# Coefficient of Variation

**Meaning and definition of Coefficient of Variation **

The **coefficient of variation** (CV) refers to a statistical measure of the distribution of data points in a data series around the mean. It represents the ratio of the standard deviation to the mean. The coefficient of variation is a helpful statistic in comparing the degree of variation from one data series to the other, although the means are considerably different from each other.

As expressed by Investopedia, the CV enables the determination of assumed volatility as compared to the amount of return expected from an investment. Putting it simple, a lower ratio of standard deviation to mean return indicates a better risk-return trade off.

**Formula for Coefficient of Variation **

Being calculated as the ratio of standard deviation to the mean, the coefficient of variation is computed using the following formula:

Coefficient of Variation = Standard Deviation / Expected Return

**Calculating the coefficient of variation **

The main steps involved in computation of coefficient of variation are:

1. Compute the sample mean, using the formula μ = 'x_{i} / n, where n indicates the number of data point x_{i} in the sample, and the total is over all values of i. The term i is read as a subscript of x.

2. Compute the sample variance, using the formula '(x_{i} - μ)^2 / (n-1). For instance, in the above mentioned sample set, the sample variance is [0.5^2 + 1.5^2 + 0.5^2 + 1.5^2] / 3 = 1.667.

3. Identify the sample standard deviation by solving the square root of the result obtained from Step 2. Thereafter, divide it by the sample mean. The result thus obtained is the coefficient of variation.

To continue with the aforementioned example, √(1.667)/3.5 = 0.3689.

**Applications **

The coefficient of variation is common in applied probability fields like renewal theory, queuing theory, and reliability theory. In these fields, the exponential distribution is generally more important than the normal distribution. The standard deviation of an exponential distribution is equivalent to its mean, the making its coefficient of variation to equalize 1. Distributions with a coefficient of variation to be less than 1 are considered to be low-variance, whereas those with a CV higher than 1 are considered to be high variance.

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