The Story of Kurtosis

Introduction to Research Design and Statistics

The Story of Kurtois

Introduction

Kurtosis is one of the least discussed indices of normality. It seems that skewness (non-symmetry) gets a lot more press than does kurtosis. Kurtosis can be considered as an index of peakedness or flatness of a distribution. A leptokurtics distribution is more peaked than an normal while a platykurtic distribution is flatter. Or, kurtosis can be viewed as a measure of tail heaviness, that is, a leptokurtic distribution has heavier tails than a normal. This means that there are more cases far from the mean than is found in a normal distribution. Converseley, a playkurtic distribution has fewer cases in the tails then would be expected in a normal distribution.

A normal distribution is said to be mesokurtic and has a 'normal' amount of peakedness and tails that are not too heavy nor too light.

There are several different but related formulae for computing kurtosis. The one given below is the one used by Stata.

kurtosis = m₄/m₂² 

where m₂ = Σ(X - mean)²/N
      m₄ = Σ(X - mean)⁴/N

The formula uses the second and fourth moments about the mean. Moments involve powers of deviations, for example the second moment uses the sum of squared deviations and the fourth moment uses the sum of the fourth powers of deviations.

The second moment about the mean, m₂, is really a version of the variance related to the unbiased estimate of the variance that uses N-1 in the denominator. Thus, the index of kurtosis is based on ratio of the fourth moment about the mean divided by the variance squared. In a normal distribution the kurtosis is equal to 3. A platykurtic distribution has a value less than three, while leptokurtic distributions have values greater than three. Below, are some examples to give you an idea of the values that kurtosis can take on.

Some statistics packages use the following formula for computing kurtosis.

kurtosis = (m₄/m₂²) - 3

With this version of the formula, a normal distribution has a kurtosis of zero.

Examples:

Here are seven different distributions with a mean = 10 that vary in the amount of kurtosis. As the examples progress from 1 to 7 the kurtosis value decreases.

ex 1: kurtosis = 11.00 ex 2: kurtosis = 4.33  ex 3: kurtosis = 3.00
ex 4: kurtosis =  2.00 ex 5: kurtosis = 1.50  ex 6: kurtosis = 1.25
ex 7: kurtosis =  1.00

The next example is extremely leptokurtic. It has a normal curve superimpossed over the histogram.

ex 8: kurtosis = 26.65

Next we have an example that is less leptokurtic than the previous example and, in fact, does not have a terrible fit with a normal distribution.

ex 9: kurtosis = 4.89

The final example shows a normal distribution or at least as close as we can come to a normal distribution with 400 observations.

ex 10: kurtosis = 2.98

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Phil Ender, 24Oct04