Statistics - Co-efficient of Variation

Coefficient of Variation

Standard variation is an absolute measure of dispersion. When comparison has to be made between two series then the relative measure of dispersion, known as coeff.of variation is used.

Coefficient of Variation, CV is defined and given by the following function:

Formula

${CV = \frac{\sigma}{X} \times 100 }$

Where −

${CV}$ = Coefficient of Variation.
${\sigma}$ = standard deviation.
${X}$ = mean.

Example

Problem Statement:

From the following data. Identify the risky project, is more risky:

Year	1	2	3	4	5
Project X (Cash profit in Rs. lakh)	10	15	25	30	55
Project Y (Cash profit in Rs. lakh)	5	20	40	40	30

Solution:

In order to identify the risky project, we have to identify which of these projects is less consistent in yielding profits. Hence we work out the coefficient of variation.

Project X			Project y
${X}$	${X_i - \bar X}$ ${x}$	${x^2}$	${Y}$	${Y_i - \bar Y}$ ${y}$	${y^2}$
10	-17	289	5	-22	484
15	-12	144	20	-7	49
25	-2	4	40	13	169
30	3	9	40	13	169
55	28	784	30	3	9
${\sum X = 135}$		${\sum x^2 = 1230}$	${\sum Y = 135}$		${\sum y^2 = 880}$

Project X

${Here\ \bar X= \frac{\sum X}{N} \\[7pt] = \frac{\sum 135}{5} = 27 \\[7pt] and\ \sigma_x = \sqrt {\frac{\sum X^2}{N}} \\[7pt] \Rightarrow \sigma_x = \sqrt {\frac{1230}{5}} \\[7pt] = \sqrt{246} = 15.68 \\[7pt] \Rightarrow CV_x = \frac{\sigma_x}{X} \times 100 \\[7pt] = \frac{15.68}{27} \times 100 = 58.07}$

Project Y

${Here\ \bar Y= \frac{\sum Y}{N} \\[7pt] = \frac{\sum 135}{5} = 27 \\[7pt] and\ \sigma_y = \sqrt {\frac{\sum Y^2}{N}} \\[7pt] \Rightarrow \sigma_y = \sqrt {\frac{880}{5}} \\[7pt] = \sqrt{176} = 13.26 \\[7pt] \Rightarrow CV_y = \frac{\sigma_y}{Y} \times 100 \\[7pt] = \frac{13.25}{27} \times 100 = 49.11}$

Since coeff.of variation is higher for project X than for project Y, hence despite the average profits being same, project X is more risky.

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