Distributions of data and probability distributions are not all the same shape. Some are asymmetric and skewed to the left or to the right. Other distributions are bimodal and have two peaks. In other words there are two values that dominate the distribution of values. Another feature to consider when talking about a distribution is not just the number of peaks but the shape of them. Kurtosis is the measure of the peak of a distribution, and indicates how high the distribution is around the mean. The kurtosis of a distributions is in one of three categories of classification:
We will consider each of these classifications in turn.
Kurtosis is typically measured with respect to the normal distribution. A distribution that is peaked in the same way as any normal distribution, not just the standard normal distribution, is said to be mesokurtic. The peak of a mesokurtic distribution is neither high nor low, rather it is considered to be a baseline for the two other classifications.
Besides normal distributions, binomial distributions for which p is close to 1/2 are considered to be mesokurtic.
A leptokurtic distribution is one that has kurtosis greater than a mesokurtic distribution. Leptokurtic distributions are identified by peaks that are thin and tall. The tails of these distributions, to both the right and the left, are thick and heavy. Leptokurtic distributions are named by the prefix "lepto" meaning "skinny."
There are many examples of leptokurtic distributions. One of the most well known leptokiurtic distributions is Student's t distribution.
The third classification for kurtosis is platykurtic. Platykurtic distributions are those that have a peak lower than a mesokurtic distribution. Platykurtic distributions are characterized by a certain flatness to the peak, and have slender tails. The name of these types of distributions come from the meaning of the prefix "platy" meaning "broad."
Calculation of Kurtosis
These classifications of kurtosis are still somewhat subjective and qualitative. While we might be able to see that a distribution has a higher peak than a normal distribution, what if we don’t have the graph of a normal distribution to compare with? What if we want to say that one distribution is more leptokurtic than another?
To answer these kinds of questions we need not just a qualitative description of kurtosis, but a quantitative measure. The formula used is μ4/σ4 where μ4 is Pearson’s fourth moment about the mean and sigma is the standard deviation.
Now that we have a way to calculate kurtosis, we can compare the values obtained rather than shapes. The normal distribution is found to have a kurtosis of three. This now becomes our basis for mesokurtic distributions. A distribution with kurtosis greater than three is leptokurtic and a distribution with kurtosis less than three is platykurtic.
Since we treat a mesokurtic distribution as a baseline for our other distributions, we can subtract three from our standard calculation for kurtosis. The formula μ4/σ4 - 3 is the formula for excess kurtosis. We could then classify a distribution from its excess kurtosis:
- Mesokurtic distributions have excess kurtosis of zero.
- Platykurtic distributions have negative excess kurtosis.
- Leptokurtic distributions have positive excess kurtosis.
A Note on the Name
The word "kurtosis" seems odd on the first or second reading. It actually makes sense, but we need to know Greek to recognize this. Kurtosis is derived from a transliteration of the Greek word kurtos. This Greek word has the meaning "arched" or "bulging," making it a perfect description of the concept known as kurtosis.