He **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 grade students 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 has no units and is obtained by dividing the standard deviation by the mean and multiplying by one hundred.

The smaller the coefficient of variation, the less spread the data is about the mean. For example, in a variable with a mean of 10 and another with a mean of 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 on a proportion scale, that is, scales with absolute zero regardless of the unit of measurement. An example is the variable distance 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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**What is the coefficient of variation used for?**

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 the same but their realizations are very different. Example, compare 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 a reliability indicator in scientific experiments. It is said that if the coefficient of variation is 30% or greater, the results of the experiment should be discarded due to their low reliability.

– Allows predicting how clustered around the mean are the values of the variable under study even without knowing its distribution. This is of great help for estimation of errors and calculation of sample sizes.

Suppose that in a population the variables weight and height of the people are measured. Weight with a CV of 5% and height with a CV of 14%. If it is desired to take a sample from that population, the size of the sample must be greater for estimates of height than weight, since there is greater variability in the measurement of height than in that of weight.

An important observation on the utility 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 the mean cause the CV values to be very large and possibly incalculable.

**How is it calculated?**

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 the standard deviation can be calculated previously, applying the following formulas:

**examples**

**Example 1**

The weights, in kg, of a group of 6 people were measured: 45, 62, 38, 55, 48, 52. It is desired to know the coefficient of variation of the weight variable.

It begins by calculating the arithmetic mean and the standard deviation:

Now, it is substituted in the formula of the coefficient of variation:

Answer: the coefficient of variation of the weight variable 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.

**Example 2**

In the emergency room of a hospital, the body temperature is taken, in degrees Celsius, of 5 children who are being treated. The results give 39º, 38º, 40º, 38º and 40º. What is the coefficient of variation of the temperature variable?

It begins by calculating the arithmetic mean and the standard deviation:

Now, it is substituted in the formula of the coefficient of variation:

Answer: 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 on an 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:

So the temperatures of the 5 children would be: 102.2, 100.4, 104, 100.4, 104

We proceed to calculate the arithmetic mean and the standard deviation:

Now, it is substituted in the formula of 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. It is the interval measurement scale that produces these differences, and therefore care must be taken when using the coefficient of variation to compare variables on different scales.

**solved exercises**

**Exercise 1**

The weights, in kg, of the 10 employees in a post office were measured: 85, 62, 88, 55, 98, 52, 75, 70, 76, 77. It is desired to know the coefficient of variation of the weight variable.

The arithmetic mean and the standard deviation are calculated:

Now, it is substituted in the formula of 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.

**Exercise 2**

In a certain city, the heights of 9465 children from all schools in 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.

Answer: the coefficient of variation of the height variable of the first graders of the city is 12.37%.

### Exercise 3

A park ranger suspects that the black and white rabbit populations 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 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 correct in his suspicion that the variability of the weights of the two populations of rabbits are not the same.

**References**

Freund, R.; Wilson, W.; Mohr, D. (2010). Statistical methods. Third ed. Academic Press-Elsevier Inc.

Gordon, R.; Camargo, I. (2015). Selection of statistics for the estimation of experimental precision in maize trials. Mesoamerican Agronomy Magazine. Retrieved from revistas.ucr.ac.cr.

Gorgas, J.; Cardiel, N.; Zamorano, J. (2015). Basic statistics for science students. Faculty of Physical Sciences. Complutense University of Madrid.

Salinas, H. (2010). Statistics and probabilities. Recovered from mat.uda.cl.

Sokal, R.; Rohlf, F. (2000). Biometrics. The principles and practice of statistics in biological research. Third ed. Blume Editions.

Spiegel, M.; Stephens, L. (2008). Statistics. fourth ed. McGraw-Hill/Interamericana de Mexico SA

Vasallo, J. (2015). Statistics applied to health sciences. Elsevier Spain SL

Wikipedia (2019). Coefficient of variation. Retrieved from en.wikipedia.org.