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Effects of Transformations

Module by: David Lane

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.

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

Recall that to tranform the degrees Fahrenheit to degrees Centigrade, we use the formula C=0.55556F-17.7778 C 0.55556 F 17.7778 which means we multiply each temperature Fahrenheit by 0.555560.55556 and then subtract 17.77817.778. As you might have expected, you multiply the mean temperature in Fahrenheit by 0.555560.55556 and then subtract 17.77817.778 to get the mean in Centigrade. That is, 0.55556×54-17.7778=12.222 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 0.555560.55556. Since the variance is the standard deviaton squared, the variance in degrees Centigrade is equal to 0.5555620.555562 times the variance of degrees Fahrenheit.

To sum up, if a variable XX has a mean of mm, a standard deviation of ss, and a variance of s2s2 then a new variable YY created using the linear transformation Y=bX+A Y b X A will have a mean of bμ+A b μ A , a standard deviation of bs b s , and a variance of b2s2 b 2 s 2 .

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