The coefficient of variation (CV) expresses the standard deviation with respect to the mean. That is, it seeks to explain how large the value of the standard deviation is with respect to that of the mean.
For example, the variable height of fourth graders has a coefficient of variation of 12%, which means that the standard deviation is 12% of the mean value..
Denoted by CV, the coefficient of variation is unitless and is obtained by dividing the standard deviation by the mean and multiplying by one hundred.
The smaller the coefficient of variation, the less dispersed the data from the mean. For example, in a variable with mean 10 and another with mean 25, both with a standard deviation of 5, their coefficients of variation are 50% and 20% respectively. Of course there is greater variability (dispersion) in the first variable than in the second.
It is advisable to work with the coefficient of variation for variables measured in a proportion scale, that is, scales with absolute zero regardless of the unit of measurement. An example is the distance variable that does not matter if it is measured in yards or meters, zero yards or zero meters means the same thing: zero distance or displacement.
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The coefficient of variation is used to:
- Compare the variability between distributions in which the units are different. For example, if you want to compare the variability in the measurement of the distance traveled by two different vehicles in which one was measured in miles and the other in kilometers.
- Contrast the variability between distributions in which the units are equal but their realizations are very different. Example, comparing the variability in the measurement of the distance traveled by two different vehicles, both measured in kilometers, but in which one vehicle traveled 10,000 km in total and the other only 700 km.
- The coefficient of variation is frequently used as an indicator of reliability in scientific experiments. It is said that if the coefficient of variation is 30% or greater, the results of the experiment should be discarded because of their low reliability..
- It allows predicting how grouped around the mean are the values of the variable under study even without knowing its distribution. This is of great help for estimating errors and calculating sample sizes..
Suppose that the variables weight and height of people are measured in a population. Weight with a CV of 5% and height with a CV of 14%. If you want to take a sample from this population, its size must be larger for estimates of height than for weight, since there is greater variability in the measurement of height than in that of weight..
An important observation on the usefulness of the coefficient of variation is that it loses meaning when the value of the mean is close to zero. The mean is the divisor of the CV calculation and, therefore, very small values of this cause that the CV values are very large and, possibly, incalculable.
The calculation of the coefficient of variation is relatively simple, it will be enough to know the arithmetic mean and the standard deviation of a data set to calculate it according to the formula:
In case they are not known, but the data is available, the arithmetic mean and standard deviation can be calculated previously, applying the following formulas:
The weights, in kg, of a group of 6 people were measured: 45, 62, 38, 55, 48, 52. We want to know the coefficient of variation of the weight variable.
It starts by calculating the arithmetic mean and standard deviation:
Ans: the coefficient of variation of the variable weight of the 6 people in the sample is 16.64%, with an average weight of 50 kg and a standard deviation of 8.32 kg.
In the emergency room of a hospital, the body temperature is taken, in degrees Celsius, of 5 children who are being cared for. The results are 39th, 38th, 40th, 38th and 40th. What is the coefficient of variation of the variable temperature?
It starts by calculating the arithmetic mean and the standard deviation:
Now, it is substituted in the formula for the coefficient of variation:
Ans: the coefficient of variation of the temperature variable of the 5 children in the sample is 2.56%, with an average temperature of 39 ° C and a standard deviation of 1 ° C.
With temperature, care must be taken in handling the scales, since being a variable measured in the interval scale it does not have an absolute zero. In the case under study, what would happen if the temperatures were transformed from degrees Celsius to degrees Fahrenheit:
The arithmetic mean and standard deviation are calculated:
Now, it is substituted in the formula for the coefficient of variation:
Ans: the coefficient of variation of the temperature variable of the 5 children in the sample is 1.76%, with an average temperature of 102.2 ° F and a standard deviation of 1.80 ° F.
It is observed that the mean, the standard deviation and the coefficient of variation are different when the temperature is measured in degrees Celsius or in degrees Fahrenheit, even though they are the same children. The interval measurement scale is the one that produces these differences and, therefore, care must be taken when using the coefficient of variation to compare variables on different scales..
The weights, in kg, of the 10 employees in a post office were measured: 85, 62, 88, 55, 98, 52, 75, 70, 76, 77. We want to know the coefficient of variation of the variable weight.
The arithmetic mean and standard deviation are calculated:
Now, it is substituted in the formula for the coefficient of variation:
Ans: the coefficient of variation of the variable weight of the 10 people in the post office is 19.74%, with an average weight of 73.80 kg and a standard deviation of 14.57 kg.
In a certain city, the heights of the 9,465 children in all schools attending the first grade are measured, obtaining an average height of 109.90 centimeters with a standard deviation of 13.59 cm. Calculate the coefficient of variation.
Ans: the coefficient of variation of the variable height of the first-grade children in the city is 12.37%.
A park ranger suspects that the populations of black and white rabbits in his park do not have the same variability in size. To demonstrate this, he took samples of 25 rabbits from each population and obtained the following results:
- White rabbits: average weight of 7.65 kg and standard deviation of 2.55 kg
-Black rabbits: average weight of 6.00 kg and standard deviation of 2.43 kg
Is the park ranger right? The answer to the park ranger's hypothesis can be obtained by means of the coefficient of variation:
Ans: the coefficient of variation of the weights of the black rabbits is almost 7% greater than that of the white rabbits, so it can be said that the park ranger is right in his suspicion that the variability of the weights of the two populations of rabbits are not the same.
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