Poll 1,000 voters and 52% support a measure; poll another 1,000 and you might get 49%. The sample proportion $\hat{p}$ is itself a random variable. Its distribution over all possible samples, the sampling distribution, follows from the binomial distribution and is close to a normal curve. You saw this kind of variation in simulations.
Key idea
For random samples of size $n$ from a population with proportion $p$, the sample proportion $\hat{p}$ has
$$\mu_{\hat{p}} = p, \qquad \sigma_{\hat{p}} = \sqrt{\dfrac{p(1 - p)}{n}}.$$
When $np \ge 10$ and $n(1 - p) \ge 10$, the sampling distribution of $\hat{p}$ is approximately normal.
Why it works
The number of successes $X$ in the sample is binomial (when the population is large, or at least 10 times the sample), with mean $np$ and SD $\sqrt{np(1 - p)}$. Since $\hat{p} = X/n$, divide both by $n$: the mean becomes $p$ and the SD becomes $\sqrt{np(1 - p)}/n = \sqrt{p(1 - p)/n}$. With enough expected successes and failures, the binomial bars fill in a bell shape, so a normal curve fits.
Worked example: Center and spread
40% of a city's adults have a library card. For random samples of 150 adults, describe the sampling distribution of $\hat{p}$.
- Center: $\mu_{\hat{p}} = 0.4$.
- Spread: $$\sigma_{\hat{p}} = \sqrt{\dfrac{0.4 \cdot 0.6}{150}} = \sqrt{0.0016} = 0.04$$
- Shape: $np = 60$ and $n(1 - p) = 90$ are both at least 10, so approximately normal.
Answer: About normal, mean 0.4, SD 0.04.
Worked example: A probability
In that setting, how likely is a sample with $\hat{p} \ge 0.48$? ($P(Z < 2.00) = 0.9772$.)
- $z = \dfrac{0.48 - 0.4}{0.04} = 2$.
- $$P(\hat{p} \ge 0.48) = P(Z \ge 2) = 1 - 0.9772 = 0.0228$$
Answer: About $0.023$: a sample this high would be unusual.
Worked example: A bigger sample
What happens to $\sigma_{\hat{p}}$ with samples of 600 instead of 150?
- n is 4 times as large, so $\sigma_{\hat{p}}$ is divided by $\sqrt{4} = 2$.
- $\sqrt{0.24/600} = \sqrt{0.0004} = 0.02$.
Answer: It halves, to 0.02.
Common mistake
Confusing the distribution of one sample's data with the distribution of the statistic. One sample's data are just yeses and nos (0s and 1s); they are not bell-shaped at all. The bell shape describes $\hat{p}$ across many different samples. Check: is the question about individuals, or about a statistic computed from a whole sample?