Coefficient of Variation

The coefficient of variation measures the variability of a series of numbers independent of the unit of measurement used for these numbers. In order to do so, the coefficient of variation eliminates the unit of measurement of the standard deviation of a series of numbers by dividing the standard deviation by the mean of these numbers. The coefficient of variation can be used to compare distributions obtained with different units, such as the variability of the weights of newborns (measured in grams) with the size of adults (measured in centimeters). The coefficient of variation is meaningful only for measurements with a real zero (i.e., “ratio scales”) because the mean is meaningful (i.e., unique) only for these scales. So, for example, it would be meaningless to compute the coefficient of variation of the temperature measured in degrees Fahrenheit, because changing the measurement to degrees Celsius will not change the temperature but will change the value of the coefficient of variation (because the value of zero for Celsius is 32 for Fahrenheit, and therefore the mean of the temperature will change from one scale to the other). In addition, the values of the measurement used to compute the coefficient of variation are assumed to be always positive or null. The coefficient of variation is primarily a descriptive statistic, but it is amenable to statistical inferences such as null hypothesis testing or confidence intervals. Standard procedures are often very dependent on the normality assumption, [Page 170]and current work is exploring alternative procedures that are less dependent on this normality assumption.

Definition and Notation

The coefficient of variation, denoted Cv (or occasionally V), eliminates the unit of measurement from the standard deviation of a series of numbers by dividing it by the mean of this series of numbers. Formally, if, for a series of N numbers, the standard deviation and the mean are denoted respectively by S and M, the coefficient of variation is computed as

Often the coefficient of variation is expressed as a percentage, which corresponds to the following formula for the coefficient of variation:

This last formula can be potentially misleading because, as shown later, the value of the coefficient of variation can exceed 1 and therefore would create percentages larger than 100. In that case, Formula 1, which expresses Cv as a ratio rather than a percentage, should be used.

Range

In a finite sample of N nonnegative numbers with a real zero, the coefficient of variation can take a value between 0 and

(the maximum value of Cv is reached when all values but one are equal to zero).

Estimation of a Population Coefficient of Variation

The coefficient of variation computed on a sample is a biased estimate of the population coefficient of variation denoted yv An unbiased estimate of the population coefficient of variation, denoted Ĉv, is computed as

(where N is the sample size).

Testing the Coefficient of Variation

When the coefficient of variation is computed on a sample drawn from a normal population, its standard error, denoted σcνis known and is equal to

When γν is not known (which is, in general, the case), σcν, can be estimated by replacing γν by its estimation from the sample. Either Ĉν or Ĉν can be used for this purpose (Ĉν being preferable because it is a better estimate). So σcν can be estimated

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