Statistical Inference: A Short Course

3.2 The Arithmetic Mean

First, some notation:

a. We shall denote the population mean as follows:

(3.1)

(the Greek letter μ is pronounced “mu”);

b. We shall denote the sample mean as follows:

(3.2)

Clearly, Equations (3.1) and (3.2) are “simple averages.”

Example 3.1

Suppose we have a sample of n = 5 observations on a variable X: 1, 4, 10, 8, 10. What is the arithmetic mean of X or the value of ? It is readily seen that

How should we interpret 6.6? A “physical” interpretation of the mean is that it represents X's center of gravity, that is, X's absolute frequency distribution will “balance” at the mean (Fig. 3.1).

Figure 3.1 is the center of gravity of X.

So much for physics. How does the statistician interpret the word “balance?” To answer this question, let us introduce the concept of the ith deviation from the mean. Specifically, given a variable X:X₁, X₂,. . . , X_n, the ith ...