Chapter 2: Descriptive Statistics

Prerequisite: Chapter 1

2.1 Review of Univariate Statistics

The central tendency of a more or less symmetric distribution of a set of interval, or higher, scaled scores, is often summarized by the arithmetic mean, which is defined as

[pic]. (2.1)

We can use the mean to create a deviation score,

[pic] (2.2)

so named because it quantifies the deviation of the score from the mean.

Deviation is often measured by squaring, since it equates negative and positive deviations. The sum of squared deviations, usually just called the sum of squares, is given by

[pic] (2.3)

Another method of calculating the sum of squares was frequently used during the era that preceded computers when students would work with calculating machines,

[pic] (2.4)

Regardless whether one uses Equation (2.3) or Equation (2.4), the amount of deviation that exists around the mean in a set of scores can be averaged using the standard deviation, or its square, the variance. The variance is just

[pic]

with s being the positive square root of s2.

We can take the deviation scores and standardize them, creating, well; standardized scores:

[pic]. (2.5)

Next, we define a very important concept, that of the covariance of two variables, in this case x and y. The covariance between x and y may be written Cov(x, y). We have
[pic]

[pic] (2.6)

where the[pic]are the deviation scores for the x variable, and the [pic]are defined analogously for y. Note that with a little semantic gamesmanship, we can say that the variance is the covariance of a variable with itself. The product[pic]is usually called a cross product.

2.2 Matrix Expressions for Descriptive Statistics

In this section we will return to our data matrix, X, with n observations and m variables,

[pic]

We now define the mean vector[pic], such