The mode refers to the value which occurs in the data with the highest frequency. In a grouped frequency distribution, the data is arranged into class intervals so by looking at the highest frequency we can only determine which class interval the mode lies in. In order to determine the exact value for the mode, we need to use the formula given below.

**Formula for Calculating Mode for Grouped Data:**

Given a grouped frequency distribution table we first identify the class interval which has the highest frequency. This class interval is known as the modal class interval and the frequency is called the modal frequency. The mode lies somewhere within the modal class interval for which the exact value can be calculated using the formula, \text{Mode }= l + \frac{h(f_1 - f_0)}{2f_1 - f_0 - f_2}

where, l \text{ is the lower limit of the modal class }\\ h \text{ is the size of the modal class }\\ f_1 \text{ is the frequency of the modal class }\\ f_0 \text{ is the frequency of the class interval preceding the modal class }\\ f_2 \text{ is the frequency of the class interval succeeding the modal class }

**Example 1:**

We can calculate the mode for the following grouped frequency distribution table:

Class Interval | Frequency |

0-10 | 3 |

10-20 | 8 |

20-30 | 11 |

30-40 | 13 |

40-50 | 6 |

50-60 | 2 |

60-70 | 1 |

** Solution**: Here the highest frequency is 13, so the modal class is 30-40. \text{The lower limit of the modal class is }l=30.\\ \text{The size of the modal class is } h=10\\ f_1=13 \text{ is the frequency of the modal class }\\ f_0 = 11 \text{ is the frequency of the class interval preceding the modal class }\\ f_2 = 6 \text{ is the frequency of the class interval succeeding the modal class }

Substituting all these values into the above formula we get that, \text{Mode }= l + \frac{h(f_1 - f_0)}{2f_1 - f_0 - f_2} \text{Mode }= 30 + \frac{10 \times (13 - 11)}{(2\times 13) - 11 - 6} \text{Mode }= 30 + \frac{20}{9} = 30 + 2.2222 = 32.2222

**Example 2:**

We can calculate the mode for the following grouped frequency distribution table:

Class Interval | Frequency |

300-350 | 39 |

350-400 | 67 |

400-450 | 88 |

450-500 | 53 |

500-550 | 44 |

550-600 | 33 |

** Solution**: Here the highest frequency is 88, so the modal class is 400-450. We have that l=400, h=50, f_1=88, f_0=67 \text{ and }\\ f_2 = 53

Substituting all these values into the above formula we get that, \text{Mode }= l + \frac{h(f_1 - f_0)}{2f_1 - f_0 - f_2} \text{Mode }= 400 + \frac{50 \times (88 - 67)}{(2\times 88) - 67 - 53} \text{Mode }= 400 + \frac{1050}{56} = 400 +18.75 = 418.75