A coefficient of variation (CV) shall be a measure of relative variability. It is the ratio of the standard deviation to the mean (mean). For instance, the expression “The standard deviation is 15% of the mean” is a CV.
A CV is particularly useful when comparing the results of two different surveys or tests that have different measurements or values. For instance, if you compare the results of two tests that have different scoring mechanisms. When sample A has a CV of 12% and sample B has a CV of 25%, it would appear that sample B has a greater variation than its average.
Formula
The formula for the coefficient of variation is:
Coefficient of variation = (Standard Deviation / Average) * 100.
In symbols: CV = (SD/xbar) * 100.
Multiplication of the coefficient by 100 is an optional step to obtain a percentage instead of a decimal.
Example of coefficient of variation
A scientist 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. wrong answers) is randomly assigned to the test participants. The results of the two tests are:
Regular test Randomized answers
Average 59.9 44.8
SD 10.2 12.7
Comparing the results of the two tests is challenging. Comparing standard deviations does not work, because the means are also different. Calculation with the formula CV=(SD/Mean)*100 helps to make sense of the data:
Regular tests Randomized answers
Average 59.9 44.8
SD 10.2 12.7
CV 17.03 28.35
Observing the standard deviations of 10,2 and 12,7, one might think that the tests have similar results. Nevertheless, when one adjusts to the difference in means, the results have a greater significance:
Regular tests: CV = 17.03
Random answers: CV = 28.35
Coefficient of variation can also be used to compare the variability between different measurements. For instance, it is possible to compare IQ scores with Woodcock-Johnson III test scores on Cognitive Abilities.
Use the following formula to calculate the CV by hand for a population or a sample.
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.
It is possible to calculate the coefficient of variation in Excel by using the standard deviation and mean formulas. You can enter for a given column of data (i.e. A1:A10): “=stdev(A1:A10)/average(A1:A10)” and then multiply by 100.
Find a coefficient of variation by hand: Steps.
Question example: Two versions of one test are given to the students. A test has predefined responses and a second test has randomized responses. To find the coefficient of variation.
Regular Test Randomized Answers
Average 50.1 45.8
SD 11.2 12.9
Step 1: Divide the standard deviation by the average of 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 average of the second sample:
12.9 / 45.8 = 0.28166
Step 4: Multiply step 3 by 100:
0.28166 * 100 = 28.266%
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