cells have the lightest color because any variable has the strongest relationship to itself. From the heatmap in Figure 2.9, we can also see that the two MPG variables (city.mpg and highway.mpg) have strong negative relationships with many of the other numerical variables in the data set.
Figure 2.9 Heatmap of correlation for all numerical variables.
2.2 Summary Statistics
Data visualization is an effective and intuitive representation of the qualitative features of the data. Key characteristics of data can also be quantitatively summarized by numerical statistics. This section introduces common summary statistics for univariate and multivariate data.
2.2.1 Sample Mean, Variance, and Covariance
Sample Mean – Measure of Location
A sample mean or sample average provides a measure of location, or central tendency, of a variable in a data set. Consider a univariate data set, which is a data set with a single variable, that consists of a random sample of n observations x1, x2,…, xn. The sample mean is simply the ordinary arithmetic average
For a data set yi, i = 1, 2,…, n obtained by multiplying each xi by a constant a, i.e., yi = axi, i = 1, 2,…, n, it is easy to see that
Sample Variance – Measure of Spread
The sample variance measures the spread of the data and is defined as
The square root of the sample variance, s = √s2, is called the sample standard deviation. The sample standard deviation is of the same measurement unit as the observations. For yi = axi, i = 1,2,…, n, its sample variance is
Sample Covariance and Correlation – Measure of Linear Association Between Two Variables
If each of the n observations of a data set is measured on two variables x1 and x2, let (x11, x21,...,xn1) and (x12, x22,...,xn2) denote the n observations on x1 and x2, respectively. The sample covariance of x1 and x2 is defined as
where x̄1 and x̄2 are the sample means of x1 and x2, respectively. The value of sample covariance of two variables is affected by the linear association between them. From (2.2), if x1 and x2 have a strong positive linear association, they are usually both above their means or both below their means. Consequently, the product (xi1−x¯1)(xi2−x¯2) will typically be positive and their sample covariance will have a large positive value. On the other hand, if x1 and x2 have a strong negative linear association, the product (xi1−x¯1)(xi2−x¯2) will typically be negative and their sample covariance will have a negative value. If y1 and y2 are obtained by multiplying each measurement of x1 and x2 with a1 and a2, respectively, it is easy to see from (2.2) that the sample covariance of y1 and y2 is
Equation (2.3) says that if the measurements are scaled, for example by changing measurement units, the sample covariance will be scaled correspondingly. The sample covariance’s dependence on the measurement units makes it difficult to determine how large a sample covariance indicates a strong (linear) association between two variables. The sample correlation defined as follows is a measure of linear association that does not depend on the measurement units, or scaling of the variables
where s1 and s2 are the sample standard deviation of x1 and x2, respectively. The sample correlation ranges between −1 and 1, with values close to 1, −1, and 0 indicating a strong positive linear association, a strong negative linear association, and no linear association, respectively.
Example 2.2 To illustrate the calculation of summary statistics, we take a random sample of 10 observations, as shown in Table 2.1, from the auto.spec data set on the variables curb.weight, length, and width. We use xi, i =1,2,3, to represent the three variables:
Table 2.1 A random sample of 10 observations from the auto. spec data set.
| x1 | x2 | x3 |
| 3515 | 190.9 | 70.3 |
| 2300 | 168.7 | 64.0 |
| 2800 | 168.9 | 65.0 |
| 2122 | 166.3 | 64.4 |
| 2293 | 169.1 | 66.0 |
| 2765 | 176.8 | 64.8 |
| 2275 | 171.7 | 65.5 |
| 1890 | 159.1 | 64.2 |
| 2926 | 173.2 | 66.3 |
| 1909 | 158.8 | 63.6 |