In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable’s mean.
- The zeroth central moment is the total probability i.e. equal to one.
- The first central moment is the expected value or mean and equal to zero.
- The second central moment is the variance.
- The third central moment is skewness.
- The fourth central moment is kurtosis.
In the previous blog, we already covered the third central moment that is skewness. In this blog, we will talk about the fourth central moment that is kurtosis and its significance.
Kurtosis measures the flatness or peakedness of a distribution. Flat-looking distributions are referred to as platykurtic, while peaked distributions are referred to as leptokurtic. While the distribution in between them which looks more normal is referred to as mesokurtic.
A measure of kurtosis, a coefficient by Karl Pearson, is given by –
For a mesokurtic curve, the value of β2 is 3. When β2 > 3, the curve is more peaked than the mesokurtic curve and is termed as leptokurtic. While β2 < 3 implies the curve is flatter as compared to the mesokurtic curve and is called a platykurtic curve.
The below table summarises the β2 and γ2 values and the expected type of distribution we can see.
Talking about its significance, the measure of kurtosis is very helpful in the selection of an appropriate average. For the normal distribution, the mean is most appropriate; for leptokurtic distribution, the median is more appropriate; and for a platykurtic distribution, the quartile range is most appropriate.