Coefficient of Variation Formula

How to Find a Coefficient of Variation

What is the Coefficient of Variation?

The coefficient of variation (CV) is a measure of relative variability. It is the ratio of the standard deviation to the mean (average). For example, the expression “The standard deviation is 15% of the mean” is a CV.

The CV is particularly useful when you want to compare results from two different surveys or tests that have different measures or values. For example, if you are comparing the results from two tests that have different scoring mechanisms. If sample A has a CV of 12% and sample B has a CV of 25%, you would say that sample B has more variation, relative to its mean.

The coefficient of variation formula is useful particularly in those cases where we need to compare results from two different surveys having different values. In statistics, the Coefficient of variation formula (CV), also known as relative standard deviation (RSD), is a standardized measure of the dispersion of a probability distribution or frequency distribution. If the value of the coefficient of variation is lower then it indicates that the data has less variability and high stability. The general steps to find the coefficient of variation are as follows:

• Step 1: Check for the sample set.
• Step 2: Calculate standard deviation and mean.
• Step 3: Put the values in the coefficient of variation formula, CV =

Now let us understand this concept with the help of a few examples.

Example: Two plants C and D of a factory show the following results about the number of workers and the wages paid to them.

Using coefficient of variation formulas, find in which plant, C or D is there greater variability in individual wages.

Formula

The formula for the coefficient of variation is:

Coefficient of Variation = (Standard Deviation / Mean) * 100.
In symbols: CV = (SD/x̄) * 100.

Multiplying the coefficient by 100 is an optional step to get a percentage, as opposed to a decimal.

Coefficient of Variation Formula

There are two formulas for the coefficient of variation. These are the population coefficient of variation and the sample coefficient of variation. Population, in statistics, is the entire group that is under consideration. In other words, the population is used to denote the complete data set. When a specific part is chosen from this population it is known as the sample. The sample is used to represent the entire population of the study. The population mean and the sample mean will always be the same. However, as the value of the standard deviation differs thus, there are two coefficient of variation formulas. These are given below:

Coefficient of Variation Example

A researcher is comparing two multiple-choice tests with different conditions. In the first test, a typical multiple-choice test is administered. In the second test, alternative choices (i.e. incorrect answers) are randomly assigned to test takers. The results from the two tests are:

Trying to compare the two test results is challenging. Comparing standard deviations doesn’t really work, because the means are also different. Calculation using the formula CV=(SD/Mean)*100 helps to make sense of the data:

Looking at the standard deviations of 10.2 and 12.7, you might think that the tests have similar results. However, when you adjust for the difference in the means, the results have more significance:
Regular test: CV = 17.03
Randomized answers: CV = 28.35

The coefficient of variation can also be used to compare variability between different measures. For example, you can compare IQ scores to scores on the Woodcock-Johnson III Tests of Cognitive Abilities.

Note: The Coefficient of Variation should only be used to compare positive data on a ratio scale. The CV has little or no meaning for measurements on an interval scale. Examples of interval scales include temperatures in Celsius or Fahrenheit, while the Kelvin scale is a ratio scale that starts at zero and cannot, by definition, take on a negative value (0 degrees Kelvin is the absence of heat).

Solution:

To Find: Which plant has greater variability.
For this, we need to find the coefficient of variation. The plant that has a higher coefficient of variation will have greater variability.
Coefficient of variation for plant C.

Using coefficient of variation formula,

CV = (σ/μ) × 100, μ≠0

CV = (9/2500) × 100

CV = 0.36%

Now, CV for plant D

CV = (σ/μ) × 100

CV = (10/2500) × 100

CV = 0.4%

Plant C has CV = 0.36 and plant D has CV = 0.4

Answer: Hence plant D has greater variability in individual wages.

How to Find a Coefficient of Variation: Overview. 

Use the following formula to calculate the CV by hand for a population or a sample.

These formulas for coefficient of variation give percentages. To get a decimal, omit multiplying by 100.

σ is the standard deviation for a population, which is the same as “s” for the sample.
μ is the mean for the population, which is the same as XBar in the sample.

In other words, to find the coefficient of variation, divide the standard deviation by the mean and multiply by 100.

How to find a coefficient of variation in Excel.

You can calculate the coefficient of variation in Excel using the formulas for standard deviation and mean. For a given column of data (i.e. A1:A10), you could enter: “=stdev(A1:A10)/average(A1:A10)) then multiply by 100.

Co-efficient of Variation (CV) in Excel

The co-efficient of variation formula can be performed in Excel by first using the standard deviation function for a data set. Next, calculate the mean by using the Excel function provided. Since the co-efficient of variation is the standard deviation divided by the mean, divide the cell containing the standard deviation by the cell containing the mean.

How to Find a Coefficient of Variation by hand: Steps.

Example question: Two versions of a test are given to students. One test has pre-set answers and a second test has randomized answers. Find the coefficient of variation.

Step 1: Divide the standard deviation by the mean for the first sample:
11.2 / 50.1 = 0.22355

Step 2: Multiply Step 1 by 100:
0.22355 * 100 = 22.355%

Step 3: Divide the standard deviation by the mean for the second sample:
12.9 / 45.8 = 0.28166

Step 4: Multiply Step 3 by 100:
0.28166 * 100 = 28.266%

That’s it! Now you can compare the two results directly.

What Is the Co-efficient of Variation (CV)?

The co-efficient of variation (CV) is a statistical measure of the dispersion of data points in a data series around the mean. The co-efficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from one another.

What is Coefficient of Variation?

The coefficient of variation is a type of measure of dispersion. A measure of dispersion is a quantity that is used to gauge the extent of variability of data. Thus, the coefficient of variation is used to measure the dispersion of data from the average or the mean value. CV is the abbreviated form of the coefficient of variation.

Coefficient of Variation Definition

The coefficient of variation is a dimensionless relative measure of dispersion that is defined as the ratio of the standard deviation to the mean. If there are data sets that have different units then the best way to draw a comparison between them is by using the coefficient of variation.

KEY TAKEAWAYS

• The co-efficient of variation (CV) is a statistical measure of the relative dispersion of data points in a data series around the mean.
• It represents the ratio of the standard deviation to the mean.
• The CV is useful for comparing the degree of variation from one data series to another, even if the means are drastically different from one another.
• In finance, the co-efficient of variation allows investors to determine how much volatility, or risk, is assumed in comparison to the amount of return expected from investments.
• The lower the ratio of the standard deviation to mean return, the better risk-return tradeoff.

Understanding the Co-efficient of Variation (CV)

The co-efficient of variation shows the extent of variability of data in a sample in relation to the mean of the population.

In finance, the co-efficient of variation allows investors to determine how much volatility, or risk, is assumed in comparison to the amount of return expected from investments. Ideally, if the co-efficient of variation formula should result in a lower ratio of the standard deviation to mean return, then the better the risk-return tradeoff.

They’re most often used to analyze dispersion around the mean, but quartile, quintile, or decile CVs can also be used to understand variation around the median or 10th percentile, for example.

The co-efficient of variation formula or calculation can be used to determine the deviation between the historical mean price and the current price performance of a stock, commodity, or bond, relative to other assets.

Co-efficient of Variation (CV) Formula

Below is the formula for how to calculate the co-efficient of variation:

Multiplying the co-efficient by 100 is an optional step to get a percentage rather than a decimal.

Co-efficient of Variation (CV) vs. Standard Deviation

The standard deviation is a statistic that measures the dispersion of a data set relative to its mean. It is used to determine the spread of values in a single data set rather than to compare different units.

When we want to compare two or more data sets, the co-efficient of variation is used. The CV is the ratio of the standard deviation to the mean. And because it’s independent of the unit in which the measurement was taken, it can be used to compare data sets with different units or widely different means.

In short, the standard deviation measures how far the average value lies from the mean, whereas the co-efficient of variation measures the ratio of the standard deviation to the mean.

Advantages and Disadvantages of the Co-efficient of Variation (CV)

Advantages

The co-efficient of variation can be useful when comparing data sets with different units or widely different means.

That includes when the risk/reward ratio is used to select investments. For example, an investor who is risk-averse may want to consider assets with a historically low degree of volatility relative to the return, in relation to the overall market or its industry. Conversely, risk-seeking investors may look to invest in assets with a historically high degree of volatility.

Disadvantages

When the mean value is close to zero, the CV becomes very sensitive to small changes in the mean. Using the example above, a notable flaw would be if the expected return in the denominator is negative or zero. In this case, the co-efficient of variation could be misleading.

If the expected return in the denominator of the co-efficient of variation formula is negative or zero, then the result could be misleading.

How Can the Co-efficient of Variation Be Used?

The co-efficient of variation is used in many different fields, including chemistry, engineering, physics, economics, and neuroscience.

Other than helping when using the risk/reward ratio to select investments, it is used by economists to measure economic inequality. Outside of finance, it is commonly applied to audit the precision of a particular process and arrive at a perfect balance.

Example of Co-efficient of Variation (CV) for Selecting Investments

For example, consider a risk-averse investor who wishes to invest in an exchange-traded fund (ETF), which is a basket of securities that tracks a broad market index. The investor selects the SPDR S&P 500 ETF, the Invesco QQQ ETF, and the iShares Russell 2000 ETF. Then, they analyze the ETFs’ returns and volatility over the past 15 years and assumes that the ETFs could have similar returns to their long-term averages.

For illustrative purposes, the following 15-year historical information is used for the investor’s decision:

• If the SPDR S&P 500 ETF has an average annual return of 5.47% and a standard deviation of 14.68%, the SPDR S&P 500 ETF’s co-efficient of variation is 2.68.
• If the Invesco QQQ ETF has an average annual return of 6.88% and a standard deviation of 21.31%, the QQQ’s co-efficient of variation is 3.10.
• If the iShares Russell 2000 ETF has an average annual return of 7.16% and a standard deviation of 19.46%, the IWM’s co-efficient of variation is 2.72.

Based on the approximate figures, the investor could invest in either the SPDR S&P 500 ETF or the iShares Russell 2000 ETF, since the risk/reward ratios are approximately the same and indicate a better risk-return tradeoff than the Invesco QQQ ETF.

What does the co-efficient of variation tell us?

The co-efficient of variation (CV) indicates the size of a standard deviation in relation to its mean. The higher the co-efficient of variation, the greater the dispersion level around the mean.

What is considered a good co-efficient of variation?

That depends on what you’re looking at and comparing. No set value can be considered universally “good.” However, generally speaking, it is often the case that a lower co-efficient of variation is more desirable, as that would suggest a lower spread of data values relative to the mean.

How do I calculate the co-efficient of variation?

To calculate the co-efficient of variation, first find the mean, then the sum of squares, and then work out the standard deviation. With that information at hand, it is possible to calculate the co-efficient of variation by dividing the standard deviation by the mean.

The Bottom Line

The co-efficient of variation is a simple way to compare the degree of variation from one data series to another. It can be applied to pretty much anything, including the process of picking suitable investments.

Generally speaking, a high CV indicates that the group is more variable, whereas a low value would suggest the opposite.

Coefficient of Variation Uses

If two data sets having similar values need to be compared then the standard deviation can be used. However, if two data sets having different unit need to be compared then the coefficient of variation needs to be used. Some applications of coefficient of variation are as follows:

• In the finance industry if an investor wants to invest in a particular ETF, then he uses the coefficient of variation to choose the one which will give a better risk-return trade-off.
• The coefficient of variation is also used to gauge the consistency of data. A distribution with a smaller coefficient of variation is more consistent than one with a larger CV.

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