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__

__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__

__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.

Wages(Rs.) | Number of Workers |

100-200 | 1 |

120-200 | 2 |

140-200 | 6 |

160-200 | 20 |

180-200 | 11 |

200-200 | 3 |

220-200 | 2 |

**Solution:**

Wages(Rs.) | Number of Workers(f) | Mid-ptm | m-1702017020d | fdfd | fd2fd2 | fd3fd3 | fd4fd4 |

100-200 | 1 | 110 | -3 | -3 | 9 | -27 | 81 |

120-200 | 2 | 130 | -2 | -4 | 8 | -16 | 32 |

140-200 | 6 | 150 | -1 | -6 | 6 | -6 | 6 |

160-200 | 20 | 170 | 0 | 0 | 0 | 0 | 0 |

180-200 | 11 | 190 | 1 | 11 | 11 | 11 | 11 |

200-200 | 3 | 210 | 2 | 6 | 12 | 24 | 48 |

220-200 | 2 | 230 | 3 | 6 | 18 | 54 | 162 |

N=45N=45 | ∑fd=10∑fd=10 | ∑fd2=64∑fd2=64 | ∑fd3=40∑fd3=40 | ∑fd4=330∑fd4=330 |

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

__Moments about mean__

__Moments about mean__

μ2=μ′2−(μ′1)2=568.88−(4.44)2=549.16μ3=μ′3−3(μ′1)(μ′2)+2(μ′1)3=7111.11−(4.44)(568.88)+2(4.44)3=7111.11−7577.48+175.05=−291.32μ4=μ′4−4(μ′1)(μ′3)+6(μ1)2(μ′2)−3(μ′1)4=1173333.33−4(4.44)(7111.11)+6(4.44)2(568.88)−3(4.44)4=1173333.33−126293.31+67288.03−1165.87=1113162.18μ2=μ2′−(μ1′)2=568.88−(4.44)2=549.16μ3=μ3′−3(μ1′)(μ2′)+2(μ1′)3=7111.11−(4.44)(568.88)+2(4.44)3=7111.11−7577.48+175.05=−291.32μ4=μ4′−4(μ1′)(μ3′)+6(μ1)2(μ2′)−3(μ1′)4=1173333.33−4(4.44)(7111.11)+6(4.44)2(568.88)−3(4.44)4=1173333.33−126293.31+67288.03−1165.87=1113162.18

From the value of movement about mean, we can now calculate β1β1 and β2β2:

β1=μ23=(−291.32)2(549.16)3=0.00051β2=μ4(μ2)2=1113162.18(546.16)2=3.69β1=μ32=(−291.32)2(549.16)3=0.00051β2=μ4(μ2)2=1113162.18(546.16)2=3.69

From the above calculations, it can be concluded that β1β1, which measures skewness is almost zero, thereby indicating that the distribution is almost symmetrical. β2β2 Which measures kurtosis, has a value greater than 3, thus implying that the distribution is leptokurtic.