## Sampling Distribution of p

### Sampling Distribution of p

### Learning Objectives

- Compute the mean and standard deviation of the sampling distribution of
- State the relationship between the sampling distribution of and the normal distribution

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

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

As you can see, is the mean of the 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 is closely related to the binomial distribution. The binomial distribution is the distribution of the total number of successes (favoring) Candidate , for example) whereas the distribution of is the distribution of the mean number of successes. The mean, of course, is the total divided by the sample size, . Therefore, the sampling distribution of and the binomial distribution differ in that is the mean of the scores and the binomial distribution is dealing with the total number of successes (7).

The binomial distribution has a mean of:

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

The standard deviation of the binomial distribution
is:

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

Returning to the voter example, and . (Don't confuse , the population proportion and , the sample proportion.) Therefore, the mean of the sampling distribution of is . The standard error is

The sampling distribution of is a discrete rather than a continuous distribution. For example, with an of , it is possible to have a of or a of but not a of .

The sampling distribution of is approximately normally distributed if is fairly large and is not close to or . A rule of thumb is that the approximation is good if both and are greater than . The sampling distribution for the voter example is shown in Figure 1. Note that even though is only , the approximation is quite good.

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

Source: David M. Lane, https://onlinestatbook.com/2/sampling_distributions/samp_dist_p.html

This work is in the Public Domain.