KS 4 Mathematics D 5 Frequency diagrams for

KS 4 Mathematics D 5 Frequency diagrams for continuous data 1 of 40 © Boardworks Ltd 2005

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

Analysing data Tom regularly takes part in downhill cycle races. He records the race times of all the competitors in a race on a spreadsheet. Tom’s best time is 101. 6 seconds. How accurately has he measured this time? Is the data continuous or discrete? 3 of 40 © Boardworks Ltd 2005

Analysing data Here are some race times in seconds from the downhill racing event. 88. 4 91. 5 92. 1 93. 3 93. 9 94. 7 95. 0 95. 3 95. 5 95. 6 96. 3 96. 5 96. 9 97. 0 97. 3 97. 4 97. 7 97. 8 98. 0 98. 2 98. 4 98. 5 98. 9 99. 0 99. 1 99. 6 99. 8 100. 0 100. 6 101. 1 101. 4 101. 5 101. 6 101. 8 101. 9 102. 1 102. 5 102. 6 102. 7 103. 1 104. 1 105. 0 105. 2 105. 6 105. 7 105. 8 105. 9 If you wanted to analyze the performances, what could you do with the data? How easy is the data to analyze in this format? Can you draw any conclusions? 4 of 40 © Boardworks Ltd 2005

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. 110. 0 105. 0 100. 0 95. 0 90. 0 85. 0 80. 0 5 of 40 What labels could be added to the axes? What does the graph show? Is it an appropriate graph? © Boardworks Ltd 2005

Grouping data A list of results is called a data set. It is often easier to analyze a large data set if the data is put into groups. The widths of the groups are called class intervals. A frequency diagram or histogram can then be drawn. You will need to decide on the size of each class interval so that there are between 5 and 10 class intervals. What is the best size for the class intervals for the race times data? 6 of 40 © Boardworks Ltd 2005

class intervals 88. 4 91. 5 92. 1 93. 3 93. 9 94. 7 95. 0 95. 3 95. 5 95. 6 96. 3 96. 5 96. 9 97. 0 97. 3 97. 4 97. 7 97. 8 98. 0 98. 2 98. 4 98. 5 98. 9 99. 0 99. 1 99. 6 99. 8 100. 0 100. 6 101. 1 101. 4 101. 5 101. 6 101. 8 101. 9 102. 1 102. 5 102. 6 102. 7 103. 1 104. 1 105. 0 105. 2 105. 6 105. 7 105. 8 105. 9 The times range from about 85 to about 110 seconds. 110 – 85 = 25 seconds. Suppose we decide to use class intervals with a width of 5 seconds. 25 ÷ 5 = 5 class intervals 7 of 40 © Boardworks Ltd 2005

Notation for class intervals How should the class intervals be written down? Times in seconds 85 – 90 90 – 95 Frequency What is wrong with this table? 95 – 100 – 105 - 110 8 of 40 © Boardworks Ltd 2005

Notation for class intervals Can you explain what the symbols in the middle column mean? Time in seconds 85 – 90 but not including 90 Time in seconds 85 ≤ t < 90 90 – 95 but not including 95 90 ≤ t < 95 95 – 100 but not including 100 95 ≤ t < 100 – 105 but not including 105 100 ≤ t < 105 – 110 but not including 110 105 ≤ t < 110 9 of 40 Frequency © Boardworks Ltd 2005

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”. 2) 105 ≤ t < 110 “times larger than or equal to 105 seconds and less than 110 “from 105 up to but not including 110”. seconds” or 10 of 40 © Boardworks Ltd 2005

Notation for class intervals 11 of 40 © Boardworks Ltd 2005

Class intervals 88. 4 91. 5 92. 1 93. 3 93. 9 94. 7 95. 0 95. 3 95. 5 95. 6 96. 3 96. 5 96. 9 97. 0 97. 3 97. 4 97. 7 97. 8 98. 0 98. 2 98. 4 98. 5 98. 9 99. 0 99. 1 99. 6 99. 8 100. 0 100. 6 101. 1 101. 4 101. 5 101. 6 101. 8 101. 9 102. 1 102. 5 102. 6 102. 7 103. 1 104. 1 105. 0 105. 2 105. 6 105. 7 105. 8 105. 9 Time in seconds 85 ≤ t < 90 90 ≤ t < 95 95 ≤ t < 100 ≤ t < 105 ≤ t < 110 12 of 40 Frequency 1 5 28 19 7 Use the data to fill in the table. © Boardworks Ltd 2005

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

Frequency diagrams can be used to display grouped continuous data. For example, this frequency diagram shows the distribution of heights for a group of students: Heights of students Frequency 35 30 25 20 15 10 5 0 155 160 165 170 175 180 185 Height (cm) This type of frequency diagram is often called a histogram. 14 of 40 © Boardworks Ltd 2005

Drawing frequency diagrams When drawing frequency diagrams for grouped continuous data, remember the following points: 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 below: 80 85 90 The bars must be joined together, to indicate that the data is continuous. 15 of 40 © Boardworks Ltd 2005

Frequency diagram of cycling data The cycling data we looked at earlier can be displayed in the following frequency diagram: 30 Frequency 25 20 15 10 5 0 80 85 90 95 100 105 Times in seconds What conclusions can you draw from the graph? 16 of 40 © Boardworks Ltd 2005

Changing the class interval When the class intervals are changed, the same data produces the following graph: Frequency 20 15 10 5 0 85 87. 5 90 92. 5 95 97. 5 100 102. 5 107. 5 Times in seconds What size class intervals have been used? What additional information is available from this graph? Which graph is more useful? 17 of 40 © Boardworks Ltd 2005

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

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. What are the midpoints of each class interval for the race times data? Times in seconds 85 ≤ t < 90 90 ≤ t < 95 95 ≤ t < 100 ≤ t < 105 ≤ t < 110 19 of 40 Midpoint 87. 5 92. 5 97. 5 102. 5 107. 5 To find the midpoint of two numbers, add them together and divide by 2. © Boardworks Ltd 2005

Midpoints 20 of 40 © Boardworks Ltd 2005

Line graph of midpoints If we plot the midpoints at the top of each bar and join them together the following graph is produced: 30 Frequency 25 20 15 10 5 0 75 80 85 90 95 100 105 110 Times in seconds 21 of 40 © Boardworks Ltd 2005

Frequency polygon of cycling data Removing the bars leaves a frequency polygon. 30 Frequency 25 20 15 10 5 0 75 80 85 90 95 100 105 110 Times in seconds 22 of 40 © Boardworks Ltd 2005

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 10 8 15 6 4 2 0 85 20 Senior category 10 5 90 95 100 105 110 115 120 125 130 135 0 85 90 95 100 105 110 115 120 125 130 135 For each category, find the size of the class intervals the modal class interval the range. Compare the performances in the two categories. 23 of 40 © Boardworks Ltd 2005

Comparing frequency polygons The same data has been used in these graphs. Junior category 20 Senior category 30 20 10 0 85 10 0 95 105 115 125 135 85 95 105 115 125 135 For each category, find the size of the class intervals 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? 24 of 40 © Boardworks Ltd 2005

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. 25 of 40 © Boardworks Ltd 2005

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

Histograms This frequency diagram represents the race times for the Youth category, which is for 14 to 16 year olds. 12 Frequency 10 8 6 4 There are __ 3 times as many people in the 105 ≤ t < 110 interval as there are in the 95 ≤ t < 100 interval. 2 0 95 100 105 110 115 120 125 130 135 Time in seconds How many people are represented by each square on the grid? 27 of 40 © Boardworks Ltd 2005

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. ” 12 Frequency 10 8 6 4 2 0 28 of 40 95 100 105 110 115 120 125 130 135 Time in seconds © Boardworks Ltd 2005

Combining intervals Some of the intervals are very small, which makes any conclusions about them unreliable. 12 Frequency 10 8 6 Which intervals would you combine? 4 2 0 29 of 40 It is sometimes sensible to combine intervals together. 95 100 105 110 115 120 125 130 135 Time in seconds © Boardworks Ltd 2005

Histograms with bars of unequal width Frequency This graph represents the same data as the previous one. What has changed? The first two intervals both had 12 a frequency of 2. The first bar 10 now represents an interval 8 twice as big. 6 How many people are in this interval? 4 2 0 95 100 105 110 115 120 125 130 135 Time in seconds How many people does one square represent? Do the numbers along the vertical axis still represent frequency? 30 of 40 © Boardworks Ltd 2005

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 The frequency for 105 ≤ t < 110 is the same as the frequency ≤ t < 135 for 120 ______. Are the areas of the bars the same? 0 95 100 105 110 115 120 125 130 135 Time in seconds Each square stills represents two people. In a histogram, the frequency is equal to the area of the bar. 31 of 40 © Boardworks Ltd 2005

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

Frequency density In a histogram, the frequency is given by the area of each bar. frequency density 4 people 95 110 105 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, frequency Height of the bar = width of interval This height is called the frequency density. 33 of 40 © Boardworks Ltd 2005

Frequency density frequency Frequency density = width of interval In our example, each square represents 2 people. 0. 4 4 people 95 110 105 What scale do we need for the vertical axis? Width of interval = 10 Area = 4 Height = 4 ÷ 10 = 0. 4 Frequency density = 0. 4 34 of 40 © Boardworks Ltd 2005

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 2. 4 2. 0 Time in seconds 1. 6 Frequency Area density × (frequency) width 1. 2 95 ≤ t < 105 0. 4 × 10 4 0. 8 105 ≤ t < 110 1. 2 × 5 6 0. 4 110 ≤ t < 115 1. 4 × 5 7 115 ≤ t < 120 2. 2 × 5 11 120 ≤ t < 135 0. 4 × 15 6 0 95 100 105 110 115 120 125 130 135 Time in seconds 35 of 40 © Boardworks Ltd 2005

Calculating the frequency density frequency Frequency density = width of interval Complete the table for this data and draw a histogram. Time in Frequency seconds 95 ≤ t < 100 8 100 ≤ t < 105 5 105 ≤ t < 115 8 115 ≤ t < 130 12 130 ≤ t < 150 2 36 of 40 Frequency ÷ width of interval 8÷ 5 5÷ 5 8 ÷ 10 12 ÷ 15 2 ÷ 20 Frequency density 1. 6 1. 0 0. 8 0. 1 © Boardworks Ltd 2005

Calculating the frequency density Your histogram should look like this: 1. 6 Frequency density 1. 4 1. 2 1. 0 0. 8 0. 6 0. 4 0. 2 0 95 100 105 110 115 120 125 130 135 140 145 150 Time in seconds 37 of 40 © Boardworks Ltd 2005

Calculating the class intervals This is a histogram of race times from a longer race. Frequency density 8 The first bar represents 40 people. 6 4 The lowest time was 100 seconds. 2 0 100 Time in seconds Work out the scale along the bottom and the frequencies for each interval. 38 of 40 © Boardworks Ltd 2005

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. Frequency density Length of interval class interval 40 1 40 ÷ 1= 40 100 ≤ t < 140 4 × 40 = 160 4 40 140 ≤ t < 180 80 × 6 = 480 6 80 180 ≤ t < 260 20 × 5 = 100 5 20 260 ≤ t < 280 20 × 3 = 60 3 20 280 ≤ t < 300 39 of 40 © Boardworks Ltd 2005

Review Write a definition of each word below and then design a mind map outlining the key facts you have learnt. data set frequency class interval frequency polygon midpoint modal class interval range histogram axes frequency density frequency diagram Include methods for calculating and drawing; possible mistakes to avoid … 40 of 40 © Boardworks Ltd 2005