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# CV, Coefficient of variation

Go to solution Solved by Kavitha Sundar,

Coefficient of Variation (CV)

Coefficient of Variation (CV) - is a statistical measure of the dispersion of observations in a data set around the mean. It is calculated as the ratio of the standard deviation to the mean and is usually expressed in percentage. It helps in comparison of variation in two or more data sets with different means and standard deviations respectively. Higher the CV, the more is the spread of the data around its mean.

An application oriented question on the topic along with responses can be seen below. The best answer was provided by Kavitha Sundar on 24th November 2017.

## Question

Q52. Explain the use of Coefficient of Variation with examples.

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Q52. Explain the use of Coefficient of Variation with examples.

Coefficient of variation is the ratio of standard deviation to the mean. The higher the CV, the more is the spread of the data around its mean and the team or process is very unstable or ununiformed.  In simple, it is % variation in mean, where SD is total variation in mean.  It is a measure of relative variability.

This is used to compare variations of two or more data sets.

For Eg. If I have to compare results of two groups lets say Group A & Group B. Group A has CV of  25% and Group B has CV of 18%. This says that the Group A has more variability to its mean.

Formula for CV = SD / mean

It can be expressed as in percentage %. Hence the formula for CV can be multiplied by 100.

Benefits of CV –

1.      Measure of Precision – It is used to describe the level of variations existing within the population independently from the absolute values of the individual observations. If the population is same, where you have to find out the variation, then use Standard deviation. If the population is different, use this CV to estimate the spread or variability from its relative mean.

Eg. If Male and female elephant group is compared, then use SD to find out the variation.

If you have to compare the male elephant population with male mice population, then use CV.

In simple, when the two groups differ significantly, use CV as  a measure. It is to assess the precision of the measurement technique.

2.      Measure of Repeatability -  CV is used to measure the repeatability within the group and not the validity / reproducibility. It is used in a way to tell you the degree of association but not agreement. Measuring repeatability with out validity is a useful analysis. When assessing the measurement error, CV value depends on both the variability between sampling units and variability between repeated readings from the same user. If we have to select the variable group of sampling units, then the repeatability CV would be higher than taking up for a homogenous group. The aim is to be maximize the repeatability within the given situation.

Eg. Used by Microbiologists and pharmacist to evaluate the intra assay and inter assay CV, in order to bring down the CV value to make it acceptable.

3.      Consistency of data – CV is used to understand and confirm the consistency of data. Consistency means uniformity in the values of the data set. How consistent the values are from the mean of the data set is measured. As small as the CV means the data is uniform or consistent.

Eg. If the temperature of an adult is to be compared to the same of a newborn, certain values are recorded In the real time for some time. Hence CV for adult is 10% and CV for newborn is of 2%. As for Newborn the CV is smaller, the variation in the data is very minimal. Means the data for Newborn is consistent than adult.

4.      Indicator for Risk Assessment – It is a better indicator for all levels of risk assessment. In any type of situation, if we were to assess the risk, this would be the right tool.

Eg. If Bank A gives a rate of interest at 20% and Bank B gives u at 10%, with a standard deviation of 10% and 5% respectively. Which bank is better to take a loan?

As Bank B has SD of 5%, the Rate of interest is minimal for a longer run to balance his needs by the customer. Hence customer would prefer Bank B.

5.      Decision making: If the team has to downsize due to high cost, the decision is to eliminate some of the team members. CV Is a useful tool where it tells us in which team ,there is more of variability, which team receives higher cost , etc to make strategic decisions.

Eg. Organization has two functions – coding and billing with 40 and 65 employees in it. They earn around \$450 and \$350 respectively with SD as 7 and 9.

Q – A) which section has a higher salary package? Which function has highest variability?

a)    Salary for Coding = 40 *450 = 18000

Salary for billing = 65 * 350 = 22750

So, Salary for billing is higher.

CV for Coding =( 7/450) *100 = 1.6%

CV for billing =( 9/350) *100 = 2.6%

Billing is more of variability since it has more CV.

CV is useful only for the calculations, when the mean of sample population is not zero.

Lets assume, if the sample mean is equal to zero, then the denominator would become zero. Hence the CV gets nullified.

Yes. CV is useful if all the data points or atmost of the data points share the same value as of plus or minus sign.

Conclusion:

CV has its own use and limitations. Hence it should used to carefully in

1.      Estimating the variation 2 different populations

2.      Estimating the 2 set of categories variations.

3.      Risk assessment indicator

4.      Decision making

Thanks

Kavitha

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The Coefficient of variation (CV) is basically the ratio of the standard deviation to the mean in a given data set. It is used as a measure of relative variability and allows to compare the range or spread of many data sets.

Just to understand it very easily let us take the examples of a QSR which is trying to find the best bet to open outlet  , between 2 territories  with favourable traits, proport- Traits like population, SocioEconomic level of population, Competition, Prospect growth in the territory etc. The Real Estate team has cited 20  locations and their rentals in both territories. NOW, the decision is narrowed down on the rentals of the sites as the sales projected in both the territories is more or less proportionate to the respective rentals. It would now be prudent to open the outlets in the territory where the difference in rentals are not very high amongst the outlets. This helps to budget the costs and the disparity of rentals is not much and hence the allocations of budgets for project work become almost even for all outlets.  The management wants to understand in which territory the variation in rentals is higher. Then the territory with lesser variation in rentals will be the choice to open outlets.

 Territory 1 Territory 2 Average Rentals (Mean)- Average Rentals (Mean)- 120000 200000 Standard Deviation in Rentals Standard Deviation in Rentals 2000 3000 Coefficient of Variation Coefficient of Variation =2000/120000= 0.016 =3000/200000 = 0.015

In territory 1, The Average or mean of the rentals is Rs. 1.20 Lacs and the standard deviation is 2000/- . Ie. Rentals of most of the outlets are in the range from 1.22 Lacs to 1.18 Lacs

In Territory 2 , The Average or Mean of the rentals is 2 Lacs and the standard deviation is 3000/-ie. Rentals of most of the outlets are in the range from 2.03 lacs to 1.97 Lacs.

It is obvious from the coefficient of variation that the range of rentals are higher in Territory 1. The management will hence try to work on the Territory #2

Likewise, CV is used in many other situations like:

-          To compare relative risks in process during the Design stage as it be applied to any kind of probability distribution.

-          Since it is a statistical measure that is normalized and hence has no dimension, It is used as a measure of dispersion and used instead of standard deviation to compare data sets with different measures and significantly different means.

-          Most commonly CV is used to measure the volatility in the prices of stocks and securities

In conclusion , is useful in any study that demonstrates exponential distribution i.e. It helps to show when distributions are considered low – variance and when they are said to be high – variance.

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The Coefficient of Variation (CV) that is also known as the Relative Standard Deviation (RSD) is the ratio of the Standard Deviation of a Dataset to its Mean, popularly expressed as a percentage.

The CV is a useful metric to compare variations of two datasets with different means. This metric has the advantage of all ratio coefficients in that it acts as a “common denominator” when comparing diverse data sets.

Some of the relevant features of the CV are that it is independent of the order of values in the dataset and that it is relevant with only positive values of the dataset.

The applications of the CV are many and include:

1.    Evaluation of risk of investments vis-à-vis the return – Lower the CV, better the risk – return match

2.    Assess the homogeneity of solid powder mixtures – Closer the CV to the defined norm, more homogeneous is the mixture

3.    Measuring specific properties of chemicals or proportions of specific materials in mixtures

4.    Calculation of economic disparity of a community or a group

5.    Comparison of performance of two batches in a batch processing industry

The CV can used to test hypotheses through Levene’s test.

The general interpretation of the CV is that lower the CV lesser the variation relative to the mean and therefore a lower value of CV is preferable.

While the advantages of CV are many, one of its disadvantages is that it is usable with only parameters on a ratio scale but not on an ordinal, value or categorical scales. Further if the dataset consists of both positive and negative values, the mean tends to zero and CV tends to infinity. If the two datasets being compared, contain values or on a scale related to one another, then the CV would be different for both the datasets in spite of the data sets being related. (E.g. The CVs of two data sets measuring the temperature of the same substances but expressed as Celsius in one data set and as Fahrenheit in another).

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Coefficient of variation (CV) is a measure of relative variability with respect to the mean of the population.

It is the ratio of the standard deviation to the mean (average).

COV= (SD/M) * 100

COV:- Coefficient of variation

SD:- Standard Deviation

M:- Mean

COV is used to describe relative variability within a population independently of absolute values of observation.

If the absolute values are similar, than both the samples can be compared with standard deviation.

But when both (the absolute values) population mean are different than we should use COV , or when the measured variables are completely different than we can use COV for comparing tow population.

COV should not be used to compare population data which are on interval scale, which can have negative values.

As an example,

For packaging machine, some data were taken for a 2.5 Kg pack size, and 3 kg pack size.

We want to know for which pack size machines is more stable. We can use COV for this analysis.

For 2.5 Kg

Mean:- 2502.2321, SD:- 4.7695, COV = (SD/M)*100 = 0.1906%

For 3 Kg Pack size

Mean:- 3002.5667

SD:- 5.94635

COV = (SD/M)*100 = 0.19%.

In this example we can see that there is difference in SD for both the pack size, but when we compare COV, there is not is very marginal difference. Hence we can say that machine is stable for both 3 Kg pack size and 2.5 Kg pack size.

Example 2:-

In a process industry there is a machine which gives actual shape to the product. Different products have different shapes and different weight.

Now we want to check whether the machine performs equally for both the type of products or not. We can compare COV for this.

For Product A:-

Mean:- 13.61, SD:- 0.383, COV:- 2.81 %

For product B

Means:- 59.82, SD:- 2.844, COV:- 4.72%

So for product A we got COV 2.81% and for product B we got COV 4.72%.

Hence, we can say that machine gives less variation when we run product A.

Similar studies can also be done when there are different variables.

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Coefficient of Variation (CV) is used to compare variability between different measures. Coefficient of variation is the ratio of standard deviation to mean values.  CV helps to assess the variability between tests conducted. Is sample X has 10% CV against sample Y which has 15% CV, then sample Y is having higher variance completed to sample X in relation to their mean. CV comes in handy in cases where the samples under comparison have mean values of different range. In this case just going by standard deviation may not provide better insight, so CV values will give better perspective as they account ratio between standard deviation and mean. One point to consider is CV will help only in case of positive values of ratio data, it will not be meaningful in case of interval scales like temperature measures in Celsius or Fahrenheit.

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Coefficient of Variation (COV) is defined the ratio between standard deviation and the mean. COV helps in comparing the relative dispersion between two data sets, when the means are different.

The precision for a data will differ depending upon the largeness of the mean.

For instance, consider the following 2 situations:

1.    We are dealing with wheels whose mean diameter is 5mm and it has a standard deviation of 0.5mm.

2.    We are dealing with wheels whose mean diameter is 500mm and has a standard deviation of 2mm.

Which of the above has higher dispersion? The standard deviation is higher for the second instance. However, the COV for the 1st instance is 0.5 / 5 = 0.1; whereas the COV for the 2nd instance is 2 / 500 = 0.004.

From a practical significance, standard deviation of 2mm for a mean diameter of 500mm is more tolerable than standard deviation of 0.5 for a mean diameter of 5mm

This illustrates that going by just the standard deviation, we could not have meaningfully compared the dispersion.

The COV gives a quantified and comparable measure of dispersion.

Another example:

1.     I have stock whose mean value is Rs.75 with a standard deviation of Rs.6.

2.     I have stock whose mean value is Rs.800 with a standard deviation of Rs.64

Which of the above carries more risk?

The second stock obviously has a higher standard deviation. However,

COV for 1st stock = 6 / 75 = 0.08

COV for 2nd stock = 64 / 800 = 0.08

The COVs for both the stock is equal to 0.08, and hence the risks based on dispersion for both these stocks are comparable.

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COV:

The coefficient of variation (COV) is a measure of relative variability. It is the ratio of the standard deviation to the mean.

COV = STD Dev/ average ( %)

like the formula says, if the STD Dev is higher, then the COV will also be higher.

Hence in usual terms, less the variance, less is the coefficient.

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Coefficient of Variation (CV) is a statistical measure of the dispersion or spread of data points in a data set around the mean value of the data set.

It is the ratio of standard deviation to the mean. It is also called relative standard deviation (RSD).

Uses of Coefficient of Variation:

Example 1:  Comparing the spread around mean in two or more data sets when the Means of the individual data sets vary.

CV especially helps in comparing the degree of variation from one data set to another, when Standard deviation (SD) between data the sets may be similar but mean of the data sets may be drastically different.

Set A: Let Mean = 60 and SD = 10; Then CV = 6

Set B: Let Mean = 30 and SD = 11; Then CV = 2.7

Looking at SD Sets A and B may seem similar but looking at the CV it can be seen that Set A has a much higher spread around mean than Set B.

Example 2: Evaluating and pick stocks in the Investment Market.

Volatility in stock market indicates the stock’s value spread. High Volatility means the spread could be over a large range of values, while Low Volatility means the value does not change dramatically, but changes at a steady pace over a period. Higher the volatility, the riskier the stock. Volatility can either be measured by using the standard deviation or variance between returns from the same stock.

CV helps investors assess the Volatility (risk) to expected returns for their portfolio.

Analyzing Data for past 10 years for three stocks A, and C as indicated below:

Stock A: Standard Deviation is 15.5% and Average Annual Returns is 3.5%, Then CV = 15.5/4.5 = 3.4

Stock B: Standard Deviation is 27.3% and Average Annual Returns is 6.4%, Then CV = 27.3/6.4 = 4.3

Stock C: Standard Deviation is 16.7% and Average Annual Returns is 5.1%, Then CV = 16.7/5.1 = 3.3

A Risk averse investor may pick Stock A or C based on similar CV, while a risk taking investor might go for Stock B.

Example 3: Selection of a material between A and B, that has uniform behaviour at different temperatures for thermal applications.

Data can be collected for the materials A and B under various temperatures in the required range. Subsequently, Mean, SD and CV can be calculated from the collected data sets for Materials A and B. The Material with the lower CV would be more suited for the application.

Example 4: Out of 2 Restaurants, evaluating which one has a more consistent delivery time.

Delivery Time sets for the two restaurants can be collected and the respective CV’s calculated. The one having a lower CV is more consistent.

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Coefficient of Variation(cv) Definition:  CV is the percentage variation in mean. Standard Deviation(SD) is the total variation in in the mean. When we want to compare more than one series then we use CV.  the more large CV is, the more variable the series is that is less stable/uniform, and the small CV is the less variable the series is i.e more stable/uniform.

Formula: CV = SD/Mean  that is it the ratio of SD and Mean.

Only for NON-ZERO mean CV gets calculated.

Example:

Series A= (5, 9.5, 4.9, 1.85, 5.25, 7.05, 6.0)

No.of Samples 7

Mean 5.6499

Standard Deviation 2.327

Coefficient of Variance 0.4118

Step by Step Calculation:

Input: 5, 9.5, 4.9, 1.85, 5.25, 7.05, 6.0

Mean(µ) = (5 + 9.5 + 4.9 + 1.85 + 5.25 + 7.05 + 6.0)/7
Mean = 39.55/7
µ = 5.6499

= √( (1/7-1) * (5-5.6499)2+( 9.5-5.6499)2+( 4.9-5.6499)2+( 1.85-5.6499)2+( 5.25-5.6499)2+( 7.05-5.6499)2+( 6.0-5.6499)2)
= √( (1/6) * (-0.64992 + 3.85012 + -0.74992+ -3.79992 + -0.39992 + 1.40012 + 0.35012))
= √( (1/6) * (0.42237001 + 14.82327001 + 0.56235001 + 14.43924001 + 0.15992001 + 1.96028001 + 0.12257001))
= √ 5.414929
σ= 2.327

Coefficient of Variance = σ/µ
= 2.327 / 5.6499
Coefficient of Variance = 0.4118

Series B: (4,59.5, 4.9, 4.85, 5.25, 6.05, 6.0)

No.of Samples 7

Mean 12.9357

Standard Deviation 20.5451

Coefficient of Variance 1.5882

Step by Step Calculation:

Input: 4, 59.5, 4.9, 4.85, 5.25, 6.05, 6.0

Mean(µ) = (4 + 59.5 + 4.9 + 4.85 + 5.25 + 6.05 + 6.0)/7
Mean = 90.55/7
µ = 12.9357

= √( (1/7-1) * (4-12.9357)2+(59.5-12.9357)2+( 4.9-12.9357)2+( 4.85-12.9357)2+( 5.25-12.9357)2+( 6.05-12.9357)2+( 6.0-12.9357)2)
= √( (1/6) * (-8.93572 + 46.56432 + -8.03572 + -8.08572 + -7.68572 + -6.88572+ -6.93572))
= √( (1/6) * (79.84673449 + 2168.23403449 + 64.57247449 + 65.37854449 + 59.06998449 + 47.41286449 + 48.10393449))
= √ 422.10113401
σ= 20.5451

Coefficient of Variance = σ/µ
= 20.5451 / 12.9357
Coefficient of Variance = 1.5882

Series C: (4,59.5, 18.9, 20.85, 5.25, 6.05, 1.0)

No.of Samples 7

Mean 16.5071

Standard Deviation 20.4366

Coefficient of Variance 1.238

Step by Step Calculation:

Input: 4, 59.5, 18.9, 20.85, 5.25, 6.05, 1.0

Mean(µ) = (4 + 59.5 + 18.9 + 20.85 + 5.25 + 6.05 + 1.0)/7
Mean = 115.55/7
µ = 16.5071

= √( (1/7-1) * (4-16.5071)2+(59.5-16.5071)2+( 18.9-16.5071)2+( 20.85-16.5071)2+( 5.25-16.5071)2+( 6.05-16.5071)2+( 1.0-16.5071)2)
= √( (1/6) * (-12.50712 + 42.99292 + 2.39292 + 4.34292 + -11.25712 + -10.45712 + -15.50712))
= √( (1/6) * (156.42755041 + 1848.38945041 + 5.72597041 + 18.86078041 + 126.72230041 + 109.35094041 + 240.47015041))
= √ 417.65461956
σ= 20.4366

Coefficient of Variance = σ/µ
= 20.4366 / 16.5071
Coefficient of Variance = 1.238

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Coefficient of variation is the ratio of standard deviation and the mean. It can be applied to any kind of quantitative distribution. Lesser the cv is better one. Its main use to compare relative risk.  We can use in investment and finance . the main advantage of cover is that it is unit less.

This quality of unit less makes it different an more useful tha  standard deviation analysis. It can help demonstrate when distributions are considred low variance and when  they are considered high variance. In in eland finance Coventry can be used to evaluate risk where we should invest money. Cover is better indicator of relative  risk particularly among different level of risk of different securities. For example we are thinking to invest money in two stocks A and B,  which are having different return suppose Stock A has expected return of 20% and stock B has expected return of 15 %.  Now we are confused which is better.. Let's see standard deviation.  Stock has SD of 10% and stock B has 5%. So COV of stock B is less(5/15)  isess than Coventry of stock A (10/20). Lesser the COV is better,  so we will prefer to invest in stock B.

Other use of Coventry is in relative theory,  queuing theory,

Reliability theory,  exponential distribution,  in biological labs and in measure of economic inequality etc...

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Coefficient of Variation (CoV) is the ratio of Standard Deviation and the Mean. It is a unitless ratio. CoV is an overall indicator of relative risk. For example, there are two different investment options. Stock A has an expected return of 15% and Stock B has an expected return of 10%. Stock A has a standard deviation of 10% whereas Stock B has a standard deviation of 5%. Which one is a better investment? If we compare the CoV of both the options, it shows that Stock B is a better option, since CoV of Stock B is 5/10 i.e. 0.5 whereas CoV for Stock A is 10/15 i.e. 0.67. Lesser the CoV more consistent are the returns.

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Coefficient of Variation:

The Coefficient of Variation is a measure of Relative variability. It is the Ratio of the Standard Deviation to the Mean. It is a useful method for comparing the Degree of Variation from One Data Series to another, even if the means are drastically different from one another. It has the following characteristics:

-Measure of relative variation

- Shows Variation relative to mean

- Used to compare 2 or more groups

How to calculate Coefficient of Variation:

The main purpose of finding the Coefficient is used to study of Quality assurance by measuring the spread of the Population data of a Probability or frequency distribution or by determining the content or Quality of the sample data of substances

Calculation of Coefficient of variation:

- Calulate the Mean of the Data Set

- Calculate the Sample SD of the Data Set

- Finding the Ratio of the Sample SD to mean brings the Coefficient of Variation [CV] of the Data set

Formulas to calculate coefficient of variation:

Examples for Coefficient of Variation:

Calculate the relative variability (coefficient of variance) for the samples 60.25, 62.38, 65.32, 61.41, and 63.23 of a population

Solution:
Step by step calculation:
Step 1: calculate mean
Mean = (60.25 + 62.38 + 65.32 + 61.41 + 63.23)/5
= 312.59/5
= 62.51

Step 2: calculate standard deviation
= √( (1/(5 - 1)) * (60.25 - 62.51799)2 + (62.38 - 62.51799)2 + (65.32 - 62.51799)2 + (61.41 - 62.51799)2 + (63.23 - 62.51799)2)
= √( (1/4) * (-2.267992 + -0.137989992 + 2.802012 + -1.107992 + 0.712012))
= √( (1/4) * (5.14377 + 0.01904 + 7.85126 + 1.22764 + 0.50695))
= √ 3.68716
σ = 1.92

Step 3: calculate coefficient of variance
CV = (Standard Deviation (σ) / Mean (μ))
= 1.92 / 62.51
= 0.03071

2. A company has two sections with 40 and 65 employees respectively. Their average weekly wages are \$450 and \$350. The standard deviation are 7 and 9. (i) Which section has a larger wage bill?. (ii) Which section has larger variability in wages?
Solution:
(i) Wage bill for section A = 40 x 450 = 18000
Wage bill for section B = 65 x 350 = 22750
Section B is larger in wage bill.
(ii) Coefficient of variance for Section A = 7/450 x 100 =1.56 %
Coefficient of variance for Section B = 9/350 x 100 = 2.57%
Section B is more consistent so there is greater variability in the wages of section A.

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Coefficient of variance can be used to compare the spread of two different populations, with values varying over different ranges. Being a unitless value it finds application in several areas as highlighted in several answers. The biggest drawback is that it cannot be used when the mean of a sample is zero.

The three best answers are for Kavitha, Rajesh and Mohan. The chosen best answer is Kavitha's which outlines several examples in addition to the pros and cons.

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