The median of a distribution is the value of the variable which divides it into two equal parts. It is the value such that the number of observations above it is equal to the number of observations below it The median is thus a positional average.

**Steps to Calculate the Median For Grouped Data:**

- The grouped frequency distribution consists of class intervals and corresponding frequencies.
- Sum up the values of all the frequencies and denote it by the letter N. We have that
*Σ*f = N. - Calculate the (less than) cumulative frequencies for the given distribution. The cumulative frequencies can be obtained by summing up the current and preceding frequencies for each class interval.
- Calculate the value N/2 and note down the value of the cumulative frequency just greater than N/2.
- The class corresponding to the cumulative frequency just greater than N/2 is called the median class and the value of the median is obtained by the formula given below.

**Formula for Median of Grouped Data:**

The median for data given in the form of a grouped frequency distribution table can be calculated using the formula,

Median = L + h/f*(N/2 – c).

- where L is the lower limit of the median class.
- h is the size of the median class.
- f is the frequency of the median class.
- c is the cumulative frequency of the class interval preceding the median class.

**Example:**

Suppose we are given a grouped frequency distribution table as follows,

Class Interval | Frequency |

100-110 | 9 |

110-120 | 11 |

120-130 | 13 |

130-140 | 18 |

140-150 | 19 |

150-160 | 14 |

160-170 | 6 |

** Solution**: We first sum up all the frequencies to find N. We also calculate the cumulative frequencies as shown below.

Class Interval | Frequency | Cumulative Frequency |

100-110 | 9 | 9 |

110-120 | 11 | 9+11 = 20 |

120-130 | 13 | 20 +13 = 33 |

130-140 | 18 | 33 + 18 = 51 |

140-150 | 19 | 51 + 19 = 70 |

150-160 | 14 | 70 + 14 = 84 |

160-170 | 6 | 84 + 6 =90 |

N = 90 |

The value of N/2 is equal to N/2 = 90/2 = 45.

The cumulative frequency just greater than N/2 is 51.

Hence the corresponding class interval 130-140 is the median class.

We have that L = 130, h = 10 and, f = 18.

The cumulative frequency of the class just before 130-140 is 33. Hence we have that c=33.

On substituting these values in the above formula we have that,

Median = L + h/f*(N/2 – c).

Median = 130 + 10/18*(51 – 33) = 130 + 6.6667 = **136.6667**.

We have thus found the required value of the median. Out of the 90 values in the above data set around half of them lie below 136.6667 and around half of these values lie above the median value.