":0.010461,"ci_lower":0.0157,"ci_upper":0.1843,"significant":true}

20/100 vs 15/100: {"successes_a":20,"trials_a":100,"successes_b":15,"trials_b":100} → {"difference":0.05,"pooled_proportion":0.175,"z_statistic":0.9305,"p_value_two_sided":0.35212,"ci_lower":-0.0551,"ci_upper":0.1551,"significant":false}

GET https://tttkmbb.com/api/v1/calculate/proportion-z-test?successes_a=60&trials_a=200&successes_b=40&trials_b=200

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/proportion-z-test (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/proportion-z-test · Markdown: https://tttkmbb.com/statistics/proportion-z-test.md · JSON definition: https://tttkmbb.com/statistics/proportion-z-test.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="proportion-z-test"
OpenAPI operationId: calculate_two_proportion_z_test
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Z-test (reference)
NIST/SEMATECH e-Handbook of Statistical Methods, 7.2.4.1 Confidence intervals for a proportion (government)

FAQ

Why does the test use a pooled proportion but the interval does not?

Under H0 both groups share one proportion, so the test pools them for the best estimate of the null standard error; the interval describes the actual difference and therefore uses each group's own variance.

Is this the same as an A/B test significance calculator?

Yes for a two-sided test of conversion rates; z² equals the chi-square statistic of the corresponding 2×2 table without continuity correction.

Related calculators

Chi-Square Test Calculator — Same comparison as a 2×2 table, or more than two groups.
Margin of Error Calculator — Confidence interval for a single proportion.
Sample Size Calculator — Sample size needed for a target margin of error.