The standard deviation of a sample proportion

The sample proportion is a random variable. This equation relates the size of its variability to the underlying population proportion and the size of the sample. The assumption is that the proportion arises from a Binomial setting.

Suppose we take a sample of 100 students and record the proportion who have tried marijuana. Assuming the proportion of all students is 38.2%, the standard deviation of the sample proportion is 4.9%. Since the Central Limit Theorem tells us this statistic is roughly Normal, we can be 95% confident that the sample proportion will be within about 9.8% of the true proportion.

It is in this way that the standard deviation of the sample proportion is a measure of its precision as an estimator of the population proportion. In particular, if we want precision better than 9.8% then we should increase our sample size, and this formula can also be used to tell us by how much it should be increased. For example, if we want to be 95% confident that we are within 1% of the true proportion we would need a sample of around 9000 students.

Interestingly, to get 1% precision we would need to sample 9000 people whether we were just looking at UQ students or whether we were looking at the whole of Australia. The population size does not effect precision.

Note that in practice we do not know the population proportion and so need to use the standard error to measure precision instead.