# Applied Statistics Lesson of the Day – The Coefficient of Variation

In my statistics classes, I learned to use the variance or the standard deviation to measure the variability or dispersion of a data set.  However, consider the following 2 hypothetical cases:

1. the standard deviation for the incomes of households in Canada is $2,000 2. the standard deviation for the incomes of the 5 major banks in Canada is$2,000

Even though this measure of dispersion has the same value for both sets of income data, $2,000 is a significant amount for a household, whereas$2,000 is not a lot of money for one of the “Big Five” banks.  Thus, the standard deviation alone does not give a fully accurate sense of the relative variability between the 2 data sets.  One way to overcome this limitation is to take the mean of the data sets into account.

A useful statistic for measuring the variability of a data set while scaling by the mean is the sample coefficient of variation: $\text{Sample Coefficient of Variation (} \bar{c_v} \text{)} \ = \ s \ \div \ \bar{x},$

where $s$ is the sample standard deviation and $\bar{x}$ is the sample mean.

Analogously, the coefficient of variation for a random variable is $\text{Coefficient of Variation} \ (c_v) \ = \ \sigma \div \ \mu,$

where $\sigma$ is the random variable’s standard deviation and $\mu$ is the random variable’s expected value.

The coefficient of variation is a very useful statistic that I, unfortunately, never learned in my introductory statistics classes.  I hope that all new statistics students get to learn this alternative measure of dispersion.

### 3 Responses to Applied Statistics Lesson of the Day – The Coefficient of Variation

1. Nicholas Horton says:

Nice blog post.

I thought that your readers might appreciate seeing how to bootstrap a confidence interval for the CV in R:

covfun = function(x) { # multiply CV by 100
return(100*sd(x)/mean(x))
}
x = rnorm(1000, mean=1, sd=1)
covfun(x)
library(mosaic)
res = do(2000) * covfun(resample(x))
quantile(res\$result, c(.025, .975))

Nick

2. Psych n Stats Tutor says:

Reblogged this on Psychology & Statistics Tutor:Mentor and commented:
Understanding the theory behind the stats, makes it easier to understand what you are doing with them~