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.

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

12.220
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 0.556^{2}
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 σ^{2}, 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 b^{2}σ^{2}.

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