Sampling and Standard Error

You poll 1000 voters and 52% say they back candidate A. Ask a different 1000 people and you might get 50.7% or 53.4%. The 52% you report is one draw from a distribution, and that distribution is called the sampling distribution of the estimate.

The setup: a population has mean $\mu$ and variance $\sigma^2$. You draw $X_1, \dots, X_n$ independently from it. The sample mean is $\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$. Since $\bar{X}$ is a function of random variables, it is a random variable too, with its own mean and variance:

The first fact says $\bar{X}$ is unbiased: averaged over many repeated samples, it lands on $\mu$. The second says how tightly it clusters. The variance step uses independence to drop every covariance term. Remember that, because independence is the assumption that most often fails in real data. The LLN & Central Limit Theorem lesson adds the shape: for large $n$, $\bar{X}$ is close to normal.

For a poll, each $X_i$ is 1 if the voter says yes and 0 otherwise, a Bernoulli($p$) variable with $\sigma^2 = p(1-p)$. With $p = 0.52$ that is $0.52 \times 0.48 = 0.2496$. The sample proportion $\hat{p}$ is just $\bar{X}$ for these 0/1 variables, so everything above applies to it directly.