1. D5 Frequency diagrams for continuous data
KS4 Mathematics
D5 Frequency diagrams for continuous data

2. D5.1 Grouping continuous data
Contents
D5 Frequency diagrams for continuous data
D5.1 Grouping continuous data
D5.2 Frequency diagrams
D5.3 Frequency polygons
D5.4 Histograms
D5.5 Frequency density

3. Analysing data
Tom is a sixteen-year-old who regularly takes part in downhill cycle races. He records the competitors’ race times on a spreadsheet.
His best time is 101.6
seconds.
How accurately has he measured this time?
The real data used in this presentation was collected by Tom Pye at a race at Aston Hill, Bucks.
Encourage pupils to refer back to previous work on averages, graphs and types of data.
Emphasise the size of the data set; guide them towards grouping the data.
Remind pupils that the larger the number the slower the speed.
Is the data continuous or discrete?

4. Analysing data Here are some race times in seconds from a downhill
racing event.
It is advisable to print out slides with large sets of data for ease of use by pupils.
Encourage pupils to refer back to previous work on averages and graphs if appropriate.
Emphasise the size of the data set; guide them towards grouping the data.
Remind pupils that the larger the number the slower the speed.
If you wanted to analyze the boys’ performance, what could you do with the data?
How easy is the format of the data to analyze at the moment? Can you draw any conclusions?

5. Choosing the right graph
In a piece of GCSE coursework, a student used a spreadsheet program to produce a graph of the race data.
This is the graph he printed.
80.0
85.0
90.0
95.0
100.0
105.0
110.0
What labels could be added to the axes?
What does the graph show?
Pupils should realize that each bar represents one data item. Since the data is in ascending order, the bars gradually increase in height. The horizontal axis represents the riders 1 – 60; the vertical axis the race times.
The graph should illustrate the problems of using a spreadsheet without full understanding. Suggest that grouping the data might help to see general trends and an overall picture. Ask what the class intervals should be.
Point out that when the data is grouped, the times will go on the horizontal axis and frequency on the vertical axis.
Is it an appropriate graph?

6. What is the best size for the class intervals for the race times data?
Grouping data
A list of results is called a data set.
It is often easier to analyze a large data set if we put the data into groups. These are called class intervals.
A frequency diagram or histogram can then be drawn.
You will need to decide on the size of the class interval so that there are roughly between 5 and 10 class intervals.
On the next page the results are shown again to aid discussion.
Ask pupils what the shortest and fastest times are.
It is appropriate for the interval to be a multiple of 5 or 10 if possible. The best interval here would be 5 seconds.
What is the best size for the class intervals for the race times data?

7. class intervals The times roughly range from 85 to 110 seconds.
The times roughly range from 85 to 110 seconds.
110 – 85 = 25 seconds.
Suppose we decide to use class intervals with a width of 5 seconds.
25 ÷ 5 = 5 class intervals

8. Notation for class intervals
How should the class intervals be written down?
Times in seconds
Frequency
85 – 90
90 – 95
95 – 100
100 – 105
What is wrong with this table?
Discuss where 90 and 105 should go. The table is ambiguous. For discrete data it would be possible to edit the table to say 86 – 90, 91 – 95, 96 – 100, 101 – 105 etc (or 85 – 89, 90 – 94 etc) but for continuous data this would not work.

9. Notation for class intervals
Can you explain what the symbols in the middle column mean?
100 ≤ t < 105
105 ≤ t < 110
95 ≤ t < 100
90 ≤ t < 95
85 ≤ t < 90
Times in seconds
85 – 90 but not including 90
Frequency
90 – 95 but not including 95
95 – 100 but not including 100
Discuss where 90, 95 etc would go in this table. Represent the inequality on a number line. Ask pupils where numbers such as would go.
Note that the data has been rounded off to 1 d.p. so it could be argued that we could write the intervals as 85.0 – 89.9, 90.0 – 94.9 etc but this would imply that the data is discrete, so this would be incorrect.
100 – 105 but not including 105
105 – 110 but not including 110

10. Notation for class intervals
85 ≤ t < 90 means “times larger than or equal to 85 seconds and less than 90 seconds”
Another way to say this is “from 85 up to but not including 90”
Can you say these in both ways?
1) 90 ≤ t < 95
“times larger than or equal to 90 seconds and less than 95 seconds” or
“from 90 up to but not including 95”.
Pupils could work together in pairs to practise the correct use of vocabulary.
2) 105 ≤ t < 110
“times larger than or equal to 105 seconds and less than 110 seconds” or
“from 105 up to but not including 110”.

11. Notation for class intervals
This activity involves pupils deciding which class interval a number belongs in. The numbers should be dragged into the right interval.

12. Use the data to fill in the table.
Class intervals
100 ≤ t < 105
105 ≤ t < 110
95 ≤ t < 100
90 ≤ t < 95
85 ≤ t < 90
Times in seconds
Frequency 1
Pupils would benefit from having a print out of the data to avoid missing out data items. A tally could be used if required.
They should check they have 60 items in total in the frequency column.
Use the data to fill in the table. 5 28 19 7

13. D5 Frequency diagrams for continuous data
Contents
D5 Frequency diagrams for continuous data
D5.1 Grouping continuous data
D5.2 Frequency diagrams
D5.3 Frequency polygons
D5.4 Histograms
D5.5 Frequency density

14. Frequency diagrams
Frequency diagrams can be used to display grouped continuous data.
For example, this frequency diagram shows the distribution of heights for a group students:
Frequency
Height (cm) 5 10 15 20 25 30 35
150
155
160
165
170
175
180
185
Heights of students
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 where as 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 is not make any difference when the class intervals are equal as in this example.
This type of frequency diagram is often called a histogram.

15. Drawing frequency diagrams
When drawing a frequency diagrams for grouped continuous data remember the following points:
The time intervals go on the horizontal axis.
The frequencies go on the vertical axis.
The highest and lowest times in each interval go at either end of the bar, as shown below:
The bars must be joined together, to indicate that the data is continuous.

16. Frequency diagram of cycling data
The cycling data we looked at earlier can be displayed in the following frequency diagram:
Frequency 80 5 10 15 20 25 30 85 90 95
100
105
Times in seconds
What conclusions can you draw from the graph?

17. Changing the class interval
When the class intervals are changed the same data produces the following graph:
Times in seconds 85
87.5 90
92.5 95
97.5
100
102.5
105
107.5
Frequency 5 10 15 20
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?

18. D5 Frequency diagrams for continuous data
Contents
D5 Frequency diagrams for continuous data
D5.1 Grouping continuous data
D5.2 Frequency diagrams
D5.3 Frequency polygons
D5.4 Histograms
D5.5 Frequency density

19. What are the midpoints of each class interval for the race times data?
As well as a frequency diagram, it might also be appropriate to construct a frequency polygon.
This plots the midpoints of each bar and joins them together.
What are the midpoints of each class interval for the race times data?
Times in seconds
Midpoint
85 ≤ t < 90
90 ≤ t < 95
95 ≤ t < 100
100 ≤ t < 105
105 ≤ t < 110
To find the midpoint of two numbers, add them together and divide by 2.
87.5
Pupils may recall the midpoint of grouped data from previous work on estimating the mean from grouped data.
92.5
97.5
102.5
107.5

20. Midpoints
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.

21. Line graph of midpoints
If we join together the midpoints for each bar the following graph is produced:
Frequency 80 5 10 15 20 25 30 85 90 95
100
105
Times in seconds
110 75
This graph is an illustration of how to produce a frequency polygon from a histogram/ frequency diagram.
Notice that the empty class intervals at each end have been included.

22. Frequency polygon of cycling data
Removing the bars leaves a frequency polygon.
Frequency 80 5 10 15 20 25 30 85 90 95
100
105
Times in seconds
110 75
Frequency 80 5 10 15 20 25 30 85 90 95
100
105
Times in seconds
110 75
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.

23. Comparing frequency polygons
Here are the race times for two age categories. Juniors are aged from 17 to 18 and seniors are aged from 19 to 30.
Junior category 2 4 6 8 10 85 90 95
100
105
110
115
120
125
130
135
Senior category 5 10 15 20 85 90 95
100
105
110
115
120
125
130
135
For each category, find
A print out of this page would be useful for pupils to analyze.
Discuss how to work backwards from the midpoint to the class intervals.
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.
the size of the class intervals
the modal class interval
the range.
Compare the performances in the two categories.

24. Comparing frequency polygons
The same data has been used in these graphs.
Junior category
Senior category 20 30 20 10 10 85 95
105
115
125
135 85 95
105
115
125
135
For each category, find
the size of the class intervals
It would be useful to print this page for pupils.
the number of class intervals
the modal class interval.
Compare these graphs with the previous ones. Which do you find more useful for analyzing the race times and why?

25. 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.
The wider intervals used for the averages in the second set of graphs illustrates the loss of information when the number of intervals is reduced.
Emphasise the importance of using the phrase “on average” in the final sentence.
It might be interesting to discuss why younger bikers are faster than older ones!
This means that on average Juniors are faster than Seniors.

26. D5 Frequency diagrams for continuous data
Contents
D5 Frequency diagrams for continuous data
D5.1 Grouping continuous data
D5.2 Frequency diagrams
D5.3 Frequency polygons
D5.4 Histograms
D5.5 Frequency density

27. Is the bar three times as big?
Histograms
This frequency diagram represents the race times for the Youth category, which is 14 to 16 year olds.
Frequency 2 4 6 8 10 12
Time in seconds 95
100
105
110
115
120
125
130
135
There are __ times
as many people in the 105 ≤ t < 110 interval than there are in the 95 ≤ t < 100 interval. 3
Is the bar three times as big?
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?

28. Discuss this statement. Do you agree or disagree?
Histograms
Discuss this statement. Do you agree or disagree?
“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.”
Frequency 2 4 6 8 10 12
Time in seconds 95
100
105
110
115
120
125
130
135
The area of the bars is proportional to the frequency.
Because all the bars are the same width, we only need to look at the height of the bars to work out the frequency.

29. Which intervals would you combine?
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 95
100
105
110
115
120
125
130
135
It is sometimes sensible to combine intervals together.
Which intervals would you combine?
Encourage pupils to combine the first two intervals and the last three.

30. Histograms with bars of unequal width
This graph represents the same data as the previous one.
What has changed?
The first two intervals both had a frequency of 2. The first bar now represents an interval twice as big.
Frequency 2 4 6 8 10 12
Time in seconds 95
100
105
110
115
120
125
130
135
How many people are in this interval?
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.
How many people does one square represent?
Do the numbers along the vertical axis still represent frequency?

31. 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?
Frequency
Time in seconds 95
100
105
110
115
120
125
130
135
The frequency for 105 ≤ t < 110 is the same as the frequency for ___________.
120 ≤ t < 135
Are the areas of the bars the same?
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.

32. D5 Frequency diagrams for continuous data
Contents
D5 Frequency diagrams for continuous data
D5.1 Grouping continuous data
D5.2 Frequency diagrams
D5.3 Frequency polygons
D5.4 Histograms
D5.5 Frequency density

33. Frequency density
In a histogram, the frequency is given by the area of each bar.
frequency density
4 people 95
105
110
It follows that the height of the bar × the width of 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,
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.
Height of the bar =
frequency
width of interval
This height is called the frequency density.

34. What scale do we need for the vertical axis?
Frequency density
Frequency density =
frequency
width of interval
In our example, each square represents 2 people.
0.4
4 people 95
105
110
What scale do we need for the vertical axis?
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

35. 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.
Frequency density
0.4
0.8
1.2
1.6
2.0
2.4
Time in seconds 95
100
105
110
115
120
125
130
135
Frequency density × width
110 ≤ t < 115
115 ≤ t < 120
105 ≤ t < 110
120 ≤ t < 135
95 ≤ t < 105
Time in seconds
Area
(frequency)
0.4 × 10 4
The results obtained should be compared with the original graph.
1.2 × 5 6
1.4 × 5 7
2.2 × 5 11
0.4 × 15 6

36. Calculating the frequency density
width of interval
Complete the table for this data and draw a histogram. 2
130 ≤ t < 150 12
115 ≤ t < 130 8
105 ≤ t < 115 5
100 ≤ t < 105
95 ≤ t < 100
Frequency density
Frequency ÷ width of interval
Frequency
Time in seconds
8 ÷ 5
1.6
5 ÷ 5
1.0
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 type 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.
8 ÷ 10
0.8
12 ÷ 15
0.8
2 ÷ 20
0.1

37. Calculating the frequency density
Your histogram should look like this:
Frequency density
0.2
0.4
0.6
0.8
1.0
1.2
Time in seconds 95
100
105
110
115
120
125
130
135
140
145
150
1.4
1.6
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.

38. Calculating the class intervals
This is a histogram of race times from a longer race.
Time in seconds
Frequency density 2 4 6 8
The first bar represents 40 people.
The lowest time was 100 seconds.
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.
100
Work out the scale along the bottom and the frequencies for each interval.

39. Calculating the frequency density
We can use the formula,
Frequency = frequency density × width of interval
to complete the following table for the data in the histogram.
Length of interval
Frequency density
Frequency
class interval 40 1
40 ÷ 1= 40
100 ≤ t < 140
4 × 40 = 160 4 40
140 ≤ t < 180
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 type 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.
80 × 6 = 480 6 80
180 ≤ t < 260
20 × 5 = 100 5 20
260 ≤ t < 280
20 × 3 = 60 3 20
280 ≤ t < 300

40. Review
Write a definition of each word below and then design a mind map outlining the key facts you have learnt.
data set
class interval
midpoint
range
axes
frequency diagram
frequency
frequency polygon
modal class interval
histogram
frequency density
Include methods for calculating and drawing; possible mistakes to avoid …