# Difference between using a z-score or a percentiles to summarize data in normal and skewed distributions

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?

• There's a one-to-one mapping between z-score and percentile, so there's no fundamental difference. – Moss Murderer Jan 17 '18 at 4:49
• @Moss That depends on what you mean. The relationship between the percentiles of empirical data and their Z-scores is not given by the Normal probability function: it's specific to each dataset. – whuber Jan 17 '18 at 16:54
• @whuber: Yes, the theoretical relation between z-score and percentile given by the normal pdf need not apply to any given dataset, but the empirical relationship between the two measures is guaranteed to be a monotone function, so the two measures are equivalent in the sense that you can freely go back and forth between the two, no? – Moss Murderer Jan 17 '18 at 20:28
• @Moss That's a correct mathematical statement--but it doesn't sound practically useful. Regardless, it's a definite and important difference, which seems to me like it contradicts your original opinion of "no fundamental difference." – whuber Jan 17 '18 at 20:44

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.