I am taking a Probability and Statistics course. We know that:

coefficient of variation = (standard deviation / mean) * 100%

Geometrically, what does it mean that the coefficient of variation = 100%. I know that in that case, the standard deviation must be equal to the mean, but I can't imagine the case in which that equality occurs.

## Best Answer

The coefficient of variation simply expresses the standard deviation as a fraction of the mean value, and is best applied to ratio-scale data that has a meaningful zero value. A low CV indicates that the distribution is narrow about the mean, while a high CV indicates that the distribution has more variability about the mean. A CV at or above 1 is usually considered high variability, as the standard deviation is equal to or higher than the mean.

An exponential distribution is an example of a distribution that has a coefficient of variation equal to 1, as it has mean equal to its standard deviation. CVs above 1 may be seen in ratio-scale data with heavy tails, as in a hyperexponential distribution. In ratio-scale data, a value cannot be further from the mean than zero on the low end, but can be arbitrarily high on the upper end - these upper tail values can contribute greatly to the standard deviation.