SOUMYA VIPIN: NOTE ON KURTOSIS

LEARNING MATERIAL FOR KURTOSIS

Kurtosis

In probability theory and statistics, kurtosis is the measure of the tailedness of the probability distribution of a real valued random variable. In a similar way to the concept of skeweness, kurtosis is a description of the shape of a probability distribution and, just as for skewness, there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Kurtosis is a statistical measure that is used to describe the distribution. Kurtosis measures extreme values in either tail. The degree of tailedness of a distribution is measured by kurtosis. It tells us the extent to which the distribution is more or less outlier-prone than the normal distribution. Kurtosis is a measure of the combined sizes of the two tails. It measures the amount of probability in the tails. The value is often compared to the kurtosis of the normal distribution, which is equal to 3. If the kurtosis is greater than 3, then the dataset has heavier tails than a normal distribution (more in the tails). If the kurtosis is less than 3, then the dataset has lighter tails than a normal distribution (less in the tails). Kurtosis is sometimes reported as “excess kurtosis.” Excess kurtosis is determined by subtracting 3 form the kurtosis. This makes the normal distribution kurtosis equal 0. Kurtosis originally was thought to measure the peakedness of a distribution. Though you will still see this as part of the definition in many places, this is a misconception.

Definition of kurtosis

Kurtosis is defined by may eminent persons major definitions are given below. “Kurtosis is the degree of peakedness of a distribution” – Wolfram MathWorld. “We use kurtosis as a measure of peakedness (or flatness)” – Real Statistics Using Excel. You can find other definitions that include peakedness or flatness when you search the web.

The problem is these definitions are not correct. Dr. Peter Westfall published an article that addresses why kurtosis does not measure peakedness (link to article). He said: “Kurtosis tells you virtually nothing about the shape of the peak – its only unambiguous interpretation is in terms of tail extremity.” Dr. Westfall includes numerous examples of why you cannot relate the peakedness of the distribution to the kurtosis. Dr. Donald Wheeler also discussed this in his two-part series on skewness and kurtosis. He said: “Kurtosis was originally thought to be a measure the “peakedness” of a distribution. However, since the central portion of the distribution is virtually ignored by this parameter, kurtosis cannot be said to measure peakedness directly. While there is a correlation between peakedness and kurtosis, the relationship is an indirect and imperfect one at best.” Dr. Wheeler defines kurtosis as: “The kurtosis parameter is a measure of the combined weight of the tails relative to the rest of the distribution.” So, kurtosis is all about the tails of the distribution – not the peakedness or flatness. It measures the tail-heaviness of the distribution.

Types of kurtosis.The kurtosis are of three different types mesokurtic, leptokurtic and platykurtic. All of these types explains ho the kurtosis value is deviating from the normal distribution. Mesokurtic distributions are technically defined as having kurtosis of zero, although the distribution doesn’t have to be exactly zero in order for it to be classified as mesokurtic. A leptokurtic distribution has excess positive kurtosis, here the kurtosis is greater than 3. The tails are fatter than the normal distribution. Platykurtic distributions have negative kurtosis. The tails are very thin compared to the normal distribution.