The Central Limit Theorem

Let $X_1, X_2, \ldots, X_n$ be independent, identically distributed (iid) random variables with mean $\mu$ and finite variance $\sigma^2$. The sample mean is

$$\bar{X}_n = \dfrac{X_1 + X_2 + \cdots + X_n}{n}.$$

The Central Limit Theorem (CLT) states that for large $n$, the sample mean is approximately normally distributed:

$$\bar{X}_n \;\approx\; N\!\left(\mu,\; \dfrac{\sigma^2}{n}\right).$$

That is, $\bar{X}_n$ has approximate mean $\mu$ and approximate variance $\sigma^2/n$. The remarkable feature is that this holds regardless of the distribution of the $X_i$ -- it could be skewed, discrete, bimodal -- as long as $\sigma^2 < \infty$.

A common rule of thumb is that $n \geq 30$ is large enough for the approximation to be useful. (For sample means drawn from a normal population, the result is exact for every $n$.)

For example, suppose the daily number of customers at a coffee shop has mean $\mu = 200$ and variance $\sigma^2 = 144$. Then the average daily count over $n = 36$ days has approximate distribution

$$\bar{X}_{36} \;\approx\; N\!\left(200,\; \dfrac{144}{36}\right) = N(200,\, 4),$$

so the standard deviation of $\bar{X}_{36}$ is $\sqrt{4} = 2$.