Coefficient of Variation Formula

In statistic, the Coefficient of variation formula (CV), also known as relative standard deviation (RSD), is a standardized measure of the dispersion of a probability distribution or frequency distribution. When the value of the coefficient of variation is lower, it means the data has less variability and high stability.

The formula for coefficient of variation is given below:

\(\mathbf{coefficient\ of\ variation = \frac{Standard \ Deviation}{Mean}\times 100 \%}\)

As per sample and population data type, the formula for standard deviation may vary.

\(Sample\ Standard\ Deviation={\sqrt\frac{\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}{n-1}}\) \(Population\ Standard\ Deviation ={\sqrt\frac{\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}{n}}\)
Where,
xi = Terms given in the data
$\overline{x}$ = Mean
n = Total number of terms.

Solved Examples

Example 1: A researcher is comparing two multiple-choice tests with different conditions. In the first test, a typical multiple-choice test is administered. In the second test, alternative choices (i.e. incorrect answers) are randomly assigned to test takers. The results from the two tests are:
Regular Test                 Randomized Answer
Mean 59.9 44.8
SD 10.2 12.7

Trying to compare the two test results is challenging. Comparing standard deviations doesn’t really work, because the means are also different. Calculation using the formula CV=(SD/Mean)*100 helps to make sense of the data:

Regular Test       Randomized Answer
Mean 59.9 44.8
SD 10.2 12.7
CV 17.03 28.35


Looking at the standard deviations 10.2 and 12.7, you might think that the tests have similar results. However, when you adjust for the difference in the means, the results have more significance:

Regular test: CV = 17.03

Randomized answers: CV = 28.35

Example 2: Find the coefficient of variation of the following sample set of numbers.

{1, 5, 6, 8, 10, 40, 65, 88}.

Solution:

Given sample set: {1, 5, 6, 8, 10, 40, 65, 88}.

Sample mean = (1 + 5 + 6 + 8 + 10 + 40 + 65 + 88)/8 = 223/8 = 27.875

\(\sum _{i=1}^n\left(x_i-\bar{x}\right)^2=\left(1-27.875\right)^2+\left(5-27.875\right)^2+\left(6-27.875\right)^2+\left(8-27.875\right)^2+\left(10-27.875\right)^2+\left(40-27.875\right)^2+\left(65-27.875\right)^2+(88-27.875)^2\\=7578.875\)

Variance:

\(\frac{\sum _{i=1}^n\left(x_i-\bar{x}\right)^2}{n-1}=\frac{7578.875}{7}=1082.696\)

Standard deviation:

\(\sigma=\sqrt{\sum _{i=1}^n\frac{\left(x_i-\bar{x}\right)^2}{n-1}}=\sqrt{1082.696}=32.904\)

Coefficient of variation = 32.901/27.875=1.180

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