### Data Handling: Cumulative Frequency, Quartiles and IQR

#### Cumulative Frequency

The marks obtained by a group of students for maths are as follows:

Marks (x) | frequency (f) |

0 - 20 | 3 |

21 - 40 | 6 |

41 - 60 | 9 |

61 - 80 | 8 |

81 - 100 | 4 |

Now, we can rearrange the table with a running-total as frequency. This frequency is called **cumulative frequency**. A cumulative frequency table consists of classes
up to a certain upper value and cumulative frequencies, which takes the following form.

Marks (x) | cumulative frequency (f) |

up to 20 | 3 |

up to 40 | 9 |

up to 60 | 18 |

up to 80 | 26 |

up to 100 | 30 |

Now you can plot a graph to represent the above data and it looks like the following:

From the graph, we can find the the following:

**Lower Quartile**- the class value for the 1/4^{th}cumulative frequency = 36**Median**- the class value for the 1/2 of the cumulative frequency = 55**Upper Quartile**- the class value for the 3/4^{th}cumulative frequency = 68**Inter Quartile Range**- the difference between the quartiles - 68 - 36 = 32

Now, we can draw a box-plot; it shows the minimum-value, LQ, Median, UQ and maximum value in that order as shown in the image.

#### Averages and Spread of Data with Excel

In the following animation, a set of random data is generated with Excel and then averages and spread of data are calculated - automatically.

If you would like to practise more, please visit this page.

**Now, in order to complement what you have learnt so far, work out the following:**

- The time taken for a certain test by a group of students are is as follows:

**Marks (x)****frequency (f)**11 - 20 3 21 - 30 7 31 - 40 18 41 - 50 5 51 - 60 2 Find the following:

- the median
- quartiles
- IQR
- the pass mark, if 4/7
^{th}of the students are passed - the number of students who scored more than 50

- The time taken by 25 kids to finish their lunch, in minutes, is as follows:

8, 9, 7, 11, 7, 12, 15, 9, 9, 10, 11, 17, 16, 10, 11, 6, 19, 12, 17, 15, 16, 12, 13, 13, 17

Construct an appropriate grouped-frequency table for this data and then plot a cumulative frequency graph for the same. Then calculate the mean, median, quartiles and IQR for the data.