Sampling Distributions

Let $X_1, X_2, \ldots, X_n$ be independent observations drawn from a population with mean $\mu$ and standard deviation $\sigma$. The sample mean is

$$\bar X = \frac{1}{n}\sum_{i=1}^n X_i.$$

Because the values $X_i$ are random, $\bar X$ is itself a random variable. Its probability distribution -- the distribution of values $\bar X$ takes across all possible samples of size $n$ -- is called the sampling distribution of $\bar X$.

The mean and standard deviation of this sampling distribution are

$$E[\bar X] = \mu, \qquad \mathrm{SD}(\bar X) = \frac{\sigma}{\sqrt{n}}.$$

The quantity $\sigma/\sqrt{n}$ is called the standard error of $\bar X$, denoted $\mathrm{SE}(\bar X)$.

For instance, if a population has $\mu = 50$ and $\sigma = 8$, then a sample mean based on $n = 16$ observations has

$$E[\bar X] = 50, \qquad \mathrm{SE}(\bar X) = \frac{8}{\sqrt{16}} = 2.$$

Notice that the standard error shrinks as $n$ grows: larger samples produce sample means that cluster more tightly around $\mu$.