There arises an issue though when comparing two distributions which requires some additional insight. For instance, if you examine the average corn price and its associated standard deviation over two different time periods, you may want to judge which period had the greatest variability. Simply comparing the standard deviations would be the first thought, however, that would describe the absolute difference in variation. When comparing two different distributions we are usually interested in knowing their relative variability.
One way is to compare the ratio of the standard deviations to the means. This is a form of indexing and can reveal which variability is more significant (when compared to the average value). Why is this important? When I was a lot younger, I was a carpenter's assistant which meant that I would often have the job of cutting 2x4's to stated lengths called out by my boss. At first, I was not very good at this and would often be off a small amount. One day the boss was angry at me over this and told me the board I just cut was 1/2 inch off. At first blush that sounded trivial to me so I told him 1/2 inch was not very much, and asked why was he so mad? His reply was "if your nose was 1/2 inch longer you would think it was a big deal."
He made a very interesting statistical point without knowing it. 1/2 inch may be a minor amount if you are thinking about the length of a road a contractor bid to pave but if a brain surgeon is 1/2 inch off in directing some surgical tool during brain surgery, it could be very very important. Therefore, it is not simply a matter of the amount of the variance from standard but how much the variance is compared to the consequences of the range.
Holding all other things equal, as the mean of a process increases so will its variance. However, the relative variation may be decreasing if the variance increases at a slower rate than the mean. This principle of relative variation can be evaluated with the coefficient of variation.
The coefficient of variation is a unitless measure which is calculated as mentioned, by dividing the standard deviation by the mean. If you multiply the result by 100 it can be expressed in percentage terms. You can compare the coefficients of variation for two time periods or two processes to judge their relative variability. We will look at a couple of examples of this shortly and why it can make a big difference when thinking about risk. Since risk is often defined as the probability of a negative outcome, it is often measured with respect to the standard deviation, variance and coefficient of variation.
