The Kurtosis of a distribution gives us some idea about the shape of the distribution. Depending on the value of the Kurtosis we can classify our data as leptokurtic, mesokurtic, and platykurtic. The Kurtosis of a given set of ungrouped data values can be calculated using the formula, \text{Kurtosis }= \frac{\sum (x_i-\bar{x})^4}{n\sigma^4} where, \bar{x} denotes the mean and \sigma denotes the standard deviation.

**Example 1: Kurtosis for Ungrouped Data**

Consider the following set of ungrouped data values, 23, 34, 38, 47, 59, 63, 84. We first calculate the values of the mean and the standard deviation. \bar{x}=\frac{\sum x_i}{n} = \frac{23+34+38+47+59+63+84}{7}= \frac{348}{7} = 49.7143.

x_i | (x_i-\bar{x}) | (x_i-\bar{x})^2 | (x_i-\bar{x})^3 | (x_i-\bar{x})^4 |

=(x_i-49.7143) | =(x_i-49.7143)^2 | =(x_i-49.7143)^3 | =(x_i-49.7143)^4 | |

23 | -26.714 | 713.653 | -19065 | 509301 |

34 | -15.714 | 246.939 | -3880.5 | 60978.8 |

38 | -11.714 | 137.225 | -1607.5 | 18830.6 |

47 | -2.7143 | 7.3673 | -19.997 | 54.2778 |

59 | 9.2857 | 86.2245 | 800.656 | 7434.66 |

63 | 13.2857 | 176.51 | 2345.06 | 31155.9 |

84 | 34.2857 | 1175.51 | 40303.2 | 1381824 |

TOTAL = 348 | TOTAL = 0 | TOTAL = 2543.43 | TOTAL = 18876.2 | TOTAL = 2009579 |

We now calculate the Kurtosis using the formula,\text{Kurtosis }= \frac{\sum (x_i-\bar{x})^4}{n\sigma^4} \text{Kurtosis }= \frac{2009579}{7\times 19.0617^4} = 2.1745

**Example 2: Grouped Data Kurtosis Calculation**

Consider the following set of data values given in the form of a grouped frequency distribution table.

Class Intervals | Frequency |

0-5 | 2 |

5-10 | 3 |

10-15 | 1 |

15-20 | 4 |

20-25 | 5 |

25-30 | 9 |

30-35 | 6 |

35-40 | 12 |

40-45 | 8 |

45-50 | 7 |

We calculate the values required to calculate the mean and the standard deviation,

Classes | Class Mark (x_i) | Frequency (f_i) | f_ix_i | x_i-\bar{x} | f_i(x_i-\bar{x})^2 | f_i(x_i-\bar{x})^4 |

0 – 5 | 2.5 | 2 | 5 | -28.86 | 1665.76 | 1387376 |

5-10 | 7.5 | 3 | 22.5 | -23.86 | 1707.85 | 972249 |

10-15 | 12.5 | 1 | 12.5 | -18.86 | 355.686 | 126513 |

15 – 20 | 17.5 | 4 | 70 | -13.86 | 768.36 | 147594 |

20 – 25 | 22.5 | 5 | 112.5 | -8.8596 | 392.467 | 30806.1 |

25 – 30 | 27.5 | 9 | 247.5 | -3.8596 | 134.072 | 1997.26 |

30 – 35 | 32.5 | 6 | 195 | 1.1404 | 7.8024 | 10.1462 |

35 – 40 | 37.5 | 12 | 450 | 6.1404 | 452.447 | 17059 |

40 – 45 | 42.5 | 8 | 340 | 11.1404 | 992.859 | 123221 |

45 – 50 | 47.5 | 7 | 332.5 | 16.1404 | 1823.58 | 475062 |

n=57 | \sum f_ix_i=1787.5 | \sum f_i(x_i-\bar{x})^2 =8300.877 | \sum f_i(x_i-\bar{x})^4= 3281887 |

The formula for calculating kurtosis for a set of grouped data values is as follows,

\text{Kurtosis } = \frac{\sum_{i=1}^{n}f_i(x_i-\bar{x})^4}{n\sigma^4}= \frac{3281887.0786}{57 \times 12.0677^4}= 2.7149