The most common measures of dispersion are the variance and the standard deviation. These two measures indicate how data are scattered about the mean. However, the unit of variance is the square of the units being observed, which makes it an impractical measure to interpret. For example, if distances measured in meters are analyzed, the variance is measured in meters squared. As a result, you would generally choose to calculate the standard deviation because it represents the average distance each observation is from the mean.

Sample Variance and Standard Deviation

For a sample with n observations, variance and standard deviation are calculated as follows:

Another stock has had the following returns over the last five years:

8

13

17

20

22

Their mean is and n = 5.

Calculating variance and standard deviation, you get the following:

s2s2

= =

[(816)2 + (13 16)2 + (17 16)2 + (20 16)2 + (22 16)2] / (5 1)[8162 + 13 162 + 17 162 + 20 162 + 22 162] / 5 1

= =

126 / 4 = 31.5126 / 4 = 31.5

ss

= =

(31.5)1/2 = 5.61(31.5)1/2 = 5.61

Population Standard Deviation

If your data represent a population, the variance and standard deviation are calculated slightly differently:

In a population standard deviation, you divide by the number of observations (N), not N 1.

Coefficient of Variation

The coefficient of variation standardizes a variables dispersion (standard deviation) relative to its mean. Imagine two variables, each with a standard deviation of 20. If Variable 1 has a mean of 100 and Variable 2 has a mean of 10, it is obvious that has more relative uncertainty.

The coefficient of variation, the amount of risk per unit of the mean, is found by dividing the standard deviation by the mean, as follows:

CV = Standard Deviation / Mean

Check Your Understanding

A vehicles fuel efficiency is being tested. Four trials yielded the following fuel efficiencies (in miles per gallon):

24.8

25.3

29.1

26.4

The mean fuel efficiency is 26.4 miles per gallon.

The data from this fuel efficiency test represents a The variance of these test results is and the standard deviation is

The coefficient of variation for the fuel efficiency results is