Processing, representing and interpreting data
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1 Processing, representing and interpreting data 21 CHAPTER 2.1 A head CHAPTER polygons A diagram can be drawn from grouped discrete data. A diagram looks the same as a bar chart except that the label underneath each bar represents a group. The information in this table can be represented as a diagram. Mark Mark A histogram can be drawn from grouped continuous data. A histogram is similar to a bar chart but represents continuous data so there is no gap between the bars. There is a scale on the horizontal axis rather than a label under each bar. Example 1 The grouped table shows information about the heights of 42 students. a Write down the modal class interval. b Jenny s height is cm. In which class interval is her height recorded? c Charlie s height is exactly 18 cm. In which class interval is his height recorded? d Draw a histogram to represent this information. Solution 1 a The modal class is 165 h 17 Height (h cm) 16 h h h h h h 19 2 This class interval has the highest, 14 b Jenny s height is in the class interval 175 h 18 c Charlie s height is in the class interval 18 h cm is greater than 175 cm but less than 18 cm. 18 is shown at the end of one class interval and at the beginning of another. The sign for less than or equal to ( ) shows that 18 should go in the class interval 18 h
2 CHAPTER 21 Processing, representing and interpreting data d The histogram has a scale on the horizontal axis and no gaps between the bars Height (cm) A polygon is another graph which shows data. To draw a polygon for the data in Example 1, mark the midpoint of the top of each bar and join these points with straight lines Height (cm) This is the polygon Height (cm) Example 2 The table gives information about the weights, in grams, of 3 apples. a Write down the modal class. b Use the information to draw a histogram. c Draw a polygon to represent the information. Weight (w g) 9 w w w w w w
3 21.1 polygons CHAPTER 21 Solution 2 a The modal class is 1 w 15 b and c As the question has asked for both a histogram and a polygon to be drawn, draw the polygon on the histogram Weight (grams) More than one polygon can be drawn on the same grid to compare data. Example 3 The two polygons show the heights of a group of girls and the heights of a group of boys Girls heights Boys heights Height (cm) Compare the heights of the two groups. Give a reason for your answers. Solution 3 The boys are generally taller than the girls. There are more tall boys than tall girls. There are more short girls than short boys. The line showing the boys heights is above the line for the girls heights towards the right of the graph. There are two girls and three boys in the cm class interval. There are ten girls and one boy in the cm class interval. 331
4 CHAPTER 21 Processing, representing and interpreting data Example 4 This graph shows the percentage of Mathstown buses that arrived on time each year from 2 to 23 Explain why this graph is misleading. 96 Percentage (%) of buses on time Year Solution 4 The steep line between 2 and 21 suggests that a much higher percentage of buses were on time in 21 than in 2 but it is only 2% more. If the vertical scale starts from the lines are all less steep, showing that the increases are small. The disadvantage of this is that a large area of the graph is empty Percentage (%) of buses on time Year A zig-zag in the vertical axis can be used to show that the scale does not start at Percentage (%) of buses on time Year
5 21.1 polygons CHAPTER 21 Exercise 21A 1 The grouped table shows the test marks of a class of 32 students. a Write down the modal class. b Draw a diagram for this information. Test marks The histogram shows information about the times taken by some girls to run 1 m. a Write down the modal class. b Work out the total number of girls Some boys also ran 1 m. The grouped table shows information about their times. c Draw a histogram to show this information. d Draw a polygon to show this information Time (seconds) Time (t seconds) 1 t t t t t In one month Julie went to the post office 2 times and to the bank 2 times. The polygons show 1 information about the amount Post 9 office of time Julie spent waiting in a 8 Bank queue at the post office and at 7 the bank. a At the bank how many times did Julie wait for between 15 and 2 minutes? b At the post office how many times did Julie wait for between 5 and 1 minutes? c For what fraction of the times Julie went to the post office did she wait for less than 1 minutes? Time (minutes) Give your fraction in its simplest form. d Where did Julie generally have to wait the longest time, the bank or the post office? You must give a reason for your answer. 333
6 CHAPTER 21 Processing, representing and interpreting data 4 The graph shows the number of cars sold in 3 years. Give two reasons why the graph is misleading. Number of cars sold The grouped table shows information about the Time (x minutes) amount of time 16 students spent doing homework one x 1 4 evening. 1 x 2 12 The information in the table can be used to find, for example, the total number of students who spent up to and including 2 x minutes which is 16 ( 4 12). 3 x is the cumulative for the interval x 2 4 x 5 2 (In other words cumulative is the running total.) Similarly the cumulative for the interval 5 x 6 1 x 3 is 62 ( ) Here is the complete cumulative table. Time (x minutes) The last number in the cumulative x 1 4 column is 16, the total number of students. x 2 (4 12 ) 16 A cumulative table can be used to draw a cumulative graph. x 3 (16 46 ) 62 The cumulative 4 for the interval x 4 (62 68 ) 13 x 1 is plotted at (1, 4), that is, at the x 5 (13 2 ) 15 top end of the interval to ensure that all 4 students have been included. x 6 (15 1 ) 16 Similarly the cumulative 16 for the interval x 2 is plotted at (2, 16). The remaining points are plotted at (3, 62), (4, 13), (5, 15) and (6, 16). The points can be joined with a smooth curve or with line segments to give a cumulative graph Time (x minutes) Time (x minutes)
7 21.2 CHAPTER 21 The table gives information about the times in minutes 3 people in a doctor s surgery waited. a Complete the cumulative table. b Draw a cumulative graph for your table. Solution 5 a Example 5 Time (t minutes) t t t t t 25 2 Time (t minutes) t 5 1 t 1 6 t t 2 28 t 25 3 Time (t minutes) b t 5 t 1 t 15 t 2 t 25 Time (t minutes) The values at the end of each class interval are 5, 1, 15, 2 and 25 So the coordinates used for plotting this cumulative graph are (5, 1), (1, 6), (15, 18), (2, 28) and (25, 3). graphs can be used to find estimates for the number of items up to a certain value. Example 6 4 students took a test. The cumulative graph gives information about their marks O O Mark a Use the graph to find an estimate for the number of students whose mark is less than 3 b Use the graph to work out an estimate for the number of students whose mark is more than 45 c 21 students passed the test. Work out the pass mark for the test. 335
8 CHAPTER 21 Processing, representing and interpreting data Solution O Mark a 24 students have a mark less than 3 Find a mark of 3 on the x-axis. Move vertically up to the curve. Read the value off the cumulative axis. b 37 students have a mark less than students have a mark more than 45 Find a mark of 45 on the x-axis. Move vertically up to the curve. Read the value off the cumulative axis. The number of students whose mark was more than 45 is required so subtract 37 from 4 c 21 students passed the test so students did not pass the test. The pass mark for the test is 26 Find 19 on the cumulative axis. Move horizontally across to the curve. Read the value off the Mark axis. Example 7 The grouped table gives information about the number of people in a bus on 3 mornings. Number of passengers a Complete the cumulative table. b Draw a cumulative diagram for your table. Number of passengers
9 21.2 CHAPTER 21 Solution 7 a Number of passengers b O 1 2 Number of passengers The data in the table is discrete. The coordinates used for plotting this cumulative graph are (4, 4), (9, 11), (14, 22), (19, 27) and (24, 3). Exercise 21B 1 The grouped table gives information about the number of minutes 6 music students practised last week. Minutes (m) m m m m m m m m 12 2 a Copy and complete the cumulative table. b Draw a cumulative graph for your table. c Use your graph to find an estimate for the number of music students who practised for i less than 4 minutes ii more than 4 minutes. Minutes (m) m 15 m 3 m 45 m 6 m 75 m 9 m 15 m
10 CHAPTER 21 Processing, representing and interpreting data 2 The grouped table gives information about the time taken for 32 people to solve a mathematics problem. Time (t seconds) t t t t t 5 2 a Copy and complete the cumulative table. b Use your table to draw a cumulative graph. Time (t seconds) t 1 t 2 t 3 t 4 t 5 3 The grouped table gives information about the number of people waiting at a bus stop on 2 mornings. a Copy and complete the cumulative table. b Use your table to draw a cumulative graph. Number of people Number of people The graph shows information about the weights in grams of 8 plums O Weight (grams) a Use the graph to find an estimate for the number of plums that weigh less than 25 grams. b Use the graph to work out an estimate for the number of plums that weigh more than 3 grams. 338
11 21.2 CHAPTER 21 5 The cumulative diagram below gives information about the prices of some houses Prices of houses ( ) Use the graph to find a the total number of houses b an estimate for the number of houses that cost less than 13 c an estimate for the number of houses that cost more than 1 but less than 15 6 The cumulative graph shows information about the English exam marks of 1 students Mark a Use the graph to find an estimate for the number of students who scored less than 4 marks. b 32 students passed the English exam. Use the graph to work out an estimate for the pass mark. Median The grouped table from the start of Section 21.2 shows information about the amount of time 16 students spent doing homework one evening. To find the class interval containing the median time, the position in the table of the n 1 2 th student is needed (see Section 17.3). 339
12 CHAPTER 21 Processing, representing and interpreting data The median time is therefore the time of the th student. To find the class interval containing the th student use the table along with the cumulative. Time (x minutes) 62 students had a time of 3 minutes or less. x x x x x x students had a time of 4 minutes or less. So the th student will be in the 3 x 4 class interval. The class interval containing the median is 3 x 4 The cumulative graph can be used to find an estimate for the median. As this graph has been drawn from grouped data the individual time of each student is not known. So it is only possible to find an estimate for the median rather than the actual median. The estimate for the median will be found from the cumulative graph. The scale makes it difficult to tell the difference between 8 and on the cumulative axis. When finding an estimate for the median from a cumulative graph it is acceptable to use 1 th value since n is large. the n 2 th value rather than the n 2 To find an estimate for the median in this example use the graph to find the time of the 8 th 16 2 student. Quartiles and interquartile range Quartiles and interquartile range were introduced in Section 17.3 Estimates for the lower quartile and upper quartile can be read off a cumulative graph. The values for the upper and lower quartiles are estimates so the ( ) 4 th value can be used to give an estimate for the lower quartile and the ( ) 12 th value can be used to give an estimate for the upper quartile. When finding estimates from a cumulative graph for the median, lower quartile and upper quartile, for a large data set containing n values read off the cumulative axis at 1 2 n for the median 1 4 n for the lower quartile 3 4 n for the upper quartile. The estimates for the lower quartile and the upper quartile from a cumulative graph can be used to find an estimate for the interquartile range. 34
13 21.2 CHAPTER Start at 12 to read off the upper quartile Start at 8 to read off the median Start at 4 to read off the lower quartile For this example an estimate for the lower quartile is 26 median is 32 upper quartile is 38 Interquartile range O Interquartile range Time (x minutes) Exercise 21C 1 Use the graph in Exercise 21B question 4 to find estimates for the i median ii lower quartile iii upper quartile. 2 Use the graph in Exercise 21B question 5 to find estimates for the i median ii lower quartile iii upper quartile. 3 Use the graph in Exercise 21B question 6 to find estimates for the i median ii interquartile range. 4 8 workers were asked what distance they travelled to get to work. The grouped table shows this information. a Copy and complete the cumulative table. b Find the class interval in which the median lies. c Use your table to draw a cumulative graph. d Use your cumulative graph to find an estimate for the median. Distance (d kilometres) d 5 d 1 d 15 d 2 d 25 d 3 Distance (d kilometres) e Use your cumulative graph to find an estimate for the interquartile range. d d d d d d
14 CHAPTER 21 Processing, representing and interpreting data 5 The table gives information about the amount of time in minutes 1 adults spent preparing last night s meal. a Use the information in the table to draw a cumulative graph. b Use your cumulative graph to find estimates for the i lower quartile ii upper quartile iii interquartile range. 6 The table gives information about the luggage weight of 2 passengers. a Use the information in the table to draw a cumulative graph. b Use your cumulative graph to find estimates for the i number of passengers whose luggage weighs more than 15 kg ii median iii interquartile range. Time (t minutes) t 1 3 t 2 19 t 3 66 t 4 9 t 5 97 t 6 1 Luggage weight (w kg) w 4 6 w 8 22 w w w w Box plots Box plots (sometimes called box and whisker diagrams) are diagrams that show the spread of a set of data. The median, lower and upper quartiles along with the minimum and maximum value are used to draw a box plot. For example, for some data about times (in minutes) the minimum value 8 maximum value 57 lower quartile 26 median 32 upper quartile 38 Here is the box plot for this data. Minimum value Lower quartile Median Upper quartile The box shows the spread over Time (minutes) the middle 5% of the data The difference between the end points shows the range of data Maximum value Example 8 The times in seconds taken by 15 students to solve a problem are listed in order Draw a box plot for this data. Solution 8 Median is the (15 1) 8 th The numbers are in order. n is not large so number 2 the median is the 1 2 (n 1) th value
15 21.4 Comparing distributions CHAPTER 21 Median 23 Lower quartile is the 1 4 (15 1) 4 th number Lower quartile 11 Upper quartile is the 3 4 (15 1) 12 th number Upper quartile 27 Minimum value 5 Maximum value = 31 Lower quartile is the 1 4 (n 1) th value. Upper quartile is the 3 4 (n 1) th value Time (seconds) 21.4 Comparing distributions Box plots are useful for comparing the distribution of data sets. 8 seedlings were divided into 2 groups. Group A were grown in a greenhouse. Group B were grown outside. After a period of time the heights of the seedlings were measured. The heights were used to draw two cumulative graphs. Example Group B Group A Group A Group B Shortest seedling (cm) Tallest seedling (cm) a Use the information provided in the table and the cumulative graphs to draw a box plot of the heights of seedlings in group A and a box plot of the heights of the seedlings in group B. b Compare the heights of the seedlings in the two groups. O Length (cm) Solution Group A Group B Shortest seedling (cm) Tallest seedling (cm) Lower quartile Median Upper quartile O Length (cm) 343
16 CHAPTER 21 Processing, representing and interpreting data Group A Group B Length (cm) The heights of the seedlings in group B are more spread out than the heights of the seedlings in group A. The seedlings in group A are generally taller than the seedlings in group B. The middle 5% of the seedlings in group A have a wider spread than the middle 5% of the seedlings in group B. Range for A cm. Range for B cm. Median for A 3.3 cm. Median for B 2.4 cm. Interquartile range for A Interquartile range for B Exercise 21D 1 Sarah measured the lengths in centimetres of the hands of some of her friends. The table shows some information about the lengths. Draw a box plot to show this information. Length of shortest hand Lower quartile Median Upper quartile Length of longest hand 15.2 cm 16.7 cm 17.5 cm 18.1 cm 19.8 cm 2 Some students took a test. The table shows information about their marks. Use this information to draw a box plot. Minimum mark 13 Lower quartile 29 Interquartile range 32 Median mark 41 Range 53 3 The number of letters delivered to an office for each of 11 days are listed in order a Find i the median ii the interquartile range. b Draw a box plot for this data. 4 Ahmed recorded the heights of some of his friends. He used some of this information to draw a box plot Height (cm) 344
17 21.4 Comparing distributions CHAPTER 21 Copy and complete the table. Minimum height Lower quartile Median height Upper quartile Maximum height 5 The box plot shows some information about the number of patients seen in a day by some dentists. a Write down the lower quartile. b Work out the interquartile range Number of patients 6 Students in class 8P and class 8Q took the same test. Their results were used to draw the following box plots. Class 8P Class 8Q Mark a In which class was the student who scored the highest mark? b In which class did the students perform better in the test? You must give a reason for your answer. 7 6 boys and 6 girls each answered a number of questions. The information about the times in seconds that they took was used to draw two cumulative graphs Boys Girls Time (seconds) 345
18 CHAPTER 21 Processing, representing and interpreting data The table shows the minimum and maximum time taken by the boys and the girls to complete the questions. Girls Boys Minimum time to complete questions (seconds) Maximum time to complete questions (seconds) a Draw a box plot to represent the girls times and a box plot to represent the boys times. b Make two comparisons between the girls times and the boys times density and histograms Section 21.1 introduced histograms. All the histograms drawn in Section 21.1 had class intervals of equal width and so the bars were of equal width. Histograms can be drawn with unequal class intervals. The vertical axis is labelled density where density c lass width For example, the table gives some information about the ages of the audience at a concert. Age (x) in years x x x x x 7 6 To draw a histogram to represent this information work out the width of each class interval (the class width) divide the by the class width to find the density which gives the height of each bar. Age (x) in years Class width density c lass width x x x x x On a grid, label the horizontal axis Age (years) and the vertical axis density. Scale the horizontal axis from to 7 and the vertical axis from to 1 Draw the bars with no gaps between them. The first bar goes from to 15 and has a height of.8 The second bar goes from 15 to 25 and has a height of 6.5 and so on. 346
19 21.5 density and histograms CHAPTER density Rearranging density gives c lass width class width density For each bar the width is the class width and the height is the density. So the area of each bar gives the. Example 1 O Age (years) Use the information in the table and the histogram to complete the table. density Time (t seconds) t 4 4 t t 8 8 t t 2 O Time (seconds) Solution 1 Method 1 density The for the class interval 4 t 6 is given as 16 The width of this class interval is 2 ( 6 4) density c lass width That is, the height of this bar 8 The vertical scale can now be inserted so that the density for each bar can be read off. For each class interval, work out class width density to give the. 347
20 CHAPTER 21 Processing, representing and interpreting data 1 density Time Class (t seconds) width density t (4 3 ) 12 4 t t (2 11 ) 22 8 t (4 6 ) t (8 2 ) 16 Method 2 The for the class interval 4 t 6 is given as 16 On the grid the area of this bar is 4 cm 2 So an area of 4 cm 2 represents a of 16 An area of 1 cm 2 represents a of For the class interval t 4 the area of the bar is 3 cm 2 so the for this class interval is For the class interval 6 t 8 the area of the bar is cm 2 so the for this class interval is For the class interval 8 t 12 the area of the bar is 6 cm 2 so the for this class interval is For the class interval 12 t 2 the area of the bar is 4 cm 2 so the for this class interval is A key is sometimes used with a histogram to show how many data items an area on the histogram represents. Example 11 O Time (seconds) This histogram shows information about the distances in metres that a number of people threw a ball. Complete the table. density represents 2 people Distance (d metres) d 1 1 d 2 2 d 3 3 d d 7 O Distance (d metres) 348
21 21.5 density and histograms CHAPTER 21 Solution 11 Each square represents 2 people. To work out the for each class interval find the number of squares in each bar multiply the number of squares by 2 Exercise 21E 1 The histogram shows information about the number of hours of television some students watched one evening. Distance Number of (d metres) squares d d d d d represents 12 students density O Time (h hours) Copy and complete the table. Time (h hours) h 1 1 h 2 2 h h 4 4 h 7 2 The histogram shows information about the ages of the customers in a restaurant one evening. Copy and complete the table. Age ( x years) 2 x x x 4 4 x x 65 density Age (x years) 349
22 CHAPTER 21 Processing, representing and interpreting data 3 The table gives information about the amount of time, in minutes, taken by some passengers to check in at the airport. Draw a histogram to illustrate this information. Time (m minutes) m m m m m The table gives information about the lengths, in centimetres, of 45 leaves. Draw a histogram to illustrate this information. Length (x centimetres) 5 x x x x x The table gives information about the times, in minutes, that 6 runners took to complete a cross country race. Draw a histogram to illustrate this information. Time (x minutes) 1 x x x x x The histogram gives information about the weights of some apples. The shaded bar represents 4 apples. a Work out how many of the apples weigh 1 grams or less. b Work out how many of the apples weigh more than 11 grams. density Weight (grams) 35
23 Chapter 21 review questions CHAPTER 21 Chapter summary You should now know: that diagrams can be used to represent grouped discrete data how to draw and interpret diagrams that histograms can be used to represent continuous data how to draw and interpret histograms that polygons can be used to compare two or more sets of data how to draw polygons that the cumulative is the running total of the how to draw a cumulative graph from a cumulative table how to use a cumulative graph to find estimates for the lower quartile, median and upper quartile of a data set that a box plot is a diagram showing the minimum, maximum, lower quartile, median and upper quartile of a set of data how to use box plots to compare two or more sets of data that in a histogram the area of a bar gives the of the class interval how to work out the densities for a histogram, using density c lass width Chapter 21 review questions 1 The table gives information about the number of computer games sold by a shop each day for a month. Draw a diagram to illustrate this information. 2 The table shows the distribution of student absences for a year. On the resource sheet draw a polygon for this distribution. Absences (d days) d d d d d d Number of computer games Absences (days) (1385 June 21) 351
24 CHAPTER 21 Processing, representing and interpreting data 3 A railway company wanted to show the improvements in its train service over 3 years. This graph was drawn. Explain why this graph may be misleading. Percentage of trains on time students took an examination. The grouped table shows information about their results. Mark (x) x 1 x 2 x 3 x 4 x 5 x 6 Mark (x) Year x x x x x x 6 3 (1385 November 21) a Complete the cumulative table. b Draw a cumulative graph for your table. c Use your graph to find an estimate for the median mark. The pass mark for the examination was 22 d Use your graph to find an estimate for the number of students who passed the examination. 5 6 office workers recorded the number of words per minute they could type. 7 The grouped table gives information about the number of words per minute they 6 could type. Number of words (w) per minute w w w w w w 12 2 a Find the class interval in which the median lies. The cumulative graph for this information has been drawn on the grid. Number of words (w) per minute b Use this graph to work out an estimate for the interquartile range of the number of words per minute. c Use this graph to work out an estimate for the number of workers who could type more than 7 words per minute. (1388 January 25)
25 Chapter 21 review questions CHAPTER 21 6 Mary recorded the heights in centimetres of the girls in her class. She put the heights in order a Find i the lower quartile ii the upper quartile. b On the grid on the resource sheet draw a box plot for this data Height (cm) (1387 June 23) 7 3 students took part in a National Science quiz. The quiz was in two parts. The cumulative graph on the grid below gives information about the marks scored in Part One. The lowest mark was 5 and the highest mark was 47 3 Part One Mark a Use the copy of the grid on the resource sheet to draw a box plot using the cumulative graph for the results of Part One. Part One Marks Below there is also a box plot for the results of Part Two. Use the box plots to compare the two distributions. b Give two differences between them. Part Two Marks (1387 November 25) 353
26 CHAPTER 21 Processing, representing and interpreting data 8 The table gives information about the heights, in centimetres, of some 16 year old students. Use the table to draw a histogram. Height (h cm) 145 h h h h A teacher asked some students how much time they spent using a mobile phone one week. The histogram was drawn from this information. 3 density (students per hour) 2 1 O Time (hours) Use the histogram to complete the table. Time (t) hours t t 1 1 t t 3 3 t 5 (1388 March 25) 354
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