Effects of Linear Transformations
- Define a linear transformation
- Compute the mean of a transformed variable
- 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
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.