Sampling Distribution of a Proportion

Prerequisite
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

Assume that in an election race between Candidate A and Candidate B, 0.60 of the voters prefer Candidate A. If random sample of 10 voters were polled, it is unlikely that exactly 60% of them (6) would prefer Candidate A. By chance the proportion in the sample preferring Candidate A could easily be a little lower than 0.60 or a little higher than 0.60. The sampling distribution of p is the distribution that would result if you repeatedly sampled 10 observations and determined the proportion (p) that favored Candidate A.

The sampling distribution of p is a special case of the sampling distribution of the mean. Table 1 shows a hypothetical random sample of 10 voters. Those who prefer Candidate A are given scores of 1 and who prefer Candidate B are given scores of 0. Note that seven of the voters prefer candidate A so the sample proportion (p) is

p = 7/10 = 0.70

As you can see, p is the mean of the 10 preference scores.

Table 1. Sample of voters.

Voter	Preference
1	1
2	0
3	1
4	1
5	1
6	0
7	1
8	0
9	1
10	1

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 M is the distribution of the mean number of successes. The mean, of course, is the total divided by the sample size, N. Therefore, the sampling distribution of p and the binomial distribution differ in that p is the mean of the scores (0.70) and the binomial distribution is dealing with the total number of successes (7).

The binomial distribution has a mean of

μ = Nπ

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

μp = π

The standard deviation of the binomial distribution is:

Dividing by N because p is a mean not a total, we find the standard error of p:

Returning to the voter example, π = 0.60 (Don’t confuse π = 0.60, the population proportion and p = 0.70, the sample proportion) and N = 10. Therefore, the mean of the sampling distribution of p is 0.60. The standard deviation is

The sampling distribution of p is a discrete rather than a continuous distribution. For example, with an N of 10, it is possible to have a p of 0.50 or a p of 0.60 but not a p of 0.55.

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. The sampling distribution for the voter example is shown in Figure 1. Note that even though N(1 – π) is only 4, the approximation is quite good.

Figure 1. The sampling distribution of p. Vertical bars are the probabilities; the smooth curve is the normal approximation.