# Intuition and uses for coefficient of variation

I'm currently attending the An Introduction to Operations Management course in Coursera.org. At some point in the course, the professor started to deal with variation in the operations' time.

The measurement he uses is the Coefficient of Variation, the ratio between the standard deviation and the mean:

$c_v = \frac{\sigma}{\mu}$

Why would this measurement be used? What are the advantages and disadvantages of working with CV besides working with, say, standard deviation? What is the intuition behind this measurement?

I think of it as a relative measure of spread or variability in the data. If you think of the statement, "The standard deviation is 2.4" it really tells you nothing without respect to the mean (and thus the unit of measure, I suppose). If the mean is equal to 104, the standard deviation of 2.4 communicates quite a different picture of spread than if the mean were 25,452 with a standard deviation of 2.4..

The same reason you normalize data (subtract the mean and divide by the standard deviation) to place data expressed in different units on a comparable or equal footing—so too this measure of variability is normalized—to aid in comparisons.

Coefficient of variation is effectively a normalized or relative measure of the variation in a data set, (e.g. a time series) in that it is a proportion (and therefore can be expressed as a percentage). Intuitively, if the mean is the expected value, then the coefficient of variation is the expected variability of a measurement, relative to the mean.

This is useful when comparing measurements across multiple heterogenous data sets or across multiple measurements taken on the same data set - the coefficient of variation between two data sets, or calculated for two sets of measurements can be directly compared, even if the data in each are measured on very different scales, sampling rates or resolutions. In contrast, standard deviation is specific to the measurement/sample it is obtained from, i.e. it is an absolute rather than a relative measure of variation.

• Can you explain this part more please: "The coefficient of variation is the expected variability of a measurement over an interval"? – B_Miner Oct 30 '12 at 20:17
• @B_Miner I meant interval in the signal processing sense and have edited above. Std dev is effectively the mean or expected variation. – BGreene Oct 30 '12 at 20:22

according to my understanding, mean is location parameter. sd/mean should not consider as coefficient of variation. why? simple argument is that statistical distance is different than euclidean distance. to measure statistical distance we use sd ; crude distance for one variable. suppose 50 is mean and 2 is sd then 4% will be cv. now is mean is 5 and sd is 2 cv= 40%. statistical variation term is independent from origin. so sd itself is good measure of variation. and remember one rule from physics that is do not compare two unit systems in single problem.

• It is hard to see any coherent argument here. We should not consider sd/mean to be the coefficient of variation? That's how it's defined. If you mean that it is not useful, explain why. (If you think it's misnamed, that's a different story.) Statistical distance differs from Euclidean distance? That's just an assertion and hinges on knowing what you mean by statistical distance. As many kinds of distance appear in statistics, the assertion remains obscure. (I didn't downvote, but I urge you to rewrite this. You may need to work with a friend with a better command of written English.) – Nick Cox Jan 1 '16 at 14:18