Effects of Linear Transformations

David M. Lane

Prerequisites

Linear Transformations

Learning Objectives
1. Define a linear transformation
2. Compute the mean of a transformed variable
3. Compute the variance of a transformed variable

This section covers the effects of linear transformations on measures of central tendency and variability. Let's start with an example we saw before in the section that defined linear transformation: temperatures of cities. Table 1 shows the temperatures of 5 cities.

Table 1. Temperatures in 5 cities on 11/16/2002.
Houston
Chicago
Minneapolis
Miami
Phoenix
54
37
31
78
70
12.22
2.78
-0.56
25.56
21.11
Mean
Median
54.000
54.000
12.220
12.220
Variance 330.00 101.852
SD 18.166 10.092

Recall that to transform the degrees Fahrenheit to degrees Centigrade, we use the formula

C = 0.556F - 17.778

which means we multiply each temperature Fahrenheit by 0.556 and then subtract 17.778. As you might have expected, you multiply the mean temperature in Fahrenheit by 0.556 and then subtract 17.778 to get the mean in Centigrade. That is, (0.556)(54) - 17.778 = 12.22. The same is true for the median. Note that this relationship holds even if the mean and median are not identical as they are in Table 1.

The formula for the standard deviation is just as simple: the standard deviation in degrees Centigrade is equal to the standard deviation in degrees Fahrenheit times 0.556. Since the variance is the standard deviation squared, the variance in degrees Centigrade is equal to 0.5562 times the variance in degrees Fahrenheit.

To sum up, if a variable X has a mean of μ, a standard deviation of σ, and a variance of σ2, then a new variable Y created using the linear transformation

Y = bX + A

will have a mean of bμ+A, a standard deviation of bσ, and a variance of b2σ2.

It should be noted that the term "linear transformation" is defined differently in the field of linear algebra. For details, follow this link.