3. Frequency diagrams
Frequency diagrams can be used to display continuous data that has been grouped together.
This frequency diagram shows the distribution of heights for a group of students.
(no. of students)
Frequency
Height (cm) 5 10 15 20 25 30 35
150
155
160
165
170
175
180
185
Heights of students
Teacher notes
Stress that the difference between this graph and a bar graph is that the bars are touching. Bar graphs can only be used to display qualitative data or discrete numerical data whereas histograms are used to show continuous data.
Strictly speaking, for a histogram we plot frequency density rather than frequency along the vertical axis. However, this does not make any difference to the graph when the class intervals are equal as in this example.
This type of frequency diagram is often called a histogram. 3

4. Drawing frequency diagrams
When drawing frequency diagrams for grouped continuous data, there are important guidelines that should be followed.
The class intervals go on the horizontal axis.
The frequencies go on the vertical axis.
The highest and lowest possible times in each interval go at either end of the bar, as shown.
The bars must be joined together, to indicate that the data is continuous.
Teacher notes
Refer students to the frequency diagram on the previous slide for confirmation of the first two points. The third point has been illustrated by using a larger section of the diagram from the previous slide.
Again, you may wish to remind students how these types of graphs differ from ‘normal’ bar charts. 4

5. Tom’s cycling data Tom regularly takes part in downhill cycle races.
He recorded all the race times from a downhill race event and put it into a table that records grouped continuous data.
Can you put this data into a histogram using the guidelines from the previous slide?
What data should go on the horizontal axis?
What data should go on the vertical axis?
100 ≤ t < 105
105 ≤ t < 110
95 ≤ t < 100
90 ≤ t < 95
85 ≤ t < 90
Time in seconds
Frequency 19 7 28 5 1
Teacher notes
The histogram that accompanies this data is displayed on the following slide. Remind students of the guidelines from the previous slide on how to draw a frequency diagram. Class intervals go on the horizontal axis and frequency goes on the vertical axis. Again, strictly speaking, for a histogram we plot frequency density rather than frequency along the vertical axis. However, this does not make any difference to the graph when the class intervals are equal as in this example.

6. Frequency diagram of cycling data
The cycling data can be displayed in a frequency diagram.
(no. of people)
Frequency 80 5 10 15 20 25 30 85 90 95
100
105
Times in seconds
Teacher notes
Ask students to express as many different conclusions as they can using the graph as a basis for their findings. They may say that the modal time taken to complete the course was 90 ≤ t < 95 seconds. They might conclude that the fastest recorded time was 80 ≤ t < 85 seconds and that a small number of people completed the course in this time. They might reach the conclusion that the maximum time difference between the slowest rider and the fastest rider was 25 seconds. Encourage all suggestions that they have and then get the students to split into groups and talk through the conclusions. What evidence is there that proves/disproves the conclusion?
What conclusions can you draw from the graph? 6

7. Changing the class interval
When the class intervals are changed, the same data gives a slightly different graph.
Times in seconds 85
87.5 90
92.5 95
97.5
100
102.5
105
107.5
Frequency 5 10 15 20
Teacher notes
This graph has used smaller class intervals to further break down the data. The class intervals are at 2.5 second intervals rather than 5 second intervals.
The more class intervals there are, the more information you gain about the breakdown of the race times. However, this can disguise general trends and be too complex for immediate analysis.
Discuss different purposes the graph might be used for, such as sponsors targeting certain racers or planning strategies to enable racers to improve; evaluating different bicycles, saddles, tyres etc; comparing age groups.
What size class intervals have been used?
What additional information is available from this graph?
Which graph is more useful? 7

9. Midpoints
As well as a frequency diagram, it might also be appropriate to construct a frequency polygon.
A frequency polygon plots the midpoints of each bar on the frequency diagram and joins them together.
To find the midpoint of two numbers, you need to add them together and divide by two.
Teacher notes
Pupils may recall the midpoint of grouped data from previous work on estimating the mean from grouped data. You may wish to give the students more examples of calculating the midpoint. The next slide provides a contextual example. Students have to calculate the midpoint of Tom’s class intervals.
What is the midpoint of 17 and 51?
What is the midpoint of 12 and 96? 9

10. Class interval midpoints

11. Midpoints Teacher notes
In this activity, pupils should add the two number together and divide by 2 to find the midpoint in preparation for drawing a frequency polygon. The data in this activity is random and does not relate to the data that the students have been looking at in the context of Tom’s downhill racing. 11

12. Line graph of midpoints
If we plot the midpoints at the top of each bar on our frequency diagram and join them up, the following graph is produced.
Frequency 80 5 10 15 20 25 30 85 90 95
100
105
Times in seconds
110 75
Teacher notes
This graph is an illustration of how to produce a frequency polygon from a histogram/frequency diagram.
Point out that the empty class intervals at each end have been included. 12

13. Frequency polygon of cycling data
Removing the bars from this frequency diagram then leaves us with a frequency polygon.
Frequency 80 5 10 15 20 25 30 85 90 95
100
105
Times in seconds
110 75
Teacher notes
Discuss with pupils the different information available on the two types of graphs. Often frequency polygon graphs are used to compare two sets of data as the shape of the graph can be more easily analyzed than bars.
Note that frequency polygons can be left open at either end, or joined to the horizontal axis like this one. 13

14. Frequency polygons Teacher notes
Students should know that the class intervals along the x-axis read: 4 ≤ t < 6.

15. Comparing frequency polygons
Teacher notes
A print out of the individual screenshots of the race time data may be useful for pupils to analyse. Discuss how to work backwards from the midpoint to the class intervals. Point out that you are assuming all class intervals to be equally large. Pupils will need to review the definition of “modal class interval” from previous work. Discuss the difference between mode and modal class interval.
Draw pupils’ attention to the fact that since we only have the grouped data frequencies, we cannot know the exact range. We must therefore use the lowest number in the lowest class interval (for example, 95 for the Juniors) and the highest in the highest class interval (135) rather than the midpoints. Encourage pupils to discuss the similarities and differences between the two groups. Provide pupils with model answers that refer to the range and an average. Discuss whether there is too much information with 10 intervals, and compare with the graphs on the next slide.
In each category, the class intervals are 5 seconds apart, starting at 85 ≤ t < 90 and continuing through to 130 ≤ t < 135. As stated above, we don’t know the exact range. However, using the lowest number in the lowest class interval and the highest number in the highest class interval we can see the range for the Juniors is: 135 – 95 = 40 s. The range for the Seniors is: 135 – 90 = 45 s. The modal class interval is the interval that contains the most frequently occurring value. For the Seniors, this is: 110 ≤ t < 115. For the Juniors, this value is 95 ≤ t < 100.
Photo credit: © Nowik, Shutterstock.com 15

16. Comparing frequency polygons
The same data has been used in these graphs.
For each category, find:
the size of the class intervals
the number of class intervals
the modal class interval.
Teacher notes
You may find it useful to print this page for pupils. The print out will allow the students to compare and contrast the data on this slide with that on the previous slide.
Point out that as the class intervals have changed on the horizontal axis, so the frequency values on the vertical axis have changed to reflect this.
In both categories, the class intervals have been extended from 5 seconds to 10 seconds, beginning at 85 ≤ t < 95. There are now 5 class intervals that the data has been divided into.
The modal class interval for the Juniors is now 95 ≤ t < 105. The modal class interval for the Seniors is 105 ≤ t < 115.
Compare these graphs with the previous ones. Which do you find more useful for analyzing the race times? Why? 16

17. Comparing sets of data
The range of times for the Junior category is smaller than for the Senior category.
This suggests the Seniors are less consistent.
Using the first set of graphs, the modal class interval for the Juniors is 95 ≤ t < 100, whereas the modal class interval for the Seniors is 110 ≤ t < 115.
Using the second set of graphs, the modal class interval for the Juniors is 95 ≤ t < 105, whereas the modal class interval for the Seniors is 105 ≤ t < 115.
Teacher notes
The wider intervals used for the averages in the second set of graphs illustrate the loss of information when the number of intervals is reduced.
Emphasise the importance of using the phrase “on average” in the final sentence.
This means that, on average, Juniors are faster than Seniors. 17

19. Histograms
This frequency diagram represents the race times for the Youth category, which is for 14 to 16 year olds.
Frequency 2 4 6 8 10 12
Time in seconds (t) 95
100
105
110
115
120
125
130
135
How many people are there in the 105 ≤ t < 110 interval?
How many people are there are in the 95 ≤ t < 100 interval?
Teacher notes
There are 6 people in the 105 ≤ t < 110 interval. This can be seen by reading horizontally across from the top of the bar that is between these two parameters.
There are 2 people in the 95 ≤ t < 100 interval. This can be seen by reading horizontally across from the top of the bar that is between these two parameters.
Ask pupils how many people are represented by each square (2). This will help introduce the idea that frequency is proportional to area in a histogram.
How many people are represented by each square on the grid? 19

20. Interpreting graphs

21. Histograms
After looking at the graph, one of the students thinks they have reached a conclusion about the histogram.
Frequency 2 4 6 8 10 12
Time in seconds (t) 95
100
105
110
115
120
125
130
135
“If a bar is twice as high as another, the area will be twice as big and so the frequency will be twice the size.”
Teacher notes
This statement is not correct for all histograms. The area of the bars is proportional to the frequency. As all the bars are the same width, we only need to look at the height of the bars to work out the frequency.
Discuss this statement. Do you agree or disagree? 21

22. Combining intervals
Some of the intervals are very small, which makes any
conclusions about them unreliable.
Frequency 2 4 6 8 10 12
Time in seconds (t) 95
100
105
110
115
120
125
130
135
Teacher notes
Encourage pupils to combine the first two intervals and the last three. In this way, we can combine intervals to give a better analysis of the data that has been collected.
It is sometimes sensible to combine intervals together.
Which intervals would you combine in this graph? 22

23. Histograms with bars of unequal width
This graph represents the same data as the previous one. 2 4 6 8 10 12
Time in seconds (t) 95
100
105
110
115
120
125
130
135
What has changed?
The first two intervals both had a frequency of 2.
The first bar now represents an interval twice as big.
How many people are in this interval?
Teacher notes
Establish that the first two bars and the last three bars have been combined.
There are 4 people represented by the first bar, 2 in each square. The area still represents the frequency, but the height of the bar does not.
Discuss the end bar as well. The original frequencies were 2, 3 and 1. Ask what the new frequency should be (6). Again, the area of the bar is 3 squares which is 6 people, but the height is not representative of frequency.
How many people does one square represent?
Do the numbers on the vertical axis still show frequency? 23

24. Histograms with bars of unequal width
In the original histogram, the frequency was proportional to the area. Is this still true in the new histogram?
Time in seconds (t) 95
100
105
110
115
120
125
130
135 12 10 8 6 4 2
The frequency for 105 ≤ t < 110 is the same as the frequency for 120 ≤ t < 135.
Are the areas of the bars the same?
Teacher notes
The answer to the first question is yes. The other questions on this slide should lead towards this conclusion.
These discussion questions should lead towards an appreciation that the vertical axis needs to create a relationship between the frequency and the area.
Each square stills represents two people.
In a histogram, the frequency is equal to the area of the bar. 24

25. Histograms Teacher notes
At this stage, students should be encouraged to try out the information contained within the tables for themselves. Can they use the data to create a histogram? You may wish to split the class into groups at this stage. One group could create a histogram for the data on the males, while another group could do the same for the females. If you have a third group, they could do a combined graph. Printing out screenshots of the information contained within the two buttons may prove useful at this stage.
You may also wish to return to this slide after looking at the next chapter on frequency density. You could ask students to use the histograms they created to calculate frequency density for this real data.

27. Frequency density
In a histogram, the frequency is given by the area of each bar.
It follows that the height of the bar × the width of the bar must be the area.
Therefore, the height must equal the area ÷ the width.
The area of the bar gives the frequency and so we can write:
4 people
frequency density
frequency
width of interval
Height of the bar =
Teacher notes
The frequency density also represents the constant of proportionality connecting the frequency to the length of the interval. If the interval were 1, then the frequency and the frequency density would be equal, since frequency ÷ 1 = frequency density.
Thus the wider the interval, the smaller the frequency density. 95
105
100
This height is called the frequency density. 27

28. Frequency density In our example, each square represents 2 people.95
105
100
0.4
Frequency density =
width of interval
frequency
What scale is needed for the vertical axis?
Teacher notes
Remind pupils that the area of the bar represents the frequency.
Width of interval = 10
Area = 4
Height = 4 ÷ 10 = 0.4
Frequency density = 0.4 28

29. Calculating the frequency
We can use the formula:
frequency = frequency density × width of interval
to check this scale for the other bars in the graph on slide 23.
Time in seconds
Frequency density × width
Area (frequency)
95 ≤ t < 105
105 ≤ t < 110
110 ≤ t < 115
115 ≤ t < 120
120 ≤ t < 125
0.4 × 10 4
1.2 × 5 6
Teacher notes
The results obtained should be compared with the original graph on slide 23.
1.4 × 5 7
2.2 × 5 11
0.4 × 15 6 29

30. Calculating the frequency density
Teacher notes
Discuss how it is possible for two bars with different frequencies to have the same frequency density.
Once the graph has been drawn, ask pupils to write down some questions that could be asked from it, both analytical questions and exam-style questions.
Ask them how they could use the frequency density and the intervals to work out the frequencies from their graph if they had not been given them. 30

31. Calculating the frequency density
Your histogram should take on a particular appearance.
Frequency density
0.2
0.4
0.6
0.8
1.0
1.2
Time in seconds (t) 95
100
105
110
115
120
125
130
135
140
145
150
1.4
1.6
Teacher notes
Verify that the area of each bar is proportional to the frequency it represents.
In this example, each square represents 1 person. However, pupils’ graphs will vary from the one shown depending on the scale they have used. 31

32. Calculating the class intervals
This histogram represents the race times from a longer race.
Time in seconds (t)
Frequency density 2 4 6 8
100
The first bar in the histogram represents 40 people.
The lowest time recorded in the race was 100 seconds.
Teacher notes
Ask pupils to make a copy of this histogram. As the scale along the bottom is calculated on the next slide, pupils can fill it in on their graph.
Work out the scale along the bottom and the frequencies for each interval. 32

33. Calculating the frequency density
Teacher notes
Discuss how it is possible for two bars to have the same frequency density.
Once the graph has been drawn, ask pupils to write down some questions that could be asked from it, both analytical questions and exam-style questions.
Ask them how they could use the frequency density and the intervals to work out the frequencies from their graph if they had not been given them. 33