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What is a cumulative frequency diagram?

A cumulative frequency diagram is a method in statistics of taking a distributon (shown by frequencies) and making it into an ordered distribution so that we can analyse it in detail.

 

What does cumulative frequency mean?

Cumulative comes from the same root word as accumulate which means 'to build up'. Cumulative frequencies means to 'build up the frequencies.'

 

Huh?

When we are plotting a cumulative frequency, instead of plotting the frequency for a single group, we plot all the frequencies up to that point.

 

Show me, I'm still not sure.

Example: Here are the marks for 50 students in a Maths exam.

Mark Frequency
21 - 30 1
31 - 40 6
41 - 50 6
51 - 60 8
61 - 70 8
71 - 80 6
81 - 90 7
91 - 100 6
101 - 110 1
111 - 120 1

If we wanted to find the cumulative frequency for the first row, it would just be the frequency for the first row.

If we wanted to find the cumulative frequency for the second row, we would add the frequencies from the first two rows

If we wanted to find the cumulative frequency for the third row, we would add the frequencies from the first three rows

etc

This should give the following table.

Mark Frequency Cum. Freq.
21 - 30 1 1
31 - 40 6 7
41 - 50 6 13
51 - 60 8 21
61 - 70 8 29
71 - 80 6 35
81 - 90 7 42
91 - 100 6 48
101 - 110 1 49
111 - 120 1 50

How do we plot this?

The first step is to draw the axes (unless you are given them, you usually are in an exam)

For any diagram that involves frequency (bar graphs, line graph, cumulative frequency, Histograms), we always plot the frequency up the side and the other scale (in this case marks) along the bottom.

Once, we have the axes, we are ready to plot our points.

This is where most of the mistakes are made when drawing a cumulative frequency diagram.

We plot the cumulative frequency against the UPPER bound of the group.

 

Mark Cum Freq Point to plot
21 - 30 1 (30,1)
31 - 40 7 (40,7)
41 - 50 13 (50,13)
51 - 60 21 (60,21)
61 - 70 29 (70,29)
71 - 80 35 (80,35)
81 - 90 42 (90,42)
91 - 100 48 (100,48)
101 - 110 49 (110,49)
111 - 120 50 (120,50)

Plotting these points will give...

Lastly, we have to draw a line to join these up.

DO NOT try to join them up with a curve, its difficult and you don't get any extra marks for a curve

Just join them up with a straight line, point to point.

Why isn't the first point joined to the axes?

Technically, you don't know what is happening before that first point. There are some times when you d and sometimes when you don't.

An examiner will be expecting you not to, so just leave it out.

 

The Shape:

If you look at this graph, you will notice it has a very distinctive shape (it gets steeper, then shallower and doesn't come back down.

All cumulative frequency diagrams will have this shape.

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