**Grouped data** is the data which is categorised into groups after its collection. The raw data can be categorized into various groups using a frequency table.

**Ungrouped data** which is also called raw data is the data that has not been placed in any group or category after its collection. Data is in the form of numbers or characteristics therefore, the data which has not been put in any of the categories is ungrouped.

## Median

The **median** is the middle number of a group of numbers that have been arranged in order by size. If the number of terms is even, the median is the mean of the two middle numbers.

Steps to find the median of a set of numbers

- Arrange the numbers in order by size.
- If the number of terms is odd, the median is the middle term.
- If the number of terms is even, add the two middlemost terms and then divide by 2.

## To Find The Median For Ungrouped Data

**Median = Value of ((n+1)/2) ^{th} item**

**Example 1**: The Clintons drove through 7 states in their winter holidays. The petrol prices were different from state to state. Find the median petrol price.

$1.59, $1.31, $1.96, $3.09, $1.64, $1.55, $2.61

**Solution**: Order the given data from smallest to greatest.

$1.31, $1.55, $1.59, $1.64, $1.96, $2.61, $3.09

The median petrol price is $1.64.

**Example 2**: Following is the data of time taken by 4 students to complete a running race. Find the median race time.

9.7 hr, 6.3 hr, 2.5 hr, 7.1 hr

**Solution**: Arrange the given data in ascending order.

2.5, 6.3, 7.1, 9.7

Since there are even number of observations in the data set, the median can be calculated by taking the mean of the two middlemost numbers.

6.3 + 7.1 = 13.4

13.4 / 2 = 6.7

Hence, the median race time is 6.7 hr.

## Cumulative Frequency

**Cumulative frequency** helps to find the number of observations that lie above (or below) a particular value in a data set. The cumulative frequency is found using a frequency distribution table. The sum obtained by adding each frequency from a frequency distribution table to the sum of its predecessors is called cumulative frequency. The last value will always be equal to the total number of observations since in the previous total all frequencies are already added.

## To Find The Median For Grouped Data

We find the cumulative frequencies and then find the value n/2. The median is present in the group (class) which corresponds to the cumulative frequency in which n/2 lies. The formula for finding the median of grouped data is given below.

**Median = l + (h/f)(n/2 – c) **

Here

l= Lower class interval of the modal class

f= Frequency of the median class

n=∑f= Number of values or total frequency

c= Cumulative frequency of the class preceding the median class

h= Class interval size of the modal class

Example: Calculate the median from the following data.

Class interval | Frequency |

40– 44 | 1 |

45 – 49 | 5 |

50 –54 | 9 |

55 – 59 | 12 |

60 – 64 | 7 |

65 – 69 | 2 |

Class interval | Frequency | Class interval | Cumulative Frequency |

40– 44 | 1 | 39.5 – 44.5 | 1 |

45 – 49 | 5 | 44.5 – 49.5 | 6 |

50 –54 | 9 | 49.5 – 54.5 | 15 |

55 – 59 | 12 | 54.5 – 59.5 | 27 |

60 – 64 | 7 | 59.5 – 64.5 | 34 |

65 – 69 | 2 | 64.5 – 69.5 | 36 |

n/2 = 36/2 = 18

Therefore. Median class is 54.5 – 59.5

Median = l + (h/f)(n/2 -c)

= 54.5 + (15/12)(18 – 15)

= 54.5 + (5/12)(3)

= 54.5+1.25

= 55.75

## To Find The Median For Discrete Data

A set of data is termed **discrete** if the observations belonging to the set are distinct, separate and unconnected observations. When the data follows a discrete set of observations grouped by size, use the formula **((n+1)/2) ^{th }**observation for finding the median. Form a cumulative frequency distribution, and the median is that value which corresponds to the cumulative frequency in which

**((n+1)/2)**observation lies.

^{th }Example:

The given frequency distribution is classified according to the number of cabs provided for different branches of the office. Calculate the median number of cabs.

No of cabs | No of Branches
(f) |
Cumulative Frequency |

3 | 2 | 2 |

4 | 11 | 13 |

5 | 15 | 28 |

6 | 20 | 48 |

7 | 25 | 73 |

8 | 18 | 91 |

9 | 10 | 101 |

Total | 101 |

Median = ((n+1)/2)^{th } observation

= (101+1)/2

= 102/2

51st observation

Median = 7 because 51st value corresponds to 7