Distribution of the Sample Mean (PS2 §6.4)

Core Concept

The Sample Mean as a Random Variable

Suppose $X_1, X_2, \ldots, X_n$ is a random sample of size $n$ drawn from a population with mean $\mu$ and variance $\sigma^2$. Each $X_i$ is an independent observation from the same distribution. The sample mean is defined as:

Because each $X_i$ is random, $\bar{X}$ is itself a random variable. Its distribution — how it behaves across all possible samples of size $n$ — is called the sampling distribution of the mean.

Result 1: The Mean of $\bar{X}$

Since each $X_i$ has mean $\mu$:

The sample mean is an unbiased estimator of the population mean — on average, it hits the right target.

Result 2: The Variance of $\bar{X}$

Since the $X_i$ are independent and each has variance $\sigma^2$:

As $n$ increases, the variance of $\bar{X}$ decreases: larger samples produce sample means that cluster more tightly around $\mu$.

Result 3: Exact Normality (when $X$ is Normal)

If the population itself follows a normal distribution, i.e.\ $X \sim \mathrm{N}(\mu, \sigma^2)$, then for any sample size $n$:

This is an exact result, holding for all $n \geq 1$.

Result 4: The Central Limit Theorem (CLT)

If the population distribution is not normal (or is unknown), the CLT tells us that for large $n$:

The approximation improves as $n$ grows. For 9709, you need only an informal understanding: a large sample size makes the distribution of $\bar{X}$ approximately normal, regardless of the shape of the underlying population. As a working rule, $n \geq 30$ is generally considered sufficient.