I have a dataset with multiple variables having different distributions – some are normal and others are skewed. I want to summarize locations for some identified values on both kinds of distributions. Simple percentiles can work for both normal and skewed distributions. (http://www.dummies.com/education/math/statistics/how-to-calculate-percentiles-in-statistics/)
But what's the difference between using simple percentiles vs. z-scores for a normal distribution?
Best Answer
The Z-score is a quantile, and takes values from $-\infty$ to $\infty$. The cumulative percentile is bounded from 0 to 1. When the distribution is known, the percentile can map 1-1 to any observation for any distribution, whereas the Z-score only has this property for normally distributed data; hence these summaries are equivalent when normality is met.
Perhaps an advantage of the Z-score is that it can more accurately summarize the distance between two extreme values. An observation 6 standard deviations above the mean is more extreme than one only 4 standard deviations about the mean, but both are beyond 3 decimal places of accuracy for the percentile, so the Z-score can more accurately summarize their relative extremity.