Distribution of a single proportion
We saw earlier that a single sample proportion p has a distribution with mean and standard deviation
and the distribution is approximately normal in large samples.
This result holds separately for the proportions in each group.
Difference between two proportions
To find the mean and standard deviation of the difference between two proportions, we apply the general results about the difference between any two independent random quantities.
Since the individual proportions are approximately normal (in large samples), their difference is also approximately normal:
Illustration
The diagram below selects samples of size 100 from each of two groups. The probabilities of success in the groups can be adjusted using the sliders at the top.
The normal approximations to the distributions of the individual sample proportions are shown on the right and the normal approximation to the distribution of the difference is shown in pink at the bottom.
Click Accumulate and take several samples. Observe that the sampling variability of the proportions and their difference match the theoretical distributions reasonably.
Note also that the distribution of the difference has greater spread than that of the individual proportions.