How to Calculate the Coefficient of Variation

Calculates the coefficient of variation as the standard deviation ÷ the mean. Being unit-free, it compares the spread of data sets with different means.

Explanation

The coefficient of variation divides the standard deviation by the mean, expressing spread as a fraction of the average size.

CV = \dfrac{\sigma}{\bar{x}}

It is often multiplied by 100 and quoted as a percentage.

Why it is needed

A standard deviation on its own can be misleading when you compare different quantities.

Take height (mean 170 cm, standard deviation 6 cm) and weight (mean 60 kg, standard deviation 6 kg). Both standard deviations are 6, but the units differ, so the comparison is meaningless. The coefficients of variation are $6 \div 170 = 0.035$ and $6 \div 60 = 0.1$ , showing that weight varies far more in relative terms.

Because the units cancel, you can compare yen with dollars, or centimetres with kilograms.

Example

For the data 10, 20, 30, 40, 50 the mean is 30 and the standard deviation is $\sqrt{200} = 14.1421$ .

CV = \dfrac{14.1421}{30} = 0.4714 = 47.14\%

Watch out

A mean of zero makes the calculation impossible, since you would be dividing by zero
A mean near zero blows the coefficient up and drains it of meaning
It is intended for quantities that are positive by nature, such as lengths, weights and amounts of money. For something like temperature, which can be negative, the value depends on where you put the zero, so it does not apply