Coefficient of variation for grouped data

Coefficient of variation for grouped data

Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution.

Coefficient of variation formula is given by

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

where,

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

If coefficient of variation of one data set is lower than the coefficient of variation of other data set, then the data set with lower coefficient of variation is more consistent than the other.

Example 1

Compute coefficient of variation for the following frequency distribution.

$x$ 2 3 4 5 6
$f$ 1 15 10 5 4

Solution

$x_i$ $f_i$ $f_i*x_i$ $f_i*x_i^2$
2 1 2 4
3 15 45 135
4 10 40 160
5 5 25 125
6 4 24 144
Total 35 136 568

Sample mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{N}\sum_{i=1}^n f_ix_i\\ &=\frac{136}{35}\\ &=3.8857 \end{aligned} $$

Sample variance of $X$ is

$$ \begin{aligned} s_x^2 &=\frac{1}{N}\sum_{i=1}^n f_ix_i^2-(\overline{x})^2\\ &=\frac{1}{35}\big(568\big)-(3.8857)^2\\ &=16.2286-15.0987\\ &=1.1299 \end{aligned} $$

Sample standard deviation of $X$ is

$$ \begin{aligned} s_x&=\sqrt{s_x^2}\\ &= \sqrt{1.1299}\\ &=1.063. \end{aligned} $$

Coefficient of variation is

$$ \begin{aligned} CV &=\frac{sx}{\overline{x}}\times 100\\ &=\frac{1.063}{3.8857}\times 100\\ &=27.3567 \end{aligned} $$

Example 2

Compute coefficient of variation for the following frequency distribution.

$x$ 5-8 9-12 13-16 17-20 21-24
$f$ 2 13 21 14 5

Solution

Class Interval Class Boundries mid-value ($x_i$) $f_i$ $f_i*x_i$ $f_ix_i^2$
5-8 4.5-8.5 6.5 2 13 84.5
9-12 8.5-12.5 10.5 13 136.5 1433.25
13-16 12.5-16.5 14.5 21 304.5 4415.25
17-20 16.5-20.5 18.5 14 259 4791.5
21-24 20.5-24.5 22.5 5 112.5 2531.25
Total 55 825.5 13255.75

Sample mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{N}\sum_{i=1}^n f_ix_i\\ &=\frac{825.5}{55}\\ &=15.0091 \end{aligned} $$

Sample variance of $X$ is

$$ \begin{aligned} s_x^2 &=\frac{1}{N}\sum_{i=1}^n f_ix_i^2-(\overline{x})^2\\ &=\frac{1}{55}\big(13255.75\big)-(15.0091)^2\\ &=241.0136-225.2731\\ &=15.7406 \end{aligned} $$

Sample standard deviation of $X$ is

$$ \begin{aligned} s_x&=\sqrt{s_x^2}\\ &= \sqrt{15.7406}\\ &=3.9674. \end{aligned} $$

Coefficient of variation is

$$ \begin{aligned} CV &=\frac{sx}{\overline{x}}\times 100\\ &=\frac{3.9674}{15.0091}\times 100\\ &=26.4333 \end{aligned} $$

Hope you enjoyed the step by step solution to find Coefficient of variation for grouped data with help of coefficient of variation formula.

Do read more about step by step solution to find Coefficient of variation for ungrouped data.

If you have any doubt or queries feel free to post them in the comment section.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.

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