You are spot-on. I would just add that there's another relatively beautiful reason why, in practice, folks pick the mean and variance for their summary; it can

You are spot-on. I would just add that there's another relatively beautiful reason why, in practice, folks pick the mean and variance for their summary; it can be done online with live data, for O(1) time and space! [0] If we extend this idea, then we easily get kurtosis and skew as the next two moments, again in constant time and space, and again getting a normal distribution but now with skew and squish. This is non-trivial; it means that we have an online algorithm which sends our measurements directly to our summaries, without worrying about how detailed our measurements are (how many samples are summarized). For comparison, taking the median/quantiles/percentiles/etc. requires either fixed-size buckets which lose precision (as seen in Prometheus histograms) or around O(log log n) space [1], which is still practical but pedantically not constant. [0] https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance https://en.wikipedia.org/wiki/Algorithms_for_calculating_var... [1] https://tmc.web.engr.illinois.edu/lb.ps https://tmc.web.engr.illinois.edu/lb.ps