Kurtosis Explained With Two Friends Who Are Late in Very Different Ways

Kurtosis Is About Tails, Not Peaks, And Your Textbook Drew It Wrong

It's midnight. The last train has gone, your phone is on 4%, and you can text one friend for a lift. Steady is never exactly on time but never far off either. Spike is usually bang on the minute, except for those few nights a year when Spike vanishes for half an hour and comes back with a story. Who do you text?

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You picked Steady, didn't you? Congratulations, you just used kurtosis. You looked past the usual nights and judged each friend by their worst ones, the rare far-out values at the edges. That's all kurtosis measures. The tails.

Now open almost any textbook. It'll tell you kurtosis is about how tall and pointy the peak of a curve is. Spike's chart has the taller, pointier peak, so by the textbook Spike is the "high kurtosis" friend for the wrong reason. The number is right. The explanation is wrong, and statisticians have been complaining about it since 1945.

The textbook measures the hat. The kurtosis lives in the boots.

Two friends, same numbers on paper

Here's the setup in numbers. Over 200 meetups, both friends arrive on time on average. Both are off by about six minutes in a typical meetup…