# 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

${\beta_2 = \frac{\mu_4}{\mu_2}}$

Where −

${\mu_4 = \frac{\sum(x- \bar x)^4}{N}}$

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.

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-pt m | m-${\frac{170}{20}}$ d | ${fd}$ | ${fd^2}$ | ${fd^3}$ | ${fd^4}$ |
---|---|---|---|---|---|---|---|

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 |

${N=45}$ | ${\sum fd = 10}$ | ${\sum fd^2 = 64}$ | ${\sum fd^3 = 40}$ | ${\sum fd^4 = 330}$ |

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'

## Moments about mean

From the value of movement about mean, we can now calculate ${\beta_1}$ and ${\beta_2}$:

From the above calculations, it can be concluded that ${\beta_1}$, which measures skewness is almost zero, thereby indicating that the distribution is almost symmetrical. ${\beta_2}$ Which measures kurtosis, has a value greater than 3, thus implying that the distribution is leptokurtic.