## Coefficient of variation for ungrouped data

Coefficient of variation is an absolute measure of variation and is used for the comparison of variability between two or more frequency distribution.

Let $x_i, i=1,2, \cdots , n$ be $n$ observations.

## Formula

Coefficient of variation is given by

$CV =\dfrac{s_x}{\overline{x}}\times 100$

where,

• $\overline{x} =\dfrac{1}{n}\sum_{i=1}^{n}x_i$ is the sample mean of $X$,
• $n$ total number of observations,
• $s_x=\sqrt{V(x)}$ is the standard deviation of $X$,
• $s_x^2=V(x)=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2 -\dfrac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)$ is the variance of $X$

## Example 1

The following data gives the hourly wage rates (in dollars) of 10 employees of a company.

20,21,24,25,18,22,24,22,20,22.

Find the coefficient of variation.

### Solution

$x_i$ $x_i^2$
20 400
21 441
24 576
25 625
18 324
22 484
24 576
22 484
20 400
22 484
Total 218 4794

Sample mean

The sample mean of $X$ is

\begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{218}{10}\\ &=21.8\text{ dollars} \end{aligned}

The average of hourly wage rates is $21.8$ dollars.

Sample variance

Sample variance of $X$ is

\begin{aligned} s_x^2 &=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\frac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)\\ &=\dfrac{1}{9}\bigg(4794-\frac{(218)^2}{10}\bigg)\\ &=\dfrac{1}{9}\big(4794-\frac{47524}{10}\big)\\ &=\dfrac{1}{9}\big(4794-4752.4\big)\\ &= \frac{41.6}{9}\\ &=4.6222 \end{aligned}

Sample standard deviation

The sample standard deviation is the positive square root of the sample variance.

The sample standard deviation is

\begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{4.6222}\\ &=2.1499 \text{ dollars} \end{aligned}

Thus the sample standard deviation of hourly wage rates is $2.1499$ dollars.

Coefficient of Variation

The coefficient of variation is

\begin{aligned} CV &=\frac{s_x}{\overline{x}}\times 100\\ &=\frac{2.1499}{21.8}\times 100\\ &= 9.8621 \end{aligned}

## Example 2

The age (in years) of 6 randomly selected students from a class are

22,25,24,23,24,20.

Compute Coefficient of variation.

### Solution

$x_i$ $x_i^2$
22 484
25 625
24 576
23 529
24 576
20 400
Total 138 3190

Sample mean

The sample mean of $X$ is

\begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{138}{6}\\ &=23\text{ years} \end{aligned}

The average of age of students is $23$ years.

Sample variance

Sample variance of $X$ is

\begin{aligned} s_x^2 &=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\frac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)\\ &=\dfrac{1}{5}\bigg(3190-\frac{(138)^2}{6}\bigg)\\ &=\dfrac{1}{5}\big(3190-\frac{19044}{6}\big)\\ &=\dfrac{1}{5}\big(3190-3174\big)\\ &= \frac{16}{5}\\ &=3.2 \end{aligned}

Sample standard deviation

The sample standard deviation is the positive square root of the sample variance.

The sample standard deviation is

\begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{3.2}\\ &=1.7889 \text{ years} \end{aligned}

Thus the sample standard deviation of age of students is $1.7889$ years.

Coefficient of Variation

The coefficient of variation is

\begin{aligned} CV &=\frac{s_x}{\overline{x}}\times 100\\ &=\frac{1.7889}{23}\times 100\\ &= 7.7776 \end{aligned}