A two-proportion confidence interval estimates the difference between two population proportions, p1 − p2, from two independent samples. Enter both samples, choose a confidence level, and the calculator gives the difference, the margin of error and the interval — with the unpooled standard error and every step shown.
Two-Proportion Confidence Interval Calculator
(p̂1 − p̂2) ± z* × SE — estimate the difference between two proportions.
Conditions: independent random samples, and at least 10 successes and 10 failures in each group.
What Is a Two-Proportion Confidence Interval?
A two-proportion confidence interval estimates p1 − p2, the difference between two population proportions. It is the right tool for comparing two groups: for example, how much higher is the success rate of a new treatment than the old one, or how much more do men support a policy than women? The interval tells you both the size of the difference and how precisely you know it.
The Two-Proportion Confidence Interval Formula
The interval is (p̂1 − p̂2) ± z* × √[p̂1(1 − p̂1)/n1 + p̂2(1 − p̂2)/n2]. This standard error is unpooled — it uses each sample's own proportion — because a confidence interval does not assume the two proportions are equal. That is the key difference from the two-proportion z-test, which pools because its null hypothesis says p1 = p2.
How to Interpret the Interval for p1 − p2
Read the interval for p1 − p2 in context. If the whole interval is above zero, p1 is plausibly larger than p2. If it is entirely below zero, p1 is plausibly smaller. If it includes zero, the data are consistent with no difference at all. Always give the interval with its confidence level: “We are 95% confident that the true difference in cure rates is between 0.012 and 0.288.”
Worked Example
A new drug cured 60 of 100 patients (p̂1 = 0.60) and the old drug cured 45 of 100 (p̂2 = 0.45). The difference is 0.15. The standard error is √(0.60 × 0.40/100 + 0.45 × 0.55/100) ≈ 0.0698, and the margin of error is 1.96 × 0.0698 ≈ 0.137. The 95% confidence interval for p1 − p2 is (0.013, 0.287): we are 95% confident the new drug's cure rate is between 1.3 and 28.7 percentage points higher.
Two-Proportion Confidence Interval FAQs
Why is the confidence interval unpooled but the z-test pooled?
The z-test assumes H0 (p1 = p2) is true, so it pools both samples into one proportion. A confidence interval makes no such assumption, so it uses each sample's own proportion in the standard error.
What if the interval includes zero?
Then zero is a plausible value for the true difference — the data do not rule out the possibility that the two population proportions are equal.
How do I report a two-proportion confidence interval?
Give the confidence level, the interval and the context: “We are 95% confident that p1 − p2 is between 0.013 and 0.287 — the new drug cures between 1.3 and 28.7 percentage points more patients.”
Related tools: Two-Proportion Z-Test · Proportion Confidence Interval · Sample Proportion · Statistics