 # Draw Cumulative Frequency Diagrams

In this worksheet, students practise drawing cumulative frequency diagrams. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Statistics

Curriculum subtopic:   Statistics, Interpreting and Representing Data

Difficulty level:   ### QUESTION 1 of 10

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.

To find a cumulative frequency...

30 students were asked to estimate the length of a line. This table shows their results.

 Distance (cm) Frequency Cum. Frreq 20 < d ≤ 30 1 30 < d ≤ 40 3 40 < d ≤ 50 6 50 < d ≤ 60 12 60 < d ≤ 70 3 70 < d ≤ 80 3 80 < d ≤ 90 2

Complete the cumulative frequency column

 Cumulative Frequency

For the highlighted row, what is the point you would plot?

 Distance (cm) Frequency Cum. Frreq 20 < d ≤ 30 1 1 30 < d ≤ 40 3 4 40 < d ≤ 50 6 10 50 < d ≤ 60 12 22 60 < d ≤ 70 3 25 70 < d ≤ 80 3 28 80 < d ≤ 90 2 30

 Cumulative Frequency

For the cumulative frequency diagram, which one of these two graphs has been drawn correctly?

 Distance (cm) Frequency Cum. Frreq 20 < d ≤ 30 1 1 30 < d ≤ 40 3 4 40 < d ≤ 50 6 10 50 < d ≤ 60 12 22 60 < d ≤ 70 3 25 70 < d ≤ 80 3 28 80 < d ≤ 90 2 30 The left one

The right one

The midday temperatures at 50 European resorts are recorded and shown below.

 Temperature (° C) Frequency Cum. Frreq 5 - 7 2 8 - 10 3 11 - 13 5 14 - 16 6 17 - 19 6 20 - 22 9 23 - 25 8 26 - 28 6 29 - 31 5

Complete the cumulative frequency column

 Cumulative Frequency

The midday temperatures at 50 European resorts are recorded and shown below.

 Temperature (° C) Frequency Cum. Frreq 5 - 7 2 2 8 - 10 3 5 11 - 13 5 10 14 - 16 6 16 17 - 19 6 22 20 - 22 9 31 23 - 25 8 39 26 - 28 6 45 29 - 31 5 50

What point would you plot for the highlighted row?

 Cumulative Frequency

For the cumulative frequency diagram, which one of these two graphs have the point been plotted incorrectly?

 Temperature (° C) Frequency Cum. Frreq 5 - 7 2 2 8 - 10 3 5 11 - 13 5 10 14 - 16 6 16 17 - 19 6 22 20 - 22 9 31 23 - 25 8 39 26 - 28 6 45 29 - 31 5 50 The left one

The right one

The table shows the scores of 100 students in an English exam

 Temperature (° C) Frequency Cum. Frreq 21 ≤ x ≤ 30 3 21 ≤ x ≤ 30 9 21 ≤ x ≤ 30 12 21 ≤ x ≤ 30 15 21 ≤ x ≤ 30 22 21 ≤ x ≤ 30 16 21 ≤ x ≤ 30 10 21 ≤ x ≤ 30 8 21 ≤ x ≤ 30 3 21 ≤ x ≤ 30 2

Complete the cumulative frequency column

 Cumulative Frequency

The table shows the scores of 100 students in an English exam

 Temperature (° C) Frequency Cum. Frreq 21 ≤ x ≤ 30 3 3 31 ≤ x ≤ 40 9 12 41 ≤ x ≤ 50 12 24 51 ≤ x ≤ 60 15 39 61 ≤ x ≤ 70 22 61 71 ≤ x ≤ 80 16 77 81 ≤ x ≤ 90 10 87 91 ≤ x ≤ 100 8 95 101 ≤ x ≤ 110 3 98 111 ≤ x ≤ 120 2 100

Which of the coordinates below are correct ones that should be plotted for this data?

(30,3)

(31,12)

(50,12)

(90,87)

(110,98)

(85,87)

Could this cumulative frequency graph be correct? Yes

No

It depends on the question

• Question 1

To find a cumulative frequency...

EDDIE SAYS
Remember that for the second row, we add the frequencies from the first and second etc When we have the cumulative frequencies, we plot them at the upper bound, this is sometimes referred to as the top of the group.
• Question 2

30 students were asked to estimate the length of a line. This table shows their results.

 Distance (cm) Frequency Cum. Frreq 20 < d ≤ 30 1 30 < d ≤ 40 3 40 < d ≤ 50 6 50 < d ≤ 60 12 60 < d ≤ 70 3 70 < d ≤ 80 3 80 < d ≤ 90 2

Complete the cumulative frequency column

 Cumulative Frequency
EDDIE SAYS
All we're doing here is adding all the frequencies up to the point we're trying to find. 1 1+3 1+3+6 1+3+6+12 etc A nice little check you can do. The questions says 30 students, is your final number 30?
• Question 3

For the highlighted row, what is the point you would plot?

 Distance (cm) Frequency Cum. Frreq 20 < d ≤ 30 1 1 30 < d ≤ 40 3 4 40 < d ≤ 50 6 10 50 < d ≤ 60 12 22 60 < d ≤ 70 3 25 70 < d ≤ 80 3 28 80 < d ≤ 90 2 30

EDDIE SAYS
The key thing here is to remember to plot at the Upper point against the cumulative frequency.
• Question 4

For the cumulative frequency diagram, which one of these two graphs has been drawn correctly?

 Distance (cm) Frequency Cum. Frreq 20 < d ≤ 30 1 1 30 < d ≤ 40 3 4 40 < d ≤ 50 6 10 50 < d ≤ 60 12 22 60 < d ≤ 70 3 25 70 < d ≤ 80 3 28 80 < d ≤ 90 2 30 The left one
EDDIE SAYS
They both look the same, but did you notice that the one on the right is plotted at the lower end of the group and the one on the right is plotted at the upper end of the group? Where do we plot?
• Question 5

The midday temperatures at 50 European resorts are recorded and shown below.

 Temperature (° C) Frequency Cum. Frreq 5 - 7 2 8 - 10 3 11 - 13 5 14 - 16 6 17 - 19 6 20 - 22 9 23 - 25 8 26 - 28 6 29 - 31 5

Complete the cumulative frequency column

 Cumulative Frequency
EDDIE SAYS
All we're doing here is adding all the frequencies up to the point we're trying to find. 2 2+3 2+3+5 2 + 3 + 5 + 6 etc A nice little check you can do. The questions says 50 resorts. Is your final number 50?
• Question 6

The midday temperatures at 50 European resorts are recorded and shown below.

 Temperature (° C) Frequency Cum. Frreq 5 - 7 2 2 8 - 10 3 5 11 - 13 5 10 14 - 16 6 16 17 - 19 6 22 20 - 22 9 31 23 - 25 8 39 26 - 28 6 45 29 - 31 5 50

What point would you plot for the highlighted row?

EDDIE SAYS
The key thing here is to remember to plot at the Upper point against the cumulative frequency.
• Question 7

For the cumulative frequency diagram, which one of these two graphs have the point been plotted incorrectly?

 Temperature (° C) Frequency Cum. Frreq 5 - 7 2 2 8 - 10 3 5 11 - 13 5 10 14 - 16 6 16 17 - 19 6 22 20 - 22 9 31 23 - 25 8 39 26 - 28 6 45 29 - 31 5 50 The right one
EDDIE SAYS
Did you remember about the shape that all CF diagrams should have? They should always be getting larger. The mistake with the right hand one is that the student has plotted the frequency, not the cumulative frequency.
• Question 8

The table shows the scores of 100 students in an English exam

 Temperature (° C) Frequency Cum. Frreq 21 ≤ x ≤ 30 3 21 ≤ x ≤ 30 9 21 ≤ x ≤ 30 12 21 ≤ x ≤ 30 15 21 ≤ x ≤ 30 22 21 ≤ x ≤ 30 16 21 ≤ x ≤ 30 10 21 ≤ x ≤ 30 8 21 ≤ x ≤ 30 3 21 ≤ x ≤ 30 2

Complete the cumulative frequency column

 Cumulative Frequency
EDDIE SAYS
All we're doing here is adding all the frequencies up to the point we're trying to find.
• Question 9

The table shows the scores of 100 students in an English exam

 Temperature (° C) Frequency Cum. Frreq 21 ≤ x ≤ 30 3 3 31 ≤ x ≤ 40 9 12 41 ≤ x ≤ 50 12 24 51 ≤ x ≤ 60 15 39 61 ≤ x ≤ 70 22 61 71 ≤ x ≤ 80 16 77 81 ≤ x ≤ 90 10 87 91 ≤ x ≤ 100 8 95 101 ≤ x ≤ 110 3 98 111 ≤ x ≤ 120 2 100

Which of the coordinates below are correct ones that should be plotted for this data?

(30,3)
(90,87)
(110,98)
EDDIE SAYS
There are three common mistakes when people are plotting cumulative frequency. 1) They plot at the middle of the group 2) They lot at the start of the group 3) They plot the frequency instead of the cumulative frequency. Remember when you are plotting, you need to plot at the top (upper limit) of the group against the cumulative frequency,
• Question 10

Could this cumulative frequency graph be correct? No
EDDIE SAYS
Make sure you read carefully. This question asks if it could be correct for a cumulative frequency question. It can\'t be because it\'s the wrong shape. Remember that a cumulative frequency graph should get steeper, then shallower but never go downwards.
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