5.4 The Distribution of the Sample Mean

• 5.4.1 The Case of a Normal Population Distribution
• 5.4.2 The Central Limit Theorem
• 5.4.3 Other Applications of the Central Limit Theorem
• EXERCISES Section 5.4 (46-57)

The importance of the sample mean $\bar{X}$ springs from its use in drawing conclusions about the population mean $\mu$. Some of the most frequently used inferential procedures are based on properties of the sampling distribution of $\bar{X}$. A preview of these properties appeared in the calculations and simulation experiments of the previous section, where we noted relationships between $E\left(\bar{X}\right)$ and $\mu$ and also among $V\left(\bar{X}\right),{\sigma }^2$ , and $n$.

Proposition

Let $X_1,X_2,\dots ,X_n$ be a random sample from a distribution with

• mean value $\mu$
• standard deviation $\sigma$.

Then

1. $E\left(\bar{X}\right)={\mu }_{\bar{X}}=\mu$
2. $V\left(\bar{X}\right)={\sigma }_{\bar{X}}^2={\sigma }^2/n$ and ${\sigma }_{\bar{X}}=\sigma /\sqrt{n}$

In addition, with $T_o={\underline{X}}_1+\cdots +{\underline{X}}_n$ (the sample total),

• $E\left(T_o\right)=n\mu$,
• $V\left(T_o\right)=n{\sigma }^2$,
• ${\sigma }_{T_o}=\sqrt{n}\sigma$.

Proofs of these results are deferred to the next section. According to Result 1, the sampling (i.e., probability) distribution of $\bar{X}$ is centered precisely at the mean of the population from which the sample has been selected. Result 2 shows that the $\bar{X}$ distribution becomes more concentrated about $\mu$ as the sample size $n$ increases. In marked contrast, the distribution of $T_o$ becomes more spread out as $n$ increases. Averaging moves probability in toward the middle, whereas totaling spreads probability out over a wider and wider range of values. The standard deviation ${\sigma }_{\bar{x}}=\sigma /\sqrt{n}$ is often called the standard error of the mean; it describes the magnitude of a typical or representative deviation of the sample mean from the population mean.

EXAMPLE 5.25