Sampling Distribution of p

Prerequisites
Introduction to Sampling Distributions, Binomial Distribution, Normal Approximation to the Binomial

Learning Objectives

1. Compute the mean and standard deviation of the sampling distribution of p
2. State the relationship between the sampling distribution of p and the normal distribution

The distribution of p is closely related to the binomial distribution. The binomial distribution is the distribution of the total number of successes (favoring Candidate A, for example) whereas the distribution of p is the distribution of the mean number of successes. The mean, of course, is the total divided by the sample size, N.

The binomial distribution has a mean of

μ = Nπ

Dividing by N to adjust for the fact that we are now dealing with means instead of totals, we find the mean of the sampling distribution of p is

μp = π

The standard deviation of the binomial distribution is:



Dividing by N to get the standard error of p, we find that:



The sampling distribution of p is a discrete rather than a continuous distribution.

The sampling distribution of p is approximately normally distributed if N is fairly large and π is not close to 0 or 1. A rule of thumb is that the approximation is good if both N π and N(1 - π) are both greater than 10.