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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?

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  • $\begingroup$ There's a one-to-one mapping between z-score and percentile, so there's no fundamental difference. $\endgroup$
    – Moss Murderer
    Commented Jan 17, 2018 at 4:49
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    $\begingroup$ @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. $\endgroup$
    – whuber ♦
    Commented Jan 17, 2018 at 16:54
  • $\begingroup$ @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? $\endgroup$
    – Moss Murderer
    Commented Jan 17, 2018 at 20:28
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    $\begingroup$ @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." $\endgroup$
    – whuber ♦
    Commented Jan 17, 2018 at 20:44

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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.

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    $\begingroup$ What you're calling the cumulative percentile is often called a percentile rank (in practice, often given in percent terms, not as a proportion) and could also be called a cumulative probability. It's not the same as percentiles as defined in the OP's link (and more generally). Such percentiles are values on the scale of the data, such as the 1% point, 50% point or the 99% point for heights or weights, which would be height or weight measurements such those fractions of the data have lower values. $\endgroup$
    – Nick Cox
    Commented Jan 17, 2018 at 17:05

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