# $z$ test for the difference between two proportions: sampling distribution of the $z$ statistic

Definition of the sampling distribution of the $z$ statistic

As you may know, when we perform a $ z$ test for the difference between two proportions, we compute the $ z$ statistic
$$
z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}
$$
based on our group 1 and group 2 samples. Now suppose that we would draw many more samples. Specifically, suppose that we would draw an infinite number of group 1 and group 2 samples, each time of size $ n_1$ and $ n_2$. Each time we have a group 1 and group 2 sample, we could compute the $ z$ statistic $ z = \frac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\frac{1}{n_1} + \frac{1}{n_2}\Bigg)}}$. Different samples will give different $ z$ values. The distribution of all these $ z$ values is the |