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

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© Boardworks Ltd 2005 2 of 40 Contents A A A A A D5.1 Grouping continuous data D5 Frequency diagrams for continuous data D5.2 Frequency diagrams D5.3 Frequency polygons D5.4 Histograms D5.5 Frequency density

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© Boardworks Ltd 2005 3 of 40 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.

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© Boardworks Ltd 2005 4 of 40 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 99.8100.0100.6 100.6101.1101.4101.4101.5101.6101.6101.8101.9 102.1102.5102.6102.7103.1103.1103.1104.1105.0 105.2105.6105.6105.7105.8105.9

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© Boardworks Ltd 2005 5 of 40 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

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© Boardworks Ltd 2005 6 of 40 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. What is the best size for the class intervals for the race times data?

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© Boardworks Ltd 2005 7 of 40 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 99.8100.0100.6 100.6101.1101.4101.4101.5101.6101.6101.8101.9 102.1102.5102.6102.7103.1103.1103.1104.1105.0 105.2105.6105.6105.7105.8105.9

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© Boardworks Ltd 2005 8 of 40 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

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© Boardworks Ltd 2005 9 of 40 100 ≤ 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? Notation for class intervals 100 – 105 but not including 105 105 – 110 but not including 110 95 – 100 but not including 100 90 – 95 but not including 95

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© Boardworks Ltd 2005 10 of 40 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 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”.

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© Boardworks Ltd 2005 11 of 40 Notation for class intervals

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© Boardworks Ltd 2005 12 of 40 100 ≤ t < 105 105 ≤ t < 110 95 ≤ t < 100 90 ≤ t < 95 85 ≤ t < 90 Times in secondsFrequency 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 99.8100.0100.6 100.6101.1101.4101.4101.5101.6101.6101.8101.9 102.1102.5102.6102.7103.1103.1103.1104.1105.0 105.2105.6105.6105.7105.8105.9 Class intervals Use the data to fill in the table. 19 7 28 5 1

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© Boardworks Ltd 2005 13 of 40 A A A A A D5.2 Frequency diagrams Contents D5.3 Frequency polygons D5.4 Histograms D5.5 Frequency density D5 Frequency diagrams for continuous data D5.1 Grouping continuous data

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© Boardworks Ltd 2005 14 of 40 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) 0 5 10 15 20 25 30 35 150 155160165170175180185 Heights of students This type of frequency diagram is often called a histogram.

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© Boardworks Ltd 2005 15 of 40 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

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© Boardworks Ltd 2005 16 of 40 Frequency diagram of cycling data The cycling data we looked at earlier can be displayed in the following frequency diagram: Frequency 80 0 5 10 15 20 25 30 859095100105 Times in seconds What conclusions can you draw from the graph?

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© Boardworks Ltd 2005 17 of 40 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 seconds 8587.59092.59597.5100102.5105107.5 Frequency 0 5 10 15 20

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© Boardworks Ltd 2005 18 of 40 Contents A A A A A D5.3 Frequency polygons D5.4 Histograms D5.5 Frequency density D5 Frequency diagrams for continuous data D5.2 Frequency diagrams D5.1 Grouping continuous data

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© Boardworks Ltd 2005 19 of 40 What are the midpoints of each class interval for the race times data? Times in secondsMidpoint 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.

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© Boardworks Ltd 2005 20 of 40 Midpoints

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© Boardworks Ltd 2005 21 of 40 Line graph of midpoints If we plot the midpoints at the top of each bar and join them together the following graph is produced: Frequency 80 0 5 10 15 20 25 30 859095100105 Times in seconds 11075

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© Boardworks Ltd 2005 22 of 40 Frequency 80 0 5 10 15 20 25 30 859095100105 Times in seconds 11075 Frequency polygon of cycling data Removing the bars leaves a frequency polygon. Frequency 80 0 5 10 15 20 25 30 859095100105 Times in seconds 11075

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© Boardworks Ltd 2005 23 of 40 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 0 859095100105110115120125130135 Junior category 2 4 6 8 10 0 859095 100 105110115120125 130 135 Compare the performances in the two categories. the size of the class intervals the modal class interval the range.

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© Boardworks Ltd 2005 24 of 40 10 20 0 Comparing frequency polygons The same data has been used in these graphs. Senior category 10 20 30 0 8595105115125135 Junior category 8595105115125135 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 intervals the number of class intervals the modal class interval.

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© Boardworks Ltd 2005 25 of 40 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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© Boardworks Ltd 2005 26 of 40 Contents A A A A A D5.4 Histograms D5.5 Frequency density D5 Frequency diagrams for continuous data D5.3 Frequency polygons D5.2 Frequency diagrams D5.1 Grouping continuous data

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© Boardworks Ltd 2005 27 of 40 There are __ times as many people in the 105 ≤ t < 110 interval than there are in the 95 ≤ t < 100 interval. 3 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? Frequency 0 2 4 6 8 10 12 Time in seconds 95100105110115120125130135

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© Boardworks Ltd 2005 28 of 40 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.” Frequency 0 2 4 6 8 10 12 Time in seconds 95100105110115120125130135

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© Boardworks Ltd 2005 29 of 40 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? Frequency 0 2 4 6 8 10 12 Time in seconds 95100105110115120125130135

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© Boardworks Ltd 2005 30 of 40 Histograms with bars of unequal width Frequency 0 2 4 6 8 10 12 Time in seconds 95100105110115120125130135 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?

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© Boardworks Ltd 2005 31 of 40 The frequency for 105 ≤ t < 110 is the same as the frequency for ___________. Histograms with bars of unequal width Frequency 0 Time in seconds 95100105110115120125130135 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. Each square stills represents two people.

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© Boardworks Ltd 2005 32 of 40 Contents A A A A A D5.5 Frequency density D5 Frequency diagrams for continuous data D5.4 Histograms D5.3 Frequency polygons D5.2 Frequency diagrams D5.1 Grouping continuous data

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© Boardworks Ltd 2005 33 of 40 Therefore, the height must equalthe 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 95105110 frequency density Height of the bar = frequency width of interval

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© Boardworks Ltd 2005 34 of 40 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 95105110

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© Boardworks Ltd 2005 35 of 40 Frequency = 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 7 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 frequency Frequency density 0 0.4 0.8 1.2 1.6 2.0 2.4 Time in seconds 95100105110115120125130135 We can use the formula,

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© Boardworks Ltd 2005 36 of 40 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

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© Boardworks Ltd 2005 37 of 40 Your histogram should look like this: Calculating the frequency density Frequency density 0 0.2 0.4 0.6 0.8 1.0 1.2 Time in seconds 95100105110115120125130135140145150 1.4 1.6

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© Boardworks Ltd 2005 38 of 40 The first bar represents 40 people. Calculating the class intervals Time in seconds Frequency density 0 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

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© Boardworks Ltd 2005 39 of 40 Length of interval Frequency density Frequencyclass interval Calculating the frequency density Frequency = frequency density × width of interval to complete the following table for the data in the histogram. We can use the formula, 20 80 40 40 ÷ 1= 40 3 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

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© Boardworks Ltd 2005 40 of 40 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 Include methods for calculating and drawing; possible mistakes to avoid … Review frequency frequency polygon modal class interval histogram frequency density

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