Effects of Linear Transformations

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 1shows the temperatures of 5 cities.

Table 1. Temperatures in 5 cities on 11/16/2002
City
Degrees Fahrenheit
Degrees Centigrade
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
12.22
12.22
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 = 0.55556F - 17.7778

which means we multiply each temperature Fahrenheit by 0.55556 and then subtract -17.778. As you might have expected, you multiply the mean temperature in Fahrenheit by 0.55556 and then subtract -17.778 to get the mean in Centigrade. That is, (0.55556)(54) - 17.7778 = 12.222. 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 of degrees Centigrade is equal to the standard deviation in degrees Fahrenheit times 0.55556. Since the variance is the standard deviation squared, the variance in degrees Centigrade is equal to 0.555562 times the variance of 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.