How to calculate Coefficient of Variation?

Statistics is the process by which the data is collected and analyzed. The coefficient of variation in statistics explain 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

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 consists of the two different results. For example, if we are comparing 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 find 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%

Coefficient of Variation in the Context of 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 

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.