Editing Coefficient of variation (section)

== Examples of misuse ==
Comparing coefficients of variation between parameters using relative units can result in differences that may not be real. If we compare the same set of temperatures in [[Celsius]] and [[Fahrenheit]] (both relative units, where [[kelvin]] and [[Rankine scale]] are their associated absolute values):

Celsius: [0, 10, 20, 30, 40]

Fahrenheit: [32, 50, 68, 86, 104]

The [[Standard deviation#Sample standard deviation|sample standard deviations]] are 15.81 and 28.46, respectively. The CV of the first set is 15.81/20 = 79%. For the second set (which are the same temperatures) it is 28.46/68 = 42%.

If, for example, the data sets are temperature readings from two different sensors (a Celsius sensor and a Fahrenheit sensor) and you want to know which sensor is better by picking the one with the least variance, then you will be misled if you use CV. The problem here is that you have divided by a relative value rather than an absolute.

Comparing the same data set, now in absolute units:

Kelvin: [273.15, 283.15, 293.15, 303.15, 313.15]

Rankine: [491.67, 509.67, 527.67, 545.67, 563.67]

The [[Standard deviation#Sample standard deviation|sample standard deviations]] are still 15.81 and 28.46, respectively, because the standard deviation is not affected by a constant offset. The coefficients of variation, however, are now both equal to 5.39%.

Mathematically speaking, the coefficient of variation is not entirely linear.  That is, for a random variable <math>X</math>, the coefficient of variation of <math>aX+b</math> is equal to the coefficient of variation of <math>X</math> only when <math>b = 0</math>.  In the above example, Celsius can only be converted to Fahrenheit through a linear transformation of the form <math>ax+b</math> with <math>b \neq 0</math>, whereas Kelvins can be converted to Rankines through a transformation of the form <math>ax</math>.