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 - 203
21 - 406
41 - 609
61 - 808
81 - 1004

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 203
up to 409
up to 6018
up to 8026
up to 10030

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.

statistics with excel


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:

  1. The time taken for a certain test by a group of students are is as follows:
    Marks (x) frequency (f)
    11 - 203
    21 - 307
    31 - 4018
    41 - 505
    51 - 602

    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.






Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.


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Recommended Reading


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