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Coefficient of Variation
Coefficient of variation is a type of relative measure of dispersion. It is expressed as the ratio of the standard deviation to the mean. The coefficient of variation is a dimensionless quantity and is usually given as a percentage. It helps to compare two data sets on the basis of the degree of variation.
The coefficient of variation can be determined for both a sample as well as a population. In industries such as finance, the coefficient of variation is used to help investors assess the risk to reward ratio. In this article, we will learn more about the coefficient of variation, its formula, and various examples.
What is Coefficient of Variation?
The coefficient of variation is a type of measure of dispersion. A measure of dispersion is a quantity that is used to gauge the extent of variability of data. Thus, the coefficient of variation is used to measure the dispersion of data from the average or the mean value. CV is the abbreviated form of the coefficient of variation.
Coefficient of Variation Definition
The coefficient of variation is a dimensionless relative measure of dispersion that is defined as the ratio of the standard deviation to the mean. If there are data sets that have different units then the best way to draw a comparison between them is by using the coefficient of variation.
Coefficient of Variation Example
Suppose there is a data set [80, 90, 100]. The mean is 90 and the population standard deviation is 8.165. The coefficient of variation is 0.09. As a percentage, the coefficient of variation is 9%.
Coefficient of Variation Formula
There are two formulas for the coefficient of variation. These are the population coefficient of variation and the sample coefficient of variation. Population, in statistics, is the entire group that is under consideration. In other words, the population is used to denote the complete data set. When a specific part is chosen from this population it is known as the sample. The sample is used to represent the entire population of the study. The population mean and the sample mean will always be the same. However, as the value of the standard deviation differs thus, there are two coefficient of variation formulas. These are given below:
 Population Coefficient of Variation = (\(\frac{\sigma }{\mu }\)) * 100.
 Sample Coefficent of Variation = \(\frac{s}{\mu}\) * 100
\(\sigma\) is the standard deviation of the population. It is given by \(\sqrt{\frac{\sum (x_{i}\mu)^{2} }{N}}\)
s is the standard deviation of the sample. It is given by \(\sqrt{\frac{\sum (x_{i}\mu)^{2} }{N  1}}\)
How to Find Coefficient of Variation?
The coefficient of variation formula is useful particularly in those cases where we need to compare results from two different surveys having different values. In statistics, 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. If the value of the coefficient of variation is lower then it indicates that the data has less variability and high stability. The general steps to find the coefficient of variation are as follows:
 Step 1: Check for the sample set.
 Step 2: Calculate standard deviation and mean.
 Step 3: Put the values in the coefficient of variation formula, CV =\(\dfrac{σ}{μ}\) × 100, μ≠0,
Now let us understand this concept with the help of a few examples.
Example: Two plants C and D of a factory show the following results about the number of workers and the wages paid to them.
No. of workers  5000  6000 
Average monthly wages  $2500  $2500 
Standard deviation  9  10 
Using coefficient of variation formulas, find in which plant, C or D is there greater variability in individual wages.
Solution:
To Find: Which plant has greater variability.
For this, we need to find the coefficient of variation. The plant that has a higher coefficient of variation will have greater variability.
Coefficient of variation for plant C.
Using coefficient of variation formula,
CV = (σ/μ) × 100, μ≠0
CV = (9/2500) × 100
CV = 0.36%
Now, CV for plant D
CV = (σ/μ) × 100
CV = (10/2500) × 100
CV = 0.4%
Plant C has CV = 0.36 and plant D has CV = 0.4
Answer: Hence plant D has greater variability in individual wages.
Coefficient of Variation and Standard Deviation
The coefficient of variation and the standard deviation are both used when the spread of the values in a dataset has to be measured. The main differences between the two measures are given in the table below.
Coefficient of Variation  Standard deviation 
It is a relative measure of dispersion  It is an absolute measure of dispersion 
It measures the ratio of the standard deviation to the mean  It measures how far a data point lies from the mean 
Coefficient of variation is usually used to compare the variation of different data sets  Standard deviation is used to measure the dispersion of data in a single data set 
Coefficient of Variation Uses
If two data sets having similar values need to be compared then the standard deviation can be used. However, if two data sets having different unit need to be compared then the coefficient of variation needs to be used. Some applications of coefficient of variation are as follows:
 In the finance industry if an investor wants to invest in a particular ETF, then he uses the coefficient of variation to choose the one which will give a better riskreturn tradeoff.
 The coefficient of variation is also used to gauge the consistency of data. A distribution with a smaller coefficient of variation is more consistent than one with a larger CV.
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Important Notes on Coefficient of Variation
 Coefficient of variation is a relative measure of dispersion that is used to determine the variablity of data.
 It is expressed as a ratio of the standard deviation to the mean multiplied by 100.
 It is a dimensionless quantity.
 The formula for the coefficient of variation is given as [\(\frac{\sigma }{\mu }\) * 100] or [\(\frac{s}{\mu}\) * 100].
Examples on Coefficient of Variation

Example 1: Find the population coefficient of variation of the given data set (320, 540, 480, 540, 420, 240)
Solution: Mean = (320 + 540 + 480 + 540 + 420 + 240) / 6 = 423.33
Standard Deviation = \(\sqrt{\frac{(320423.33)^{2} + (540423.33)^{2} + (480423.33)^{2} + (540423.33)^{2} + (420423.33)^{2} + (240423.33)^{2}}{6}}\)
Standard Deviation = 111.6
Coefficient of Variation = (Standard Deviation / mean) * 100 = (111.6 / 423.33) * 100 = 26.36%
Answer: Coefficient of variation = 26.36% 
Example 2: If the coefficient of variation is given as 20.75 and the mean is 22.6 then find the standard deviation.
Solution: Coefficient of Variation = (Standard Deviation / mean) * 100
20.75 = (SD / 22.6) * 100
SD = 4.69
Answer: Standard deviation = 4.69

Example 3: Find the sample coefficient of variance of the given data set (31.9, 42.5, 55.2, 67.8)
Solution: Mean = (31.9 + 42.5 + 55.2 + 67.8) / 4 = 49.35
Standard Deviation = \(\sqrt{\frac{(31.949.35)^{2} + (42.549.35)^{2} + (55.249.35)^{2} + (67.849.35)^{2}}{41}}\) = 15.55
Coefficient of Variation = (Standard Deviation / mean) * 100 = (15.55 / 49.35) * 100 = 31.5%
Answer: Coefficient of variation = 31.5%

Example 4: If the coefficient of variation of two distributions are 60 and 70, and their standard deviations are 25 and 16, respectively, find their arithmetic means.
Solution:
To Find: Arithmetic means of given distributions.
Given: \(CV_1\) = 60, \(σ_1\) =25
\(CV_2\) = 70, \(σ_2\) = 16
Using coefficient of variation formula,
CV = (σ/μ) × 100, μ≠0
\(CV_1\) = \(\dfrac{σ_1}{μ_1}\) × 100
60 = \(\dfrac{25}{μ_1}\) × 100
\(μ_1\) = 41.66
Similarly,
\(CV_2\) =\(\dfrac{σ_2}{μ_2}\) × 100
70 = \(\dfrac{16}{μ_2}\)×100
\(μ_2\) = 22.87.
Answer: The value of \(μ_1\) = 41.66 and \(μ_2\) = 22.87.
FAQs on Coefficient of Variation
What is Coefficient of Variation in Statistics?
Coefficient of variation is a dimensionless measure of dispersion that gives the extent of variability in data. It is very useful for comparing two data sets with differing units.
What is a Good Coefficient of Variation?
A coefficient of variation less than 20% is acceptable. For lab results, a good coefficient of variation should be lesser than 10%.
How Do You Calculate the Coefficient of Variation?
To calculate the coefficient of variation the steps are as follows.
 Find the mean of the data.
 Find the standard deviation of the data.
 Divide the standard deviation by the mean and multiply this value by 100 to get the coefficient of variation.
What is the Difference Between Standard Deviation and Coefficient of Variation?
Standard deviation is an absolute measure of dispersion that is used to determine the spread of data points in a single data set. The coefficient of variation is a relative measure of dispersion that can compare two data sets with different units on the basis of variability.
How to Interpret Coefficient of Variation?
Coefficient of variation can be used to compare data sets that cannot be compared otherwise. A high coefficient of variation indicates that the level of dispersion around the mean of the data is higher.
Is Coefficient of Variation a Measure of Dispersion?
Coefficient of variation is a relative measure of dispersion. It is equal to the ratio of the standard deviation to the mean and can be expressed as a percentage.
Is Coefficient of Variation a Measure of Central Tendency?
No, the coefficient of variation is not a measure of central tendency. The measures of central tendency include mean median and mode. The coefficient of variation is a relative measure of dispersion.
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