Kurtosis for a data distribution

This post “Kurtosis for a data distribution” will certainly help you explore a data distribution.

A distribution is simply a collection of data, or scores, on a variable. Usually, these scores are arranged in order from smallest to largest and then they can be presented graphically.

Page 6, Statistics in Plain English, Third Edition, 2010.

The shape of the distribution can be studied with the help of Kurtosis also.

Kurtosis

Prof. Karl Pearson has called it the “Convexity of a Curve”. Kurtosis gives a measure of the flatness of distribution.

The degree of kurtosis of a distribution is measured relative to that of a normal curve. The curves with greater peakedness than the normal curve are called “Leptokurtic”. The curves which are flatter than the normal curve are called “Platykurtic”. The normal curve is called “Mesokurtic.”

The Figure below describes the three different curves mentioned above:

Kurtosis for a data distribution

Kurtosis is also called the fourth standardized moment of the data distribution (The third standardized moment is skewness, the second central moment is variance & the first moment is mean).

Kurtosis gives an idea of how fat/heavy are the tails of the distribution (and hence, how peaked/flat it is), i.e how frequent extreme deviations (or outliers) are from the average value.

Karl Pearson’s Measures of Kurtosis

Kurtosis for a data distribution

In the above formula, the denominator is the “Second order central moment of distribution” and the numerator is the “Fourth order central moment of distribution”.

Kurtosis for a data distribution
Kurtosis for a data distribution

where,

are the n observations. x̅ is the mean of the n observations.

If beta2= 3, then the curve is said to be mesokurtic

If beta2 < 3, then the curve is said to be platykurtic

If beta2 > 3, then the curve is said to be leptokurtic

Conclusion

Kurtosis is a statistical measure that describes the shape of a data distribution’s tails in relation to its peak. A distribution with high kurtosis has heavy tails, while a distribution with low kurtosis has light tails. Hope this article had helped in shedding some light on “kurtosis for data distribution”.

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