The Sampling Distribution of p-hat

Flip a coin 100 times and you might get 47 heads, or 53, almost never exactly 50. That's because p-hat, the sample proportion, is a random variable. It changes every time you draw a new sample, even when nothing about the population changes. If you took every possible sample of size n from the population and calculated p-hat for each one, then plotted all those values, you'd get the sampling distribution of p-hat. It's not one number, it's a whole distribution built from imagining repeated sampling.

Two facts describe this distribution completely. Its mean equals p, the true population proportion. That makes sense: p-hat has no systematic reason to run high or low, so its average value over many samples lands right on the truth. Its standard deviation is √(p(1-p)/n). This is smaller than the spread of individual outcomes in the population, because p-hat is an average of many observations (0s and 1s), and averages vary less than individual values. As n grows, that denominator grows too, so the standard deviation shrinks: larger samples give more consistent, less variable p-hat values.

This sets up everything that follows in inference for proportions. A confidence interval is built by taking one observed p-hat and adding and subtracting a margin tied to this standard deviation. A hypothesis test asks how far a sample's p-hat would have to be from the claimed p to be surprising, using this same distribution as the yardstick. When n is large enough (both np and n(1-p) at least 10), the sampling distribution is also approximately normal, which is what justifies using z-scores in those later tools.