The coefficient of variation, definition and examples

This article will be about The coefficient of variation, the use of it and the formula to calculate it, with examples.

¿What is the coefficient of variation?

The coefficient of variation is a method that is used to determine how much the elements of a set are dispersed from the arithmetic mean of the data. The calculated value with the coefficient of variation is not in a measure of unit (a measure unit could be meters, kilometers, liters, etc.), this means that if we calculate the coefficient of variation of the weight in kilograms, the result is not going to be how many kilograms is the data disperse, instead the result will be a percentage or a coefficient that shows us how disperse the data is.

Coefficient of variation

The coefficient of variation can be used to compare 2 sets to see which of the sets has the data more disperse when the units of measure are not the same or even when they are not equivalent. (the equivalent units of measure are the ones that could be converted from one into to the other, like miles and meters or tons and pounds). This can be done because even when the sets are not equivalent, we can calculate which of the two has a major coefficient of variation, and this is the way we can see which of the data is more disperse with respect to the mean, no matter if the data are completely different from each other.

For example if we would want to know how much does the height of a city vary, and we also would want to know the variation of the temperature in the same city, and we would want to know which of the two varies more, this could not be done with the standard deviation, because this variables have not equivalent variables, because the height of the city would be measured in meters, while the temperature would be measured in Fahrenheit or Celsius. Another case would be if these measures of units were equivalents because we would only have to convert them in the same unit of measure, but in this example the meters and the Celsius do not measure the same thing (one is a measure of longitude and the other is a measure of temperature).

Example of the ceofficient of variation

The coefficient of variation is found dividing the standard deviation between the arithmetic mean, so it is important to know how to find this two. Like the formula of the coefficient of variation uses the standard deviation, then there are two formulas to find it, one is to use with a sample, and another one to use with a population, this is why we present two different formulas for the standard deviation.

Formula of the coefficient of variation

The coefficient of variation is found dividing the standard deviation between the arithmetic mean, so it is important to know how to find this two. Like the formula of the coefficient of variation uses the standard deviation, then there are two formulas to find it, one is to use with a sample, and another one to use with a population, this is why we present two different formulas for the standard deviation.

S = Typical deviation, x = Mean

  • Typical deviation formula for population
  • S = √   (
    Σ (x - x)2 / n
      )
  • Typical deviation formula for sample
  • S = √   (
    Σ (x - x)2 / n - 1
      )
  • Mean formula
  • x = Σ xi / n
  • Coefficient of variation formula
  • Cv =
    S/m

Coefficient of variation examples

Example 1: Find the coefficient of variation of the following data (take the data as a population)

6 , 4 , 8 , 2 , 10 , 0

  • First we find the mean
  • x =
    6 + 4 + 8 + 2 + 10 + 0/6
  • x =
    30/6
  • x = 5
  • Now we calcualte the typical deviation
  • S2 =
    (6-5)2 + (4-5)2 + (8-5)2 + (2-5)2 + (10-5)2 + (0-5)2/6
  • S2 =
    (1)2 + (-1)2 + (3)2 + (-3)2 + (5)2 + (-5)2/6
  • S2 =
    1 + 1 + 9 + 9 + 25 + 25/6
  • S2 =
    70/6
  • S2 = 11.67
  • S = √ 11.67
  • S = 3.42
  • And last we find the coefficient of variation
  • cv =
    S/x
  • cv =
    3.42/5
  • cv = 0.684
  • To express the result in percenteges we are going to multiply the result by 100%
  • cv = 0.684 * 100%
  • cv = 68.4%

Example 2: A group of people that often go to the gym were asked about how many times the workout daily, calculate the coefficient of variation of the results (minutes)

30 + 60 + 45 + 90

Important, The problem says that they were only a group of people that assists to the gym, so we are talking about a sample and not a population, so we are going to use the formula of the standard/typical deviation for samples.

  • We calculate the mean
  • x =
    30 + 60 + 45 + 90/4
  • x =
    225/4
  • x = 56.25
  • Then we find the typical deviation
  • S2 =
    (30-56.23)2 + (60-56.23)2 + (45-56.23)2 + (90-56.23)2/4-1
  • S2 =
    (-26.25)2 + (3.75)2 + (-11.25)2 + (33.75)2 /3
  • S2 =
    689.06 + 14.06 + 126.56 + 1 139.06/3
  • S2 =
    1 968.74/3
  • S2 = 656.25
  • S = √ 656.25
  • S = 25.62
  • And finally we find the coefficient of variation
  • cv =
    S/x
  • cv =
    25.62/56.25
  • cv = 0.455
  • and convert it in percentages
  • cv = 0.455 * 100%
  • cv = 45.5%

Example 3: A group of people were weighed and measured (how tall they were), and we obtained the following data: the average weigh was 80kg with a typical deviation of 10kg, and the average height was of 172cm with a typical deviation of 5cm. Knowing this results, define which of the variables vary the most.

  • First we find the coefficient of variation of the weight
  • cv = S / x
  • cv = 10 / 80
  • cv = 0.125
  • cv = 0.125 * 100%
  • cv = 12.5%
  • and then find the coefficient of variation of the height
  • cv = S / x
  • cv = 5 / 172
  • cv = 0.029
  • cv = 0.029 * 100%
  • cv = 2.9%

In conclusion the weight of the people varied more than the height

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Mean median and mode Mean for grouped data Median for grouped data Mode for grouped data Quartiles Standard deviation Frequency table