Situations and Definitions

A coefficient of variation (CV) can be calculated and interpreted in two
different settings: analyzing a single variable and interpreting a model. 
The standard formulation of the CV, the ratio of the standard deviation to the
mean, applies in the single variable setting. In the modeling setting, the CV
is calculated as the ratio of the root mean squared error (RMSE) to the mean of
the dependent variable. In both settings, the CV is often presented as the
given ratio multiplied by 100. The CV for a single variable aims to describe the dispersion of the variable in a way that does not depend on the variable’s measurement unit.
The higher the CV, the greater the dispersion in the variable. The CV for a model aims to describe the model fit
in terms of the relative sizes of the squared residuals and outcome values.  The
lower the CV, the smaller the residuals relative to the predicted value. 
This is suggestive of a good model fit. 

The CV for a variable can easily be calculated using the information from a
typical variable summary (and sometimes the CV will be returned by default in
the variable summary).  We demonstrate below how to calculate the CV in
Stata.

use https://stats.idre.ucla.edu/stat/stata/notes/hsb1, clear
summarize math

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
        math |       200      52.645    9.368448         33         75

di 100 * r(sd) / r(mean)

17.795513

The CV for a model can similarly be calculated when it is not included in the model
output.

regress math socst

      Source |       SS       df       MS              Number of obs =     200
-------------+------------------------------           F(  1,   198) =   83.43
       Model |  5177.88866     1  5177.88866           Prob > F      =  0.0000
    Residual |  12287.9063   198   62.060133           R-squared     =  0.2965
-------------+------------------------------           Adj R-squared =  0.2929
       Total |   17465.795   199  87.7678141           Root MSE      =  7.8778

------------------------------------------------------------------------------
        math |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       socst |   .4751335    .052017     9.13   0.000      .372555     .577712
       _cons |   27.74563   2.782287     9.97   0.000     22.25891    33.23235
------------------------------------------------------------------------------

quietly summarize math
di 100 * e(rmse) / r(mean)

14.964052

Advantages

The advantage of the CV is that it is unitless.  This allows CVs to be
compared to each other in ways that other measures, like standard deviations or
root mean squared residuals, cannot be. 

In the variable CV setting: The standard deviations of two
variables, while both measure dispersion in their respective variables, cannot
be compared to each other in a meaningful way to determine which variable has
greater dispersion because they may vary greatly in their units and the means
about which they occur. The standard deviation and mean of a
variable are expressed in the same units, so taking the ratio of these two
allows the units to cancel.  This ratio can then be compared to other such
ratios in a meaningful way: between two variables (that meet the assumptions
outlined below), the variable with the smaller CV is less dispersed than
the variable with the larger CV.

In the model CV setting: Similarly, the RMSE of two models both measure
the magnitude of the residuals, but they cannot
be compared to each other in a meaningful way to determine which model provides
better predictions of an outcome. The model RMSE and mean of the predicted
variable are expressed in the same units, so taking the ratio of these two
allows the units to cancel.  This ratio can then be compared to other such
ratios in a meaningful way: between two models (where the outcome variable meets
the assumptions outlined below), the model with the smaller CV has predicted
values that are closer to the actual values.  It is interesting to note the
differences between a model’s CV and R-squared values.  Both are unitless
measures that are indicative of model fit, but they define model fit in two
different ways: CV evaluates the relative closeness of the predictions to the
actual values while R-squared evaluates how much of the variability in the
actual values is explained by the model. 

Requirements and Disadvantages

There are some requirements that must be met in order for the CV to be
interpreted in the ways we have described.  The most obvious problem arises
when the mean of a variable is zero.  In this case, the CV cannot be
calculated.  Even if the mean of a variable is not zero, but the variable
contains both positive and negative values and the mean is close to zero, then
the CV can be misleading.  The CV of a variable or the CV of a prediction
model for a variable can be considered as a reasonable measure if the variable
contains only positive values.  This is a definite disadvantage of CVs.