Statistics

Coefficient of variation

Coefficient of variation

The coefficient of variation (symbol CV), also referred to as the coefficient of mean deviation, is defined as the ratio of the standard deviation to the mean of the data set. It is used to express the standard deviation as a percentage of the mean.

Mathematically, the coefficient of variation is calculated using the following equations:

Population:
Sample:

The coefficient of variation is especially useful when comparing data set, which have different units because the coefficient of variation is a dimensionless number. So when comparing between data sets with different units or widely different means, one should use the coefficient of variation for comparison instead of the standard deviation.

A national sampling of prices for new and used houses found that the mean price for a new house is $120,000 and the standard deviation is $6100 and that the mean price for a used house is $50,000 with a standard deviation equatl to $3150. In terms of absolute deviation, the standard deviation of price for new houses is more than twice that of used houses. However, in terms of relative variation, there is more relative variation in the price of used houses that in new houses.

The CV for used houses is $CV=\frac{3150}{50,000}\;\dot\;=6.3%$

The CV for new houses is $CV=\frac{6100}{120,000}\;\dot\;=5%$

Properties:

When the mean value is near zero, the coefficient of variation is sensitive to small changes in the mean, limiting its usefulness.
The coefficient of variation is independent of change of scale but not of origin (Proof).