© Boardworks Ltd 2005 1 of 40 D5 Frequency diagrams for continuous data KS4 Mathematics

© Boardworks Ltd 2005 1 of 40

D5 Frequency diagrams for continuous data

KS4 Mathematics


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AD5.1 Grouping continuous data


D5.2 Frequency diagrams

D5.3 Frequency polygons

D5.4 Histograms

D5.5 Frequency density


Tom is a sixteen-year-old who regularly takes part in downhill cycle races. He records the competitors’ race times on a spreadsheet.

Analysing data

How accurately has he measured this time?

Is the data continuous or discrete?

His best time is 101.6 seconds.


If you wanted to analyze the performance, what could you do with the data?

Analysing data

Here are some race times in seconds from a downhill racing event.

How easy is the format of the data to analyze at the moment? Can you draw any conclusions?

88.4 91.5 92.1 93.3 93.9 94.7 95.0 95.3 95.5 95.6 95.6 96.3 96.5 96.9 97.0 97.0 97.0 97.3 97.4 97.4 97.7 97.8 98.0 98.2 98.2 98.4 98.4 98.5 98.9 99.0 99.1 99.6 99.6 99.8 100.0 100.6100.6 101.1 101.4 101.4 101.5 101.6 101.6 101.8 101.9102.1 102.5 102.6 102.7 103.1 103.1 103.1 104.1 105.0105.2 105.6 105.6 105.7 105.8 105.9


In a piece of GCSE coursework, a student used a spreadsheet program to produce a graph of the race data.

Choosing the right graph

This is the graph he printed.

What labels could be added to the axes?

What does the graph show?

Is it an appropriate graph?80.0

85.0

90.0

95.0

100.0

105.0

110.0


Grouping data

A list of results is called a data set.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.

What is the best size for the class intervals for the race times data?


class intervals

The times roughly range from 85 to 110 seconds.

Suppose we decide to use class intervals with a width of 5 seconds.

110 – 85 = 25 seconds.

25 ÷ 5 = 5 class intervals

88.4 91.5 92.1 93.3 93.9 94.7 95.0 95.3 95.5 95.6 95.6 96.3 96.5 96.9 97.0 97.0 97.0 97.3 97.4 97.4 97.7 97.8 98.0 98.2 98.2 98.4 98.4 98.5 98.9 99.0 99.1 99.6 99.6 99.8 100.0 100.6100.6 101.1 101.4 101.4 101.5 101.6 101.6 101.8 101.9102.1 102.5 102.6 102.7 103.1 103.1 103.1 104.1 105.0105.2 105.6 105.6 105.7 105.8 105.9


How should the class intervals be written down?

Times in seconds

Frequency

85 – 90

90 – 95

95 – 100

100 – 105

105 - 110

What is wrong with this table?

Notation for class intervals


100 ≤ t < 105

105 ≤ t < 110

95 ≤ t < 100

90 ≤ t < 95

85 ≤ t < 90

Times in seconds

85 – 90 but not including 90

FrequencyTimes in seconds

Can you explain what the symbols in the middle column mean?








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

2) 105 ≤ t < 110

“times larger than or equal to 90 seconds and less than 95 seconds” or

“times larger than or equal to 105 seconds and less than 110 seconds” or “from 105 up to but not including 110”.

“from 90 up to but not including 95”.




100 ≤ t < 105

105 ≤ t < 110

95 ≤ t < 100

90 ≤ t < 95

85 ≤ t < 90

Times in seconds Frequency

88.4 91.5 92.1 93.3 93.9 94.7 95.0 95.3 95.5 95.6 95.6 96.3 96.5 96.9 97.0 97.0 97.0 97.3 97.4 97.4 97.7 97.8 98.0 98.2 98.2 98.4 98.4 98.5 98.9 99.0 99.1 99.6 99.6 99.8 100.0 100.6100.6 101.1 101.4 101.4 101.5 101.6 101.6 101.8 101.9102.1 102.5 102.6 102.7 103.1 103.1 103.1 104.1 105.0105.2 105.6 105.6 105.7 105.8 105.9

Class intervals

Use the data to fill in the table.

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D5.4 Histograms



D5.1 Grouping continuous data


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:

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Height (cm)

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5

10

15

20

25

30

35

150 155 160 165 170 175 180 185

Heights of students

This type of frequency diagram is often called a histogram.


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 bars must be joined together, to indicate that the data is continuous.

The highest and lowest times in each interval go at either end of the bar, as shown below:

80 85 90


Frequency diagram of cycling data

The cycling data we looked at earlier can be displayed in the following frequency diagram:

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800

5

10

15

20

25

30

85 90 95 100 105Times in seconds

What conclusions can you draw from the graph?


Changing the class interval

When the class intervals are changed the same data produces the following graph:

What size class intervals have been used?

What additional information is available from this graph?

Which graph is more useful?

Times in seconds85 87.5 90 92.5 95 97.5 100 102.5 105 107.5

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D5.4 Histograms






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

87.5

92.5

97.5

102.5

107.5

To find the midpoint of two numbers, add them together and divide by 2.

Midpoints

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.


Midpoints


Line graph of midpoints

If we plot the midpoints at the top of each bar and join them together the following graph is produced:

Fre

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800

5

10

15

20

25

30

85 90 95 100 105

Times in seconds

11075


Fre

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800

5

10

15

20

25

30

85 90 95 100 105

Times in seconds

11075

Frequency polygon of cycling data

Removing the bars leaves a frequency polygon.

Fre

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800

5

10

15

20

25

30

85 90 95 100 105

Times in seconds

11075


For each category, find

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.

Senior category

5

10

15

20

085 90 95 100 105 110 115 120 125 130 135

Junior category

2

4

6

8

10

085 90 95 100 105 110 115 120 125 130 135

Compare the performances in the two categories.

the size of the class intervals

the modal class intervalthe range.


10

20

0

Comparing frequency polygons

The same data has been used in these graphs.

Senior category

10

20

30

085 95 105 115 125 135

Junior category

85 95 105 115 125 135

For each category, find

Compare these graphs with the previous ones. Which do you find more useful for analyzing the race times and why?

the size of the class intervalsthe number of class intervalsthe modal class interval.


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.

This means that on average Juniors are faster than Seniors.


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D5.4 Histograms







There are __ times as many people in the 105 ≤ t < 110 interval than there are in the 95 ≤ t < 100 interval.

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Histograms

This frequency diagram represents the race times for the Youth category, which is 14 to 16 year olds.

Is the bar three times as big?

How many people are represented by each square on the grid?

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10

12

Time in seconds95 100 105 110 115 120 125 130 135


Discuss this statement. Do you agree or disagree?

Histograms

“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.”

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2

4

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10

12

Time in seconds95 100 105 110 115 120 125 130 135


Some of the intervals are very small, which makes any conclusions about them unreliable.

Combining intervals

It is sometimes sensible to combine intervals together.

Which intervals would you combine?

Fre

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2

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6

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10

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Time in seconds95 100 105 110 115 120 125 130 135


Histograms with bars of unequal widthF

requ

ency

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2

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10

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Time in seconds95 100 105 110 115 120 125 130 135

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?

How many people does one square represent?

This graph represents the same data as the previous one.What has changed?

Do the numbers along the vertical axis still represent frequency?


The frequency for 105 ≤ t < 110 is the same as the frequency for ___________.

Histograms with bars of unequal widthF

requ

ency

0

Time in seconds95 100 105 110 115 120 125 130 135

In the original histogram, the frequency was proportional to the area. Is this still true in the new histogram?

120 ≤ t < 135

Are the areas of the bars the same?

In a histogram, the frequency is equal to the area of the bar.In a histogram, the frequency is equal to the area of the bar.

Each square stills represents two people.


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D5.4 Histograms





Therefore, the height must equal the area ÷ the width.

The area of the bar gives the frequency and so we can write,

This height is called the frequency density.

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 bar must be the area.

4 people

95 105110

frequency density

Height of the bar =frequency

width of interval


Frequency density

Frequency density =frequency

width of interval

In our example, each square represents 2 people.

What scale do we need for the vertical axis?

Width of interval = 10

Area = 4

Height = 4 ÷ 10 = 0.4

Frequency density = 0.4

0.4 4 people

95 105110


Frequency = frequency density × width of intervalFrequency = frequency density × width of interval

to check this scale for the other bars in the graph.

0.4 × 15

2.2 × 5

1.4 × 5

1.2 × 5

0.4 × 10

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11

6

6

4

Frequency

density × width

110 ≤ t < 115

115 ≤ t < 120

105 ≤ t < 110

120 ≤ t < 135

95 ≤ t < 105

Time in seconds

Area

(frequency)

Calculating the frequencyF

req

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0.4

0.8

1.2

1.6

2.0

2.4

Time in seconds95 100 105 110 115 120 125 130 135

We can use the formula,


Complete the table for this data and draw a histogram.

Calculating the frequency density

Frequency density =frequency

width of interval

2130 ≤ t < 150

12115 ≤ t < 130

8105 ≤ t < 115

5100 ≤ t < 105

895 ≤ t < 100

Frequency density

Frequency ÷ width of interval

FrequencyTime in seconds

0.12 ÷ 20

0.812 ÷ 15

0.88 ÷ 10

1.05 ÷ 5

1.68 ÷ 5


Your histogram should look like this:


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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


The first bar represents 40 people.

Calculating the class intervals

Time in seconds

Fre

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nsity

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2

4

6

8

This is a histogram of race times from a longer race.

The lowest time was 100 seconds.

Work out the scale along the bottom and the frequencies for each interval.

100


Length of interval

Frequency density

Frequency class interval


Frequency = frequency density × width of intervalFrequency = frequency density × width of interval

to complete the following table for the data in the histogram.

We can use the formula,

20

20

80

40

40 ÷ 1= 40

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5

6

4

1

20 × 3 = 60

20 × 5 = 100

80 × 6 = 480

4 × 40 = 160

40

180 ≤ t < 260

260 ≤ t < 280

140 ≤ t < 180

280 ≤ t < 300

100 ≤ t < 140


Write a definition of each word below and then design a mind map outlining the key facts you have learnt.

data setclass intervalmidpointrangeaxesfrequency diagram

Include methods for calculating and

drawing; possible mistakes to avoid

…

Review

frequencyfrequency polygonmodal class intervalhistogramfrequency density

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© Boardworks Ltd 2005 1 of 40 D5 Frequency diagrams for continuous data KS4 Mathematics