Doing some exercises I hit upon this wall of text:
Coefficient of Variation should only be computed on ratio scales (i.e., data where there is a "true" zero, like temperatures in Kelvin, or heights, or sizes of populations, etc). Coefficient of Variation may not be meaningful for data that does not have a "true" zero.
Temperatures measured in Celsius is not a ratio scale, and does not have a "true" zero since may have positive and negative values. Thus, computing the CoV for the daily temperature in Celsius for the city of Boston would be invalid.
The X coordinate (and Y coordinate) of a drunk in the random walk are also not ratio scales because there isn't a "true" zero. In addition, we know that the X (and Y) coordinate have a mean position of 0. That means that the denominator of the Coefficient of Variation is zero, so it can not be computed.
The fifth answer choice - "The distance a drunk ended up away from the starting point in the random walk" - does not have a mean of 0 because distance is always positive. If this was written in a way like: "The number of units a drunk ended up away from the starting point in the random walk, where a unit is positive if a drunk moves North or East and negative if a drunk moves South or West", then the mean could possibly be zero.
What the text means by "ratio scales"? why would it be "invalid" to use coefficient of variation for unit that have negative values? why is that that the random walk has X and Y coordinates mean position of 0? couldn't my random walker go all the way towards southwest?
You text is confusingly worded, but it is correct. It boils down to this:
The CV tells you your "relative error", or $\frac{\sigma}{\mu}$. If we keep $\sigma$ the same, then our CV is dependent on the value of $\mu$; however, if $\mu$ is based on an arbitary scale (e.g., Celsius), then the CV has two problems: