How to calculate Coefficient of Variation?
How to calculate the Coefficient of Variation? Statistics is the process by which the data is collected and analyzed. The coefficient of variation in statistics is explained as the ratio of the standard deviation to the arithmetic mean, for instance, the expression standard deviation is 15 % of the arithmetic mean is the coefficient variation.
This article provides the steps to calculate the coefficient of variation along with solved examples and practice problems.
Table of Content
- What is the Coefficient of variation?
- Steps to Calculate the Coefficient of Variation
- How to calculate Coefficient of Variation in context to Finance
- Sample Problems on How to calculate Coefficient of Variation
- Practice Problems on How to calculate Coefficient of Variation?
What is the Coefficient of variation?
The coefficient of variation is the measure of relative variability. The coefficient of variation is the ratio of the standard deviation to the mean.
It’s very useful if we want to compare the results from the two different research or tests that consist of the two different results. For example, if we are comparing the results of two different matches that have two completely different scoring methods. Like if sample X has a CV of 15% and sample Y has a CV of 30%, it would be said that sample Y has more variation, relative to its mean. It helps us to provide Relatively simple and quick tools that help us to compare the data of different series.
Formula to calculate the coefficient of variation:
Coefficient of Variation = (Standard Deviation / Mean) × 100
In symbols: CV = (SD/x̄) × 100
Steps to Calculate the Coefficient of Variation
For the steps to calculate the coefficient of variation let’s watch an example.
Example: Two boys are playing games cricket and football the scores scored by the boys are as follows:-
Football | Cricket | |
Mean | 24 | 46 |
SD | 13 | 35 |
Step 1: Now, divide the standard deviation by mean for sample 1 (football)
13/24 = 0.5416
Step 2: Now, multiply step 1 with 100
0.5416×100=54.16%
Step 3: Now for sample 2, divide the standard deviation by mean
35/46=0.7608
Step 4: Now, multiply step 2 with 100
0.7608×100= 76.08%
How to calculate Coefficient of Variation in context to Finance
It helps us in the investment selection process that’s why it is important in terms of finance. In the financial matric, it shows us the risk-to-reward ratio means here the standard deviation/volatility shows the risk of the investment and mean is shown as the expected reward of the investment. The investors in the company identify the risk-to-reward ratio of each one of the security to develop an investment decision. In this, the low coefficient is not favourable when the average expected return is below the value of zero
The formula for the calculation of the coefficient of variation in the context of finance :
Coefficient of variation = σ/μ × 100%
Where,
σ – the standard deviation
μ – the mean
Sample Problems on How to calculate Coefficient of Variation
Problem 1: The standard deviation and mean of the data are 9.7 and 17.8 respectively. Find the coefficient of variation.
Solution:
SD/σ = 9.7
mean/μ = 17.8
Coefficient of variation = σ/μ × 100%
= 9.7/17.8 × 100
Coefficient of variation = 54.4%
Problem 2: The standard deviation and coefficient of variation of data are 2.5 and 36.7 respectively. Find the value of the mean.
Solution:
C.V=36.7
SD/σ= 2.5
Mean/x̄=?
C.V = σ/x̄ × 100
36.7 = 2.5 / x̄ ×100
x̄ = 2.5/36.7×100
x̄ = 6.81
Problem 3: If the mean and coefficient of variation of data are 24 and 56 respectively, then find the value of standard deviation?
Solution:
C.V=56
SD/σ=?
Mean/x̄= 24
C.V= σ/x̄ × 100
56 = σ/ 24 × 100
σ = 24×56/100
σ = 13.44
The standard deviation is 13.44
Problem 4: The mean and standard deviation of marks obtained by 40 students of a class in three subjects Mathematics, English and economics are given below.
Subject | Mean | Standard Deviation |
Maths | 56 | 11 |
English | 78 | 16 |
Economics | 69 | 13 |
Which of the three subjects shows the highest variation and which shows the lowest variation in marks?
Solution:
Coefficient of variation for maths =σ/x̄ × 100
σ=11
x̄=56
C.V = 11/56×100
Coefficient of variation for maths= 19.64%
Coefficient of variation for english= σ/x̄ × 100
σ=16
x̄=78
C.V = 16/78×100
Coefficient of variation for english= 20.51%
Coefficient of variation for economics= σ/x̄ × 100
σ=13
x̄=69
C.V = 13/69×100
Coefficient of variation for economics =18.84%
The highest variation is in english.
And the lowest variation is in economics.
Problem 5: The following table gives the values of mean and variance of heights and weights of the 10th standard students of a school.
Height | Weight | |
Mean | 166cm | 65.60cm |
Variance | 85.70cm | 39.9kg |
Which is more varying than the other?
Solution:
Coefficient of variation for heights
Mean x̄1= 166cm, variance σ1² = 85.70 cm²
Therefore standard deviation σ1 = 9.25
Coefficient of variation
C.V1= σ/x̄ × 100
= 9.25/166×100
C.V1 = 5.57% (For heights)
Coefficient of variation for weights
Mean x̄2= 65.60kg , variance σ2² = 39.9 kg²
Therefore standard deviation σ2 = 6.3kg
Coefficient of variation
C.V1= σ/x̄ × 100
= 6.3 / 65.60×100
C.V2=9.54% (For weight)
C.V1 = 5.57% and C.V2 = 9.54%
Since C .V2 > C .V1 , the weight of the students is more varying than the height.
Problem 6: If the mean and coefficient of variation of data are 16 and 40 respectively, then find the value of standard deviation?
Solution:
C.V=40
SD/σ=?
Mean/x̄= 16
C.V= σ/x̄ × 100
40 = σ/ 16 × 100
σ= 16×40/100
σ= 6.4
Problem 7: The mean and standard deviation of marks obtained by 40 students of a class in three subjects Mathematics, English and economics are given below.
Subject | Mean | Standard Deviation |
Social Studies | 65 | 10 |
Science | 60 | 12 |
Hindi | 57 | 14 |
Which of the three subjects shows the highest variation and which shows the lowest variation in marks?
Solution:
Coefficient of variation for social studies = σ/x̄ × 100
σ=10. x̄=65
C.V = 10/65×100
Coefficient of variation for Social studies= 15.38%
Coefficient of variation for Science= σ/x̄ × 100
σ=12 x̄=60
C.V = 12/60×100
Coefficient of variation for science = 20%
Coefficient of variation for Hindi = σ/x̄ × 100
σ=14 x̄=57
C.V = 14/57×100
Coefficient of variation for Hindi = 24.56%
The highest variation is in economics.
And the lowest variation is in maths.
Practice Problems on How to calculate Coefficient of Variation?
1. Calculate the coefficient of variation for a data set with the following values: 10, 15, 20, 25, 30.
2. Given a data set with the following values: 8, 12, 16, 20, 24, calculate the coefficient of variation.
3. Calculate the coefficient of variation for a sample with the following standard deviation and mean: standard deviation = 4 and mean = 18.
4. If a data set has a mean of 50 and a coefficient of variation of 0.1, what is the standard deviation of the data set?
5. Given a data set with the following values: 2, 4, 6, 8, 10, find the coefficient of variation.
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