Concept of a z-score for a gamma distribution

Question

This is a somewhat general question. For the normal distribution we have the handy concept of the z-score that can be used to measure distance from the mean in a standardized way. I've encountered a situation where I'm working with more of a gamma distribution that is somewhat right-skewed. The objective is to measure individual performers relative to some central tendency (e.g. this person is 1.5 standard deviations above the mean), but I am not aware of a z-score analogue for a gamma distribution.

Firstly, is this an appropriate way to conceptualize the problem? If so, is there a z-score analogue for the gamma? Thanks in advance.

Answer 1

The $z$ score expressed how many standard deviations a given observation from a symmetric distribution is away from the mean. Negative $z$ scores indicate an observation below, positive $z$ scores above the mean.

The logic doesn't make much sense for asymmetric distributions, even when standard deviations exist. In a symmetric distribution, observations one SD above and one SD below the mean are "equally typical". For an asymmetric distribution like the gamma, there may not even be a possible observation one SD below the mean (if the shape parameter is less than one, the mean is lower than the SD, so if you subtract one SD from the mean, you end up negative), while observations one SD above the mean make perfect sense.

Instead, it may make more sense to fit a distribution like the gamma and work with percentiles. An observation at the 30th percentile could be said to be "as typical as" an observation at the 70th percentile, since both are 20 percentage points away from the median. The advantage is that this carries the exact same information as the $z$ score in the symmetric case.

Alternatively, you could work with percentages, but without fitting a distribution, by using the empirical cumulative distribution function.