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 leptokurtic distributions is Student's t distribution. The third classification for kurtosis is platykurtic. Platykurtic distributions are those that have slender tails. Many times they possess a peak lower than a mesokurtic distribution.
The name of these types of distributions come from the meaning of the prefix "platy" meaning "broad. All uniform distributions are platykurtic.
In addition to this, the discrete probability distribution from a single flip of a coin is platykurtic. These classifications of kurtosis are still somewhat subjective and qualitative. 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. 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.
We could then classify a distribution from its excess kurtosis:. 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 an apt description of the concept known as kurtosis. Whereas skewness differentiates extreme values in one versus the other tail, kurtosis measures extreme values in either tail.
Distributions with large kurtosis exhibit tail data exceeding the tails of the normal distribution e. Distributions with low kurtosis exhibit tail data that are generally less extreme than the tails of the normal distribution. This phenomenon is known as kurtosis risk. Kurtosis is a measure of the combined weight of a distribution's tails relative to the center of the distribution. When a set of approximately normal data is graphed via a histogram, it shows a bell peak and most data within three standard deviations plus or minus of the mean.
However, when high kurtosis is present, the tails extend farther than the three standard deviations of the normal bell-curved distribution.
Kurtosis is sometimes confused with a measure of the peakedness of a distribution. However, kurtosis is a measure that describes the shape of a distribution's tails in relation to its overall shape. A distribution can be infinitely peaked with low kurtosis, and a distribution can be perfectly flat-topped with infinite kurtosis. Thus, kurtosis measures "tailedness," not "peakedness.
There are three categories of kurtosis that can be displayed by a set of data. All measures of kurtosis are compared against a standard normal distribution, or bell curve. The first category of kurtosis is a mesokurtic distribution. This distribution has a kurtosis statistic similar to that of the normal distribution, meaning the extreme value characteristic of the distribution is similar to that of a normal distribution.
The second category is a leptokurtic distribution. Any distribution that is leptokurtic displays greater kurtosis than a mesokurtic distribution. Characteristics of this distribution is one with long tails outliers. The prefix of "lepto-" means "skinny," making the shape of a leptokurtic distribution easier to remember. The "skinniness" of a leptokurtic distribution is a consequence of the outliers, which stretch the horizontal axis of the histogram graph, making the bulk of the data appear in a narrow "skinny" vertical range.
Thus leptokurtic distributions are sometimes characterized as "concentrated toward the mean," but the more relevant issue especially for investors is there are occasional extreme outliers that cause this "concentration" appearance.
Examples of leptokurtic distributions are the T-distributions with small degrees of freedom. The final type of distribution is a platykurtic distribution. These types of distributions have short tails paucity of outliers.
Toggle navigation. Statistics Kurtosis What is Kurtosis? What is Kurtosis? Saul McLeod , published What kurtosis tells us? Further Information. How to reference this article: How to reference this article: McLeod, S. Back to top.
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