Why Kurtosis is Like Liposuction. And Why it Matters.
The word kurtosis sounds like a painful, festering disease of the gums. But the term actually describes the shape of a data distribution.
Frequently, you'll see kurtosis defined as how sharply "peaked" the data are. The three main types of kurtosis are shown below.
Lepto means "thin" or "slender" in Greek. In leptokurtosis, the kurtosis value is high.
Platy means "broad" or "flat"—as in duck-billed platypus. In platykurtosis, the kurtosis value is low.
Meso means "middle" or "between." The normal distribution is mesokurtic.
Mesokurtosis can be defined with a value of 0 (called its "excess" kurtosis value). Using that benchmark, leptokurtic distributions have positive kurtosis values and platykurtic distributions have negative kurtosis values.
Question: Which type of kurtosis correctly describes each of the three distributions (blue, red, yellow) shown below?
Answer: All three distributions are examples of mesokurtosis. They're all normal distributions. The (excess) kurtosis value is 0 for each distribution.
OK, that was a mean trick question. You can roast me in the comments field. But it had a good intention—to illustrate some common misconceptions about kurtosis.
Each normal distribution shown above has a different variance. Different variances can appear to change the "peakedness" of a given distribution when they're displayed together along the same scale. But that's not the same thing as kurtosis.
Think of Kurtosis Like Liposuction
In Nature, there's no such thing as a free lunch—literally. Research suggests that fat that's liposuctioned from one part of the body all returns within a year. It just moves to a different place in the body.
Something similar happens with kurtosis. The clearest way to see this is to compare probability distribution plots for distributions with the same variance but with different kurtosis values. Here's an example.
The solid blue line shows the normal distribution (excess kurtosis ≈ 0). That's the body before liposuction. The dotted red line shows a leptokurtic distribution (excess kurtosis ≈ 5.6) with the same variance. That's the body one year after liposuction.
The arrows show where the fat (the data) moves after being "sucked out" from the sides of the normal distribution. The blue arrows show that some data shifts toward the center, giving the leptokurtic distribution its characteristic sharp, thin peak.
But that's not where all the data goes. Notice the data that relocates to the extreme tails of the distribution, as shown by the red arrows.
So the leptokurtic distribution has a thinner, sharper peak, but also—very importantly— "fatter" tails.
Conversely, here's how "liposuction" of the normal distribution results in platykurtosis (excess kurtosis ≈ - 0.83).
Here, data from the peak and from the tails of the normal distribution is redistributed to the sides. This gives the platykurtic distribution its blunter, broader peak, but-— very importantly— its thinner tails.
In fact, kurtosis is actually more influenced by data in the tails of the distribution than data in the center of a distribution. It's really a measure of how heavy the tails of a distribution are relative to its variance. That is, how much the variation in the data is due to extreme values located further away from the mean.
Why Does It Matter?
Consider the three normal distributions that appeared to mimic different types of kurtosis, when in fact they had the same kurtosis, just different variances.
For each of these distributions, the same percentage of data falls within a given number of standard deviations from the mean. That is, for all three distributions, approximately 68.2% of observations are within +/- 1 standard deviation of the mean; 95.4% are within +/- 2 standards deviations of the mean; and 99.7% are within +/- 3 standard deviations of the mean.
What would happen if you tried to use this same rubric on a distribution that was extremely leptokurtic or platykurtic? You'd make some serious estimation errors due to the fatter (or thinner) tails associated with kurtosis.
You Could Lose All Your Money, Too.
In fact, something like that appears to have happened in the financial markets in the late 90s according to Wikipedia. Some hedge funds underestimated the risk of variables with high kurtosis (leptokurtosis). In other words, their models didn't take into consideration the likelihood of data located in the "fatter" extreme tails—which was associated with greater volatility and higher risk. The end result? The hedge funds went belly up and needed bailing out.
I don't have a background in financial modelling, so I can't verify that claim. But it wouldn't surprise me.
If you click on the following link to Investopedia, you'll see a definition of high kurtosis as "a low, even distribution" with fat tails. Fat tails, yes. But "low and even"?
Hmmm. I hope the investment firm managing my 401K isn't using that definition.
If so, it might be time to move my money into an investment vehicle with a much lower kurtosis risk. Like my mattress.