Cumulative Frequency & Box Plots
Introduction
A common way for displaying data is to use what is known as a box plot or a box and whisker plot.
This measures five pieces of data which include the lowest value, highest value, lower quartile, upper quartile and also the median.
A box plot is drawn against a scale so it is accurate.
A cumulative frequency is found by adding each frequency to give the sum of all frequencies. When it comes to drawing a cumulative frequency curve you need to make sure that you are plotting the upper bound of the class interval with the appropriate cumulative frequency and to join the data values with a smooth curve.
Cumulative Frequency - Example
Take a look at the following question:
In this question you mark off the points that are given based on the scale that is provided.
The lowest and highest points are straightforward to show.
The median can also be shown
You are given the IQR to be 38 and the LQ to be 32. Using IQR = UQ – LQ then:
38 = UQ – 32 and so the UQ = 70.
The completed box plot is shown below.
Example
Take a look at the following question:
You should also find the sum of the frequency column which is found by adding all values in this column. In this case the total = 12 + 15 + 18 + 12 + 3 = 60.
The cumulative frequency is found by adding in each frequency together as shown in the table below and the last value must always be the same as the sum of all frequencies.
It is important to make sure that the final value of the cumulative frequency does equal the total of the frequency. If not, then you need to go over your arithmetic.
Example
Take a look at the following question:
In order to use the graph to determine the median you need to look at the vertical axis which relates to the cumulative frequency.
You can see that the total frequency is 60 and so the median (the middle value) will be 30.
Simply draw a horizontal line from 30 to the curve and then a vertical line to the horizontal axis as shown below.
You should find that the median weight is 170g.
You will notice that the cumulative frequency curve has been drawn with a smooth curve and not a straight line. The median value is then drawn across to the cumulative frequency and an estimate of the median weight can then be obtained.
Question Practice
Try the following questions on your own before looking at the solution:
- Draw a cumulative frequency table
- Draw a cumulative frequency graph
- Use the graph to find an estimate for the interquartile range
- Use the graph to find an estimate for the number of workers with a weekly wage of more than £530.
Question Practice Solution
a) The cumulative frequency table is shown below:
\begin{array}{|c|c|c|} \hline \text { Weekly Wage (fx) } & \text { Frequency } & \text { Cumulative Freqency } \\ \hline 100<x \leq 200 & 8 & 8 \\ \hline 200<x \leq 300 & 15 & 8+15=23 \\ \hline 300<x \leq 400 & 30 & 23+30=53 \\ \hline 400<x \leq 500 & 17 & 53+17=70 \\ \hline 500<x \leq 600 & 7 & 70+7=77 \\ \hline 600<x \leq 700 & 3 & 77+3=80 \\ \hline & 80 & \\ \hline \end{array}b) The cumulative frequency graph is shown below with appropriate markings:
c) The Interquartile Range = Upper Quartile – Lower Quartile. From the graph, the total frequency is 80 and the lower quartile is ¼ of this which is 20 and this relates to a weekly age of £285. The upper quartile is ¾ of this which is 60 and this relates to a weekly wage of £435. So the IQR = 435 – 285 = £150.
d) From the graph you can see that there are 73 workers who earn less than £530. So the number of workers who earn more than £530 is 80 – 73 = 7.
Cumulative frequency and box plots can appear on either the foundation or higher papers, but are generally found on the higher paper.