In statistics, the coefficient of variation (COV) is a simple measure of relative event dispersion. It is equal to the ratio between the standard deviation and the mean. The most common use of the COV is to compare relative risk, although it can be applied to any type of quantitative likelihood or probability distribution.

There is another use and meaning of the COV. When interpreting mathematical models, COV is calculated as the ratio between root mean squared error and the mean of a separate dependent variable. This type of COV analysis is less common, but it can be constructive when determining if a model is a good fit for a specific task or type of analysis. Several other terms are synonymous with COV, including the variation coefficient, unitized risk, and relative standard deviation.

Possible Uses of the Coefficient of Variation

A COV is particularly useful in a study that demonstrates exponential distribution. In other words, it can help demonstrate when distributions are considered low-variance and when they are considered high-variance.

In investing and finance, the COV can be used to evaluate risk. A risk-based COV can be interpreted in much the same way as the standard deviation in modern portfolio theory (MPT). The only difference is that COV is a better overall indicator of relative risk, particularly among different levels of risk for different securities. 

For example, suppose two different stocks offered different returns and had different standard deviations. Stock A might have an expected return of 15% and Stock B an expected return of 10%. However, Stock A has a standard deviation of 10%, while Stock B only has a standard deviation of 5%. Which is the better investment?

Assuming that these expected returns are accurate and that the rest of the investor's portfolio is neutral to the decision, Stock B is the better investment. Its COV (5% / 10%, or 0.5) is less than the COV for Stock A (10% / 15%, or 0.67).

Advantages of the Coefficient of Variation

The principal advantage of the COV is that it is unit-less. A COV can be run for any given quantifiable data, and otherwise unrelated COVs can be compared to one another in ways that other measures cannot.

In fact, the unit-less quality of COV is what separates it from a standard deviation analysis. The standard deviation of the two variables cannot be compared in any meaningful way. By comparing the standard deviation and the mean, however, the COV makes every dispersion relative and yet independent of the underlying unit.

As a measure of risk, the COV is used to measure volatility in the prices of stocks and other securities. It allows analysts to assess and compare the risks associated with different potential investments. Therefore, it can be used to measure and manage investment risks. 

A diversified portfolio of assets is always recommended to reduce the risk of major fluctuations in returns on a single investment. Therefore, risk and diversification are negatively related; that is, as diversification increases, risk decreases.

The Zero Disadvantage

Suppose the mean of a sample population is zero. In other words, the sum of all values above and below zero are equal to each other. In this circumstance, the formula for COV is useless because it would place a zero in the denominator.

In fact, the nature of COV calculations is that any strong presence of both positive and negative values in the sample population becomes problematic. This metric is best used when nearly all of the data points share the same plus-minus sign.