Expected value of a sample proportion

The sample proportion is a random variable. This equation says that the "mean of the sample proportion is the population proportion". If the sampling process is repeated many times, the average of all the sample proportions will get closer and closer to the proportion in the population from which the sample is drawn.

Suppose that 38.2% of all UQ students have tried marijuana. This equation says that if we took a sample of any number of UQ students then we would expect the proportion in the sample who had tried marijuana to be 0.382.

Because of this, we say that the sample proportion is an unbiased estimate of the population proportion.

While the expected value of the sample proportion is the population proportion, how close a sample will be to the population proportion depends on the variability of the sample proportion. This is quantified using its standard deviation.