The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

Location A Location B

14 mph 2 mph

15 mph 9 mph

16 mph 16 mph

17 mph 23 mph

18 mph 30 mph

Which location is the better place to site your turbine? One way to approach this question

is to look at the average wind speed at each location, with the idea that the site with the

higher average speed would be expected to produce more electricity. If we calculate the

mean speed at each location, though, we can see that they are tied, with each site having a

mean wind speed of 16 mph. What about the median? We can quickly see from the data that

each location also has a median speed of 16 mph. The two sites are equivalent whichever

measure of average we use. If you based your decision only on the average, you would not

see any reason to prefer one location over another.

However, looking at the data, we can see that the two locations are definitely not the

same. At Location A, the wind speed varies, but not by all that much, while at Location B

there appears to be much more variation. Neither the mean nor the median captures this

difference between the two sites, nor would we expect them to: they are designed to mea-

sure the “average” of a set of data, not how much variation there is in the set. Statistical

measures that are used to measure this are called measures of dispersion, or of variation.

Measures of Variation

The simplest of these measures is the range. The range of a set of data is the difference

between the highest and lowest values. For Location A, the range is 18  14 
 4 mph,

while for Location B the range is 30  2 
 28 mph. Comparing these two ranges reveals

the far greater variation experienced at Location B. While range can be a useful tool, it has

its limitations. The range does not take into account what occurs between the extreme high

and low values. (See Exercise 4 for an example where the range falls short as a measure

of variation.)

The standard deviation is an alternative, commonly used measure of variation. It has

the advantage of including all pieces of the data in its calculation, though it suffers from

the disadvantage of being much more complicated than the range in both its calculation

and interpretation. The standard deviation is probably best introduced through an example

of its calculation; we will discuss its interpretation afterward.

Here is how we would calculate the standard deviation for the wind speeds at Location A.

We first look at how much each individual speed measurement differs from the mean of

16 mph. To do this, we subtract the mean from each value. For example, for the first mea-

surement we subtract 14 mph  16 mph to get 2 mph, reflecting the fact that the 14-mph

measurement was 2 mph “below average.” These differences are called the deviations.

The wind speeds from Location A together with their deviations are shown in the table

below.

Speed Deviation

14  2

15  1

16 0

17 1

18 2

The speeds that fall below the mean have negative deviations, while the ones above the

mean have positive deviations. We are interested in measuring the total amount of varia-

tion, though, regardless of whether the variation comes from above- or below-average

16.3 Measures of Variation 627