Coefficient of Variation Formula

Coefficient of Standard Deviation Formula

According to the statistics, the coefficient of variation calculator is used to obtain the ratio of standard deviation to the mean. The coefficient of variation formula is also known as the relative standard deviation formula. This is a standardized formula for calculating the dispersion of a probability distribution or frequency distribution. If the value obtained from the calculator coefficient of variation is lower, it shows that the data has less variability and high stability.  

The variation coefficient formula is given by, 

Coefficient of Variation = \[\frac{\text{Standard Deviation}}{Mean}\] * 100

The formula for standard deviation may vary as per the samples and population data type, 

Sample Standard Deviation = \[\sqrt{\frac{\sum_{i=1}^{n}(X_{i} - \bar{X})^{2}}{n - 1}}\]

Population Standard Deviation =  \[\sqrt{\frac{\sum_{i=1}^{n}(X_{i} - \bar{X})^{2}}{n}}\]

Here, 

X\[_{i}\] represents the terms given in the data

\[\bar{X}\] represents the mean value

n represents the total number of terms. 


Problems Based on Coefficient of Variance 

Example 1: An exam was conducted with two multiple-choice tests with different conditions. In the first test, a typical multiple-choice test is determined. In the second test, alternative choices (i.e. incorrect answers) are generally assigned to test takers. Calculate the Coefficient of Variance. 


The result of the two tests is given in the table. 



Regular Test

Random Answers

Mean

78.4

63.7

Standard Deviation 

10.3

7.1


Solution:

Coefficient of variation equation (CV) = \[\frac{\text{Standard Deviation}}{Mean}\] * 100


By applying the above values in the cv formula in Statistics, we can calculate the coefficient of variation. 



Regular Test

Random Answers

Mean

78.4

63.7

Standard Deviation 

10.3

7.1

Coefficient of Variation 

13.13

11.14


The coefficient of variation of the regular test is 13.13

The coefficient of variation of the random answer is 11.14


Example 2: Calculate the coefficient of variation of the following sample set of numbers.

{1, 4, 9, 11, 15, 30, 55, 98}.

Solution: 

The given sample set is {1, 4, 9, 11, 15, 30, 55, 98}

The formula for Sample standard deviation σ = \[\sqrt{\frac{\sum_{i=1}^{n}(X_{i} - \bar{X})^{2}}{n - 1}}\] 

Mean value of the sample set = (1+4+9+11+15+30+55+98) / 8 = 223/8 = 27.87

The mean value of the sample set \[\bar{X}\]= 27.87

\[\sum_{i=1}^{n}\](X\[_{i}\] - \[\bar{X}\])\[^{2}\] = (1 - 27.87)\[^{2}\] + (4 - 27.87)\[^{2}\] + (9 - 27.87)\[^{2}\] + ( 11 - 27.87)\[^{2}\] + (15 - 27.87)\[^{2}\]

+ (30 - 27.87)\[^{2}\] + (55 - 27.87)\[^{2}\] + (98 - 27.87)\[^{2}\]

= 721.99 + 569.77 + 333.79 + 284.59 + 165.63 + 4.53 + 736.03 + 4918.21

= 7734.51

σ = \[\sqrt{\frac{7734.51}{7}}\] → \[\sqrt{1104.93}\]

σ = 33.2404

Coefficient of variation =  33.2404 / 27.87 * 100

Coefficient of variation = 119.26

FAQs (Frequently Asked Questions)

Q1. What is the Coefficient of Variation in Statistics?

Ans: The coefficient of variation in statistics is the ratio of the standard deviation to the mean. The mean value is inversely proportional to the coefficient of standard deviation formula. This means, greater the level of dispersion around the mean, the higher the value obtained from the coefficient of variation calculator. The lower the value from the calculator coefficient of variation, the more precise to estimate. 

Q2. Why Do We Calculate a Coefficient of Variation?

Ans: The value obtained from a formula of coefficient of standard deviation, shows the variability of data in a sample in relation to the mean of the population. In finance, the coefficient of variance calculator helps the investor to determine how much risk or volatility is assumed in comparison to the amount of profit or returns expected from investments.

Q3. Is the High Coefficient of Variation Good or Bad?

Ans: As per the rule of thumb, the cv formula in statistics  The value of a CV ≥ 1 shows a relatively high variation, meanwhile a CV < 1  is considered as a low variance. This means that distributions obtained from the sample coefficient of variation calculator higher than 1 are considered to be high variance and the value from a coefficient of variation equation lower than 1 are considered to be low-variance.