### 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 1/4th cumulative frequency = 36
• Median - the class value for the 1/2 of the cumulative frequency = 55
• Upper Quartile - the class value for the 3/4th 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.

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

1. 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:

1. the median
2. quartiles
3. IQR
4. the pass mark, if 4/7th of the students are passed
5. the number of students who scored more than 50
2. 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.