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 illustration1 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.
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.