# Statistics – Kurtosis

The degree of flatness or peakedness is measured by kurtosis. It tells us about the extent to which the distribution is flat or peak vis-a-vis the normal curve. Diagrammatically, shows the shape of three different types of curves.

The normal curve is called Mesokurtic curve. If the curve of a distribution is more peaked than a normal or mesokurtic curve then it is referred to as a Leptokurtic curve. If a curve is less peaked than a normal curve, it is called as a platykurtic curve. Kurtosis is measured by moments and is given by the following formula:

## Formula

β2=μ4μ2β2=μ4μ2

Where −

·        μ4=∑(x−x¯)4Nμ4=∑(x−x¯)4N

The greater the value of \beta_2 the more peaked or leptokurtic the curve. A normal curve has a value of 3, a leptokurtic has \beta_2 greater than 3 and platykurtic has \beta_2 less then 3.

### Example

Problem Statement:

The data on daily wages of 45 workers of a factory are given. Compute \beta_1 and \beta_2 using moment about the mean. Comment on the results.

Solution:

Since the deviations have been taken from an assumed mean, hence we first calculate moments about arbitrary origin and then moments about mean. Moments about arbitrary origin ‘170’

μ11=∑fdN×i=1045×20=4.44μ12=∑fd2N×i2=6445×202=568.88μ13=∑fd2N×i3=4045×203=7111.11μ14=∑fd4N×i4=33045×204=1173333.33μ11=∑fdN×i=1045×20=4.44μ21=∑fd2N×i2=6445×202=568.88μ31=∑fd2N×i3=4045×203=7111.11μ41=∑fd4N×i4=33045×204=1173333.33