Mathematics and Statistics · Ch 11 — Measures of Dispersion

Coefficient of Variation

The standard deviation is an absolute measure, so it cannot fairly compare two data sets that have different averages or different units. The relative measure built from it — the one used everywhere for comparison — is the Coefficient of Variation (C.V.).

Note

Coefficient of Variation (C.V.)

$\text{C.V.} = \frac{\sigma}{\bar{x}} \times 100\quad(\%)$

the standard deviation expressed as a percentage of the mean. It is a pure number, so two series measured in different units or with different means can be compared directly.

Note

How to read the C.V. — consistency

A smaller C.V. means less variation relative to the mean — the data is more consistent, more uniform, more stable.
A larger C.V. means more variation — the data is less consistent. So when comparing two players, machines, or investments, the one with the lower coefficient of variation is the more consistent (more reliable), regardless of which has the higher average.

Example. Batsman A averages $50$ runs with $\sigma = 5$ , giving $\text{C.V.} = \dfrac{5}{50}\times 100 = 10\%$ . Batsman B averages $40$ runs with $\sigma = 6$ , giving $\text{C.V.} = \dfrac{6}{40}\times 100 = 15\%$ . Batsman A has both a higher average and a lower C.V., so A is the more consistent scorer.

Note

A caution …

Definition 14Coefficient of variation (C.V.)

$\text{C.V.} = \dfrac{\sigma}{\bar{x}}\times 100\%$ , the standard deviation as a percentage of the mean; the unit-free measure used to compare the consistency of two data sets — the lower the …

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