The coefficient of variation (CV), also known as “relative variability,” is equal to the standard deviation of a distribution divided by its mean. As discussed in John Freund’s “Mathematical Statistics,” the CV differs from the variance in that the mean “normalizes” the CV in a way, making it unitless, which facilitates comparison between populations and distributions. Of course, the CV doesn’t work well for populations symmetric about the origin, since the mean would be so close to zero, making CV quite high and volatile, regardless of the variance. You can calculate CV from sample data of a population of interest, if you don’t know the variance and mean of the population directly.

Calculate the sample mean, using the formula ? = ?x_i / n, where n is the number of data point x_i in the sample, and the summation is over all values of i. Read i as a subscript of x.

For example, if a sample from a population is 4, 2, 3, 5, then the sample mean is 14/4 = 3.5.

Calculate the sample variance, using the formula ?(x_i - ?)^2 / (n-1).

For example, in the above sample set, the sample variance is [0.5^2 + 1.5^2 + 0.5^2 + 1.5^2] / 3 = 1.667.

Find the sample standard deviation by solving the square root of the result of step 2. Then divide by the sample mean. The result is the CV.

Continuing with the above example, ?(1.667)/3.5 = 0.3689.

References

- "Mathematical Statistics"; John Freund; 1997
- UCLA: What is the Coefficient of Variation?

About the Author

Paul Dohrman's academic background is in physics and economics. He has professional experience as an educator, mortgage consultant, and casualty actuary. His interests include development economics, technology-based charities, and angel investing.

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