Sampling Distribution for a Difference of Proportions

Each sample proportion, p1-hat and p2-hat, bounces around its own population value from sample to sample. When you subtract one random quantity from another, the result is itself random, and it has its own center and spread. Even if the two population proportions are truly identical, each sample proportion lands a little high or low just by chance, so the difference you observe is rarely exactly zero. Two truly equal proportions, each estimated from a sample of 100, can easily differ by 5 or 8 points purely from sampling variability, with no real effect behind it.

Because the two samples are independent, their sampling distributions combine in a specific way. The mean of the difference is just the difference of the means: p1 - p2. The variance works differently than you might expect. For independent random quantities, variances add, even though you are subtracting the proportions themselves. So the variance of p1-hat minus p2-hat equals p1(1-p1)/n1 plus p2(1-p2)/n2, and the standard deviation is the square root of that sum. This matters because it's the engine behind every confidence interval and hypothesis test comparing two proportions: the margin of error and the test statistic both rely on this combined SD. As both sample sizes grow, each individual sampling variance shrinks, so the combined SD shrinks too, and the sampling distribution of the difference becomes tighter and more clearly normal, setting up the conditions needed for two-sample z procedures.