Statistics Definitions > Leptokurtic

*Leptokurtic is directly related to kurtosis (a guide to how “peaked” your graph is), so you may want to read this article first: What is Kurtosis?*

A leptokurtic distribution has excess positive kurtosis, where the kurtosis is greater than 3. “Lepto-” means *slender*, referring to the tall, slender peak in the distribution. The distribution looks like a normal distribution at first glance. The following illustration^{1} shows a leptokurtic distribution along with a normal distribution (dotted line).

As you can probably tell, a leptokurtic distribution has a sharper peak around the mean. In other words, there are more values closer to the mean. The opposite is a platykurtic distribution, which is broad and flat like the uniform distribution.

## The Leptokurtic T-Test

The Student’s t-test is an example of a leptokurtic distribution. The t-distribution has fatter tails than the normal (you can also look at the first image above to see the fatter tails). Therefore, the critical values in a Student’s t-test will be larger than the critical values from a z-test.

## Financial Markets

Kurtosis isn’t just a theory confined to mathematical textbooks; it has real life applications, especially in the world of economics. Fund managers usually focus on risks and returns, kurtosis (in particular if an investment is lepto- or platy-kurtic). According to stock trader and analyst Michael Harris, a leptokurtic return means that risks are coming from outlier events. This would be a stock for investors willing to take extreme risks. For example, real estate (with a kurtosis of 8.75) and High Yield US bonds (8.63) are high risk investments while Investment grade US bonds (1.06) and Small cap US stocks (1.08) would be considered safer investments.

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But kurtosis tells you nothing about the peak of the distribution. See http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4321753/pdf/nihms-599845.pdf

Thanks for your comment, Peter. It’s become widespread to think of kurtosis as peakedness. This could possibly be because it’s much easier for the average student to visualize what a tall or flat-peaked distribution looks like (as opposed to a distribution with fat tails or thin tails). I did add a little more info about the fact that fat tails leads to a flatter peak. I hope that clarifies it!

No, there is no relationship between the peak and the tails.

Here is why “peakedness” is wrong.

Suppose someone tells you that they have calculated negative excess kurtosis either from data or from a probability distribution function (pdf). According to the “peakedness” dogma (started unfortunately by Pearson in 1905), you are supposed to conclude that the distribution is “flat-topped” when graphed. But this is obviously false in general. For one example, the beta(.5,1) has an infinite peak and has negative excess kurtosis. For another example, the 0.5*N(0, 1) + 0.5*N(4,1) distribution is bimodal (wavy); not flat at all, and also has negative excess kurtosis similar to that of the uniform (U(0,1)) distribution. These are just two examples out of an infinite number of other non-flat-topped distributions having negative excess kurtosis.

Yes, the U(0,1) distribution is flat-topped and has negative excess kurtosis. But obviously, a single example does not prove the general case. If that were so, we could say, based on the beta(.5,1) distribution, that negative excess kurtosis implies that the pdf is infinitely pointy. We could also say, based on the 0.5*N(0, 1) + 0.5*N(4,1) distribution, that negative excess kurtosis implies that the pdf is “wavy.” It’s like saying, “well, I know all bears are mammals, so it must be the case that all mammals are bears.”

Now suppose someone tells you that they have calculated positive excess kurtosis from either data or a pdf. According to the “peakedness” dogma (again, started by Pearson in 1905), you are supposed to conclude that the distribution is “peaked” or “pointy” when graphed. But this is also obviously false in general. For example, take a U(0,1) distribution and mix it with a N(0,1000000) distribution, with .00001 mixing probability on the normal. The distribution, when graphed, appears perfectly flat at its peak, but has very high kurtosis.

You can play the same game with any distribution other than U(0,1). If you take a distribution with any shape peak whatsoever, then mix it with a much wider distribution like N(0,1000000), with small mixing probability, you will get a pdf with the same shape of peak (flat, bimodal, trimodal, sinusoidal, whatever) as the original, but with high kurtosis.

And yes, the Laplace distribution has positive excess kurtosis and is pointy. But you can have any shape of the peak whatsoever and have positive excess kurtosis. So the bear/mammal analogy applies again.

One thing that can be said about cases where the data exhibit high kurtosis is that when you draw the histogram, the peak will occupy a narrow vertical strip of the graph. The reason this happens is that there will be a very small proportion of outliers (call them “rare extreme observations” if you do not like the term “outliers”) that occupy most of the horizontal scale, leading to an appearance of the histogram that some have characterized as “peaked” or “concentrated toward the mean.”

But the outliers do not determine the shape of the peak. When you zoom in on the bulk of the data, which is, after all, what is most commonly observed, you can have any shape whatsoever – pointy, inverted U, flat, sinusoidal, bimodal, trimodal, etc.

So, given that someone tells you that there is high kurtosis, all you can legitimately infer, in the absence of any other information, is that there are rare, extreme data points (or potentially observable data points). Other than the rare, extreme data points, you have no idea whatsoever as to what is the shape of the peak without actually drawing the histogram (or pdf), and zooming in on the location of the majority of the (potential) data points.

And given that someone tells you that there is negative excess kurtosis, all you can legitimately infer, in the absence of any other information, is that the outlier characteristic of the data (or pdf) is less extreme than that of a normal distribution. But you will have no idea whatsoever as to what is the shape of the peak, without actually drawing the histogram (or pdf).

The logic for why the kurtosis statistic measures outliers (rare, extreme observations in the case of data; potential rare, extreme observations in the case of a pdf) rather than the peak is actually quite simple. Kurtosis is the average (or expected value in the case of the pdf) of the Z-values, each taken to the 4th power. In the case where there are (potential) outliers, there will be some extremely large Z^4 values, giving a high kurtosis. If there are less outliers than, say, predicted by a normal pdf, then the most extreme Z^4 values will not be particularly large, giving smaller kurtosis.

What of the peak? Well, near the peak, the Z^4 values are extremely small and contribute very little to their overall average (which again, is the kurtosis). That is why kurtosis tells you virtually nothing about the shape of the peak. I give mathematical bounds on the contribution of the data near the peak to the kurtosis measure in the following article:

Kurtosis as Peakedness, 1905 – 2014. R.I.P. The American Statistician, 68, 191–195.

I hope this helps.

Peter Westfall

P.S. The height of the peak is also unrelated to kurtosis; see Kaplansky, I. (1945), “A Common Error Concerning Kurtosis,” Journal of the American Statistical Association, 40, 259). But the “height” misinterpretation also seems to persist.