Coefficient of Variation: The standard deviation of data is an absolute measure of dispersion. It is expressed in terms of units that the entries in the data are stated. For example, the standard deviation for data of heights of trees in metres is also expressed in metres. But this cannot be compared with the standard deviation of the weights of students expressed in kilograms.

This is because both these values are expressed in different units. Hence, to achieve this comparison, the standard deviation must be converted to a relative measure. This relative measure of dispersion is called the coefficient of variation, also denoted as CV. Let’s explore more about the coefficient of variation in this article.

Learn About Measures of Dispersion

What is the Coefficient of Variation (CV)?

The coefficient of variation (CV) is the standardised measure of the dispersion of data points around the mean. It is also called relative standard deviation (RSD). It is used to describe variability by expressing standard deviation as a proportion of the mean. The coefficient of variation can be calculated only for data measured on a ratio scale. The ratio scale possesses an absolute zero and has equal intervals between units.

Significance of Coefficient of Variation

The coefficient of variation is the ratio of standard deviation to the mean. This makes it a non-dimensional number. The higher the value of the coefficient of variation; the greater the extent of deviation around the mean.

Definition of Coefficient of Variation

The coefficient of variation is defined as the ratio of the standard deviation \(\left( \sigma \right)\) to the mean \(\left( \mu \right).\) The coefficient of variation is generally expressed as a percentage by multiplying the ratio by \(100.\)
Coefficient of Variation \( = \frac{{{\rm{Standard}}\,{\rm{Deviation}}}}{{{\rm{Mean}}}} \times 100\% \)

Coefficient of Variation Formula

Mathematically, the formula for CV as a percentage in symbols is given as:

\(CV = \frac{\sigma }{\mu } \times 100\% \)

where,

\({\sigma \to }\) standard deviation

\({\mu \to }\) mean

This formula for the coefficient of variation gives percentage results. For decimal values, do not multiply by \(100.\)

\(CV = \frac{\sigma }{\mu }\)

Formulas Related to the Coefficient of Variation Formula

(i) Standard Deviation

If \({\sigma ^2}\) is the variance, then standard deviation σ is given by:
\(\sigma = \sqrt {\frac{{\sum {{{\left( {{x_i} – \mu } \right)}^2}} }}{N}} \)
where,
\(N \to \) size
\({x_i} \to \) individual value
\(\mu \to \) mean

(ii) Mean

The mean of \(n\) observations \({x_1},\,{x_2},\,{x_3},\,{x_4},\,……,\,{x_n}\) is given by:
\({\rm{Mean}} = \frac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{all}}\,{\rm{observations}}}}{{{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{observations}}}}\)

\({\rm{Mean}} = \frac{{{x_1} + {x_2} + {x_3} + ….. + {x_n}}}{n}\)

\({\rm{Mean}} = \frac{{\sum\limits_{i = 1}^n {{x_i}} }}{n}\)

Uses of Coefficient of Variation

Variance is a measurement of the dispersion of numbers in a data set.
1. It is used to compare the degree of variation between two or more data series that have different measures or values.
2. It is also used to calculate the likelihood or probability of distribution of an event.
3. In finance, people use this variance to assess the risk of an investment and decide its profitability.
4. One other common use of variance is to assess the accuracy of a method.

Interpretation of Coefficient of Variation

The coefficient of variation is calculated as the ratio of standard deviation to the absolute value of the mean. Hence, CV is always a positive number. This helps us determine if the standard deviation of data is small or large as compared to the mean of that data. If the value of the coefficient of variation is one or \(100\%,\) this means that the standard deviation is equal to the mean.

For values of CV less than one or \(100\%,\) it means that the standard deviation is smaller as compared to the mean, while values greater than one or \(100\%\) occurs when the standard deviation is greater than the mean.

Steps to Calculate the Coefficient of Variation

The steps to calculate the coefficient of variation for the given data is as follows.

Step 1: Calculate the mean.
\({\rm{Mean}} = \frac{{\sum\limits_{i = 1}^n {{x_i}} }}{n}\)
Step 2: Calculate the standard deviation.
\(\sigma = \sqrt {\frac{{\sum {{{\left( {{x_i} – \mu } \right)}^2}} }}{N}} \)
\(N \to \) size
\({x_i} \to \) individual value
\(\mu \to \) mean
Step 3: Calculate the coefficient of variation as a percentage, find the ratio of standard deviation to the mean and multiply the result by \(100.\)
\(CV = \frac{\sigma }{\mu } \times 100\% \)

Advantages of Coefficient of Variation

1. The coefficient of variation is independent of units.
2. It is a non-dimensional number.
3. It can be used to compare distributions.
4. Shows the relationship between standard deviation and mean.

Disadvantages of Coefficient of Variation

1. As the coefficient of variation approaches infinity, the value of the mean approaches zero.
2. Even minor changes in the mean will affect the coefficient of variation much.
3. It cannot be used to calculate logarithmic values.
4. It cannot be used to find the intervals of the mean.

Solved Examples – Coefficient of Variation

Q.1. Given that the mean and the standard deviation of a class test are \(59.9\) and \(10.2,\) find its coefficient of variation.
Solution:
The formula to calculate the coefficient of variation is:
\(CV = \frac{\sigma }{\mu } \times 100\% \)
Here,
\(\sigma \to \) standard deviation \(= 10.2\)
\(\mu \to \) mean \(= 59.9\)
\(\therefore CV = \frac{{10.2}}{{59.9}} \times 100\% \)
\(CV = 17.03\% .\)

Q.2. The standard deviation and coefficient of variation of data are \(1.2\) and \(25.6\%,\) respectively. Find the value of the mean.
Solution:
The formula to calculate the coefficient of variation is:
\(CV = \frac{\sigma }{\mu } \times 100\% \)
Here,
\(\sigma \to \) standard deviation \(= 1.2\)
\(CV \to \) coefficient of variation \(= 25.6\%\)
\(\mu \to \) mean
\(\therefore \mu = \frac{\sigma }{{CV}} \times 100\% \)
\(\mu = \frac{{1.2}}{{25.6}} \times 100\% \)
\(\mu = 4.68.\)

Q.3. The mean of the number of cars sold over a \(3\)-month period is \(87,\) and its standard deviation is \(5.\) The mean of the commissions is \(5,225,\) and the standard deviation is \(773.\) Compare the variations of these two.
Solution:
The formula to calculate the coefficient of variation is:
\(CV = \frac{\sigma }{\mu } \times 100\% \)
(i) Cars sold
\({\sigma \to }\) standard deviation \(= 5\)
\({\mu \to }\) mean \(= 87\)
\(\therefore CV = \frac{5}{{87}} \times 100\% \)
\(CV = 5.75\% \)
(ii) Commission made
\({\sigma \to }\) standard deviation \(= 773\)
\({\mu \to }\) mean \(= 5225\)
\(CV = \frac{{773}}{{5225}} \times 100\% \)
\(CV = 14.79\% \)
CV of commission made is more than the CV of cars sold.

Q.4. The total marks scored by two students Sathya and Vidya, in \(5\) subjects are \(460\) and \(480,\) with a standard deviation of \(4.6\) and \(2.4,\) respectively. Who is more consistent in their scores?
Solution:

Name of the Student	Sathya	Vidya
Number of Subjects \(\left( n \right)\)	\(5\)	\(5\)
Total Marks \(\left( {\sum x } \right)\)	\(460\)	\(480\)
Mean \(\left( \mu  \right)\)	\(\mu  = \frac{{\sum x }}{n}\)<br> \(\mu  = \frac{{460}}{5} = 92\)	\(\mu  = \frac{{\sum x }}{n}\)<br> \(\mu  = \frac{{480}}{5} = 96\)
Standard Deviation \(\left( \sigma  \right)\)	\(4.6\)	\(2.4\)
Coefficient of Variation (CV)	\(CV = \frac{\sigma }{\mu } \times 100\% \) \(CV = \frac{{4.6}}{{92}} \times 100\% \) \(CV = 5\% \)	\(CV = \frac{\sigma }{\mu } \times 100\% \) \(CV = \frac{{2.4}}{{96}} \times 100\% \) \(CV = 2.5\% \)

CV of Sathya \( > \) CV of Vidya. Therefore, Vidya is more consistent.

Q.5. The weekly sales of two products, \(A\) and \(B,\) are as shown below.

Product \(A\)	\(59\)	\(75\)	\(27\)	\(63\)	\(27\)	\(28\)	\(56\)
Product \(B\)	\(150\)	\(200\)	\(125\)	\(310\)	\(330\)	\(250\)	\(225\)

Identify which of the products shows greater sales fluctuations.
Solution:

Product \(A\)	Product \(B\)
\(x\)	\(y\)
\(59\)	\(150\)
\(75\)	\(200\)
\(27\)	\(125\)
\(63\)	\(310\)
\(27\)	\(330\)
\(28\)	\(250\)
\(56\)	\(225\)
\(\sum x  = 335\)	\(\sum y  = 1590\)
Mean, \(\overline x  = \frac{{335}}{7}\)	Mean, \(\overline y  = \frac{{1590}}{7}\)
\(\overline x  = 47.86\)	\(\overline y  = 227.14\)

\({x_i}\)	\({x_i} – \overline x \)	\({\left( {{x_i} – \overline x } \right)^2}\)	\({y_i}\)	\({y_i} – \overline y \)	\({\left( {{y_i} – \overline y } \right)^2}\)
\(59\)	\(11.14\)	\(124.10\)	\(150\)	\(-77.14\)	\(5950.58\)
\(75\)	\(27.14\)	\(736.58\)	\(200\)	\(-27.14\)	\(736.58\)
\(27\)	\(-20.86\)	\(435.14\)	\(125\)	\(-102.14\)	\(10432.58\)
\(63\)	\(15.14\)	\(229.22\)	\(310\)	\(82.86\)	\(6865.78\)
\(27\)	\(-20.86\)	\(435.14\)	\(330\)	\(102.86\)	\(10580.18\)
\(28\)	\(-19.86\)	\(394.58\)	\(250\)	\(22.86\)	\(522.58\)
\(56\)	\(8.14\)	\(66.26\)	\(225\)	\(-2.14\)	\(4.58\)
	\(\sum {{{\left( {{x_i} – \overline x } \right)}^2}} \)	\(2421.02\)		\(\sum {{{\left( {{y_i} – \overline y } \right)}^2}} \)	\(35092.86\)
Variance	\(\sigma _A^2 = \frac{{\sum {{{\left( {{x_i} – \overline x } \right)}^2}} }}{N}\)	\(\frac{{2421.02}}{7}\) \(= 345.86\)		\(\sigma _B^2 = \frac{{\sum {{{\left( {{y_i} – \overline y } \right)}^2}} }}{N}\)	\(\frac{{35092.86}}{7}\) \(= 5013.27\)
Standard Deviation	\({\sigma _A}\)	\(18.6\)		\({\sigma _B}\)	\(70.8\)
Mean	\({\mu _A} = \overline x \)	\(47.86\)		\({\mu _B} = \overline y \)	\(227.14\)
Coefficient of Variation	\(C{V_A} = \frac{{{\sigma _A}}}{{{\mu _A}}}\)	\(0.3886\)		\(C{V_B} = \frac{{{\sigma _B}}}{{{\mu _B}}}\)	\(0.3117\)
CV as \(\% \)	\(C{V_A} = \frac{{{\sigma _A}}}{{{\mu _A}}} \times 100\% \)	\(38.86\%\)		\(C{V_B} = \frac{{{\sigma _B}}}{{{\mu _B}}} \times 100\% \)	\(31.17\%\)

The coefficient of variation is higher for product \(A\) than that of product \(B.\) Hence, the sales fluctuations are higher for product \(A.\)

Frequently Asked Questions (FAQs)

Q.1. How do you calculate CV?
Ans: The coefficient of variation is the ratio of standard deviation to the mean of that data. The formula to calculate the coefficient of variation as a percentage is given by:
\(CV = \frac{\sigma }{\mu } \times 100\% \)
Here,
\(\sigma \to \) standard deviation
\(\mu \to \) mean.

Q.2. What does the coefficient of variation tell you?
Ans: The coefficient of variation is the standardised measure of the dispersion of data points around the mean. It is defined as the ratio of the standard deviation \(\left( \sigma \right)\) to the mean \(\left( \mu \right).\)

Q.3. What are the steps to calculate the coefficient of variation?
Ans: The steps to calculate the coefficient of variation for the given data are as follows.
(a) Calculate the mean.
(b) Calculate the standard deviation.
(c) Calculate the coefficient of variation as a percentage – find the ratio of standard deviation to the mean and multiply the result by \(100.\)

Q.4. Can the coefficient of variation be more than \(1?\)
Ans: Yes. If the value of the coefficient of variation is \(1\) or \(100\%,\) then the standard deviation is equal to the mean. Values less than \(1\) indicate that the standard deviation is smaller than the mean, and CV more than \(1\) shows that the standard deviation is more than the mean. CV higher than \(1\) is said to be of high variance.

Q.5. What is the use of coefficient of variation?
Ans: The coefficient of variation (CV) is also called relative standard deviation (RSD). It is used to describe variability by expressing standard deviation as a proportion of the mean.

Q.6. Is the high coefficient of variation good or bad?
Ans: In general, for the coefficient of variation, a value of more than \(1\) or \(100\%\) indicates a high variation. Values less than \(1\) or \(100\%\) are considered to have low variation. A large variance indicates that values in the data are set far apart from the mean and from each other. On the other hand, small variance indicates that the values in a data are set close to the mean and to each other.

Now you are provided with all the necessary information on the coefficient of variation and we hope this detailed article is helpful to you.