Proportions: One and Two Groups
Constructs a two-sided confidence interval for the difference between two independent population proportions, using the Pearson chi-square simple asymptotic method. Enter the observed successes and sample sizes for each group. Runs entirely in your browser.
This calculator estimates the difference between two independent population proportions, defined as p1 − p2. The point estimate is the difference between the two observed sample proportions.
For the Pearson chi-square simple asymptotic method, the two-sided confidence interval is calculated as:
Here, p̂1 = x1/n1 and p̂2 = x2/n2. The standard error uses the two sample-specific binomial variances rather than a pooled proportion. The critical value z1−α/2 is determined by the selected confidence level.
The resulting interval gives a range of values for the population difference p1 − p2 consistent with the specified confidence level under this large-sample approximation. A positive interval estimate means the observed Group 1 proportion exceeds the Group 2 proportion; a negative estimate means it is lower.
The groups should be independent, and the Pearson simple asymptotic interval is a large-sample method. For small samples or proportions close to 0 or 1, alternative score-based or other interval methods can have better coverage properties. PASS documents several alternative methods, including Yates continuity-corrected, Miettinen–Nurminen score, and Wilson score approaches.
PASS's documentation gives a validation example based on Newcombe (1998): confidence level 95%, p1 = 0.90, p2 = 0.30, and n1 = n2 = 10. For the Pearson method, the reported confidence interval is 0.2605 to 0.9395, with a width of 0.6790. The calculator reproduces these limits to four decimal places.
Newcombe, R. G. (1998). Interval estimation for the difference between independent proportions: comparison of eleven methods. Statistics in Medicine, 17(8), 873–890.
NCSS, LLC. PASS Sample Size Software: Confidence Intervals for the Difference Between Two Proportions, Chapter 216. PASS documentation describes the Pearson method as the commonly used simple asymptotic method without continuity correction and provides the validation example used above.
See the analysis this calculator plans for applied to published trial results, with the estimates, confidence intervals and interpretation explained.