Central Limit Theorem

Suppose that $ n =$ observations $ X_1,X_2,\ldots,X_n$ are drawn from a common normal distribution with mean $ \mu$ and standard deviation $ \sigma =$ . Then the sample mean

$\displaystyle \bar{X} = \frac{X_1 + X_2 + \cdots + X_n}{n}

has the normal distribution, and has the same mean $ \mu$ and a distinctly smaller standard deviation $ \displaystyle\frac{\sigma}{\sqrt{n}} =$ .

Central limit theorem. Now we shall drop the assumption of normal distribution for $ X_1,X_2,\ldots,X_n$. Instead, we will assume an adequately large sample size $ n$. Then the distribution of the sample mean $ \bar{X}$ is still approximated by the same normal distribution with the mean $ \mu$ and the standard deviation $ \displaystyle\frac{\sigma}{\sqrt{n}}$. A general rule for ``adequately large'' $ n$ is about $ n \ge 30$, but it is often good for much smaller $ n$.

Example. The daily sales of a farmer's market vary from day to day, but it is normally distributed with mean $900 and standard deviation $300. The market is open six days a week. (a) How much variability do you expect in the average sales in a week? (b) How many days in a year (6$ \times$52 = 312 days) do you expect the sales less than $600? (c) How many weeks in a year (52 weeks) do you expect the weekly average sales less than $600?