5.3 Statistics and Their Distributions

• 5.3.1 Random Samples
• 5.3.2 Deriving a Sampling Distribution
• 5.3.3 Simulation Experiments
• EXERCISES Section 5.3 (37-45)

The observations in a single sample were denoted in Chapter 1 by $x_1,x_2,\dots ,x_n$. Consider selecting two different samples of size $n$ from the same population distribution. The $x_i$’s in the second sample will virtually always differ at least a bit from those in the first sample. For example, a first sample of $n=3$ cars of a particular type might result in fuel efficiencies $x_1=30.7$, $x_2=29.4$, $x_3=31.1$, whereas a second sample may give $x_1=28.8$, $x_2=30.0$, and $x_3=32.5$. Before we obtain data, there is uncertainty about the value of each $x_i$. Because of this uncertainty, before the data becomes available we now regard each observation as a random variable and denote the sample by $X_1,X_2,\dots ,X_n$ (uppercase letters for random variables).

This variation in observed values in turn implies that the value of any function of the sample observations

• such as the sample mean, sample standard deviation, or sample fourth spread
• also varies from sample to sample.

That is, prior to obtaining $x_1,\dots ,x_n$, there is uncertainty as to the value of $\bar{x}$, the value of $s$, and so on.

EXAMPLE 5.20

Defintion (statistic)

A statistic is any quantity whose value can be calculated from sample data. Prior to obtaining data, there is uncertainty as to what value of any particular statistic will result. Therefore,

• a statistic is a random variable and will be denoted by an uppercase letter;
• a lowercase letter is used to represent the calculated or observed value of the statistic.

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Thus the sample mean, regarded as a statistic (before a sample has been selected or an experiment carried out), is denoted by $\bar{X}$; the calculated value of this statistic is $\bar{x}$. Similarly, $S$ represents the sample standard deviation thought of as a statistic, and its computed value is $s$. If samples of two different types of bricks are selected and the individual compressive strengths are denoted by $X_1,\dots ,X_m$ and $Y_1,\dots ,Y_n$, respectively, then the statistic $\bar{X}-\bar{Y}$, the difference between the two sample mean compressive strengths, is often of great interest.

Any statistic, being a random variable, has a probability distribution. In particular, the sample mean $\bar{X}$ has a probability distribution. Suppose, for example, that $n=2$ components are randomly selected and the number of breakdowns while under warranty is determined for each one. Possible values for the sample mean number of breakdowns $\bar{X}$ are 0 (if $X_1=X_2=0$ ),. 5 (if either $X_1=0$ and $X_2=1$ or $X_1=1\text{and}{X}_2=0),1,1.5,\dots$. The probability distribution of $\bar{X}$ specifies $P\left(\bar{X}=0\right)$, $P\left(\bar{X}=.5\right)$, and so on, from which other probabilities such as $P\left(1\le \bar{X}\le 3\right)$ and $P\left(\bar{X}\ge 2.5\right)$ can be calculated. Similarly, if for a sample of size $n=2$, the only possible values of the sample variance are 0,12.5, and 50 (which is the case if $X_1$ and $X_2$ can each take on only the values $40,45$, or 50 ), then the probability distribution of $S^2$ gives $P\left(S^2=0\right),P\left(S^2=12.5\right)$, and $P\left(S^2=50\right)$. The probability distribution of a statistic is sometimes referred to as its sampling distribution to emphasize that it describes how the statistic varies in value across all samples that might be selected.