Unlearning descriptive statistics

It's worth noting that most statistical packages include the ability to generate a five-number summary (https://en.m.wikipedia.org/wiki/Five-number_summary), which generates many of the described skew-independent descriptive statistics automatically. (R, for example, will generate a five-number summary for each feature of a data frame with a simple summary(df) )

The TL;DR is to just plot everything if possible to visualize skew explicitly (or tricks like the Quartet), and there are a number of tools which can do that without much LOC overhead. (e.g ggplot2)

you can also use the aptly named fivenum() func in R :)

For anyone who is interesting in correlation measures, have a look at distance correlation:

https://en.wikipedia.org/wiki/Distance_correlation http://projecteuclid.org/euclid.aoas/1267453933

It's implemented in the R package "energy", and provides a more comprehensive measure of variable dependence than Spearman's rank correlation. Distance correlation is able to detect any kind of dependence, not just monotonic relationships.

For readers who are OK with some math, I recommend John Myles White's eye-opening post about means, medians, and modes: http://www.johnmyleswhite.com/notebook/2013/03/22/modes-medi... He describes these summary descriptive stats in terms of what penalty function they minimize: mean minimizes L2, median minimizes L1, mode minimizes L0.

A single-number statistic is _going_ to leave things out, so if you must boil things down into one number, or even a few numbers, you're going to lose something that you had in the raw data. This is why I find claims along the lines of "statistics don't tell the whole story" a little bemusing - of course they don't, the very definition of a statistic is a summarization of data that is easier to work with. The question is what data is kept or lost, or more generally what importance we place on different aspects of the raw data such that it's reflected in our descriptive statistics.

The lessons for non-technical people who want to communicate with descriptive statistics are to recognize that summarization is inherent in the nature of any descriptive statistic, that they are thereby opinionated in some way in terms of what they've preserved and what they've left out, and to recognize whether those opinions are appropriate for your purpose.

Something like what you describe is more or less how statisticians do in fact reason about statistics: https://en.wikipedia.org/wiki/Sufficient_statistic#Minimal_s..., though only for i.i.d. data from a parametric distribution.