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# Maths A - Chapter 9

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### Maths A - Chapter 9

1. 1. 9syllabussyllabusrrefefererenceence Strand: Statistics and probability Core topic: Data collection and presentation In thisIn this chachapterpter 9A Types of data 9B Collecting data 9C Organising and displaying data using column and sector graphs 9D Graphical methods of misrepresenting data 9E Histograms and frequency polygons 9F Stem-and-leaf plots 9G Five-number summaries and boxplots Collecting and entering data MQ Maths A Yr 11 - 09 Page 325 Thursday, July 5, 2001 9:08 AM
2. 2. 326 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Introduction In Australia, we are fortunate to be able to drink tap water. This is not the case in many countries throughout the world where the residents must consume bottled water. If that were the case here, we would be concerned with the price of bottled water. Since we would require ongoing supplies, it would be wise to shop around for brands and sizes that suit our particular needs. In this chapter we investigate: 1. methods we could use to collect data 2. how we could prepare and organise the data 3. ways in which we could display the data 4. drawing conclusions from the data col- lected. 1 Distinguish between the terms qualitative and quantitative. 2 Consider the following temperatures (in degrees Celsius). 1, 3, 8, 6.5, −2, 25, 0, −1, 12 a Arrange the data in ascending order. b Give the smallest, the largest and the middle temperatures. 3 At Central High School there are 117 students in Year 8 and 62 students in Year 12. In Years 9 and 10, there are 102 and 92 students respectively, while there are 77 students in Year 11. a Arrange this data set in a logical table format. b Draw a pie/sector graph to display the data. c Represent the data as a column/bar graph. 4 Express the following in the units indicated. a 500 g for 98c (express in c/g). b \$1.10 for 1.25 L (in \$/L). c 1.25 kg for \$9.50 (in c/g). 5 Which is the better buy: a 375-mL can of drink for \$1, or a 1-L bottle of drink for \$2.50? MQ Maths A Yr 11 - 09 Page 326 Wednesday, July 4, 2001 5:49 PM
3. 3. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 327 Types of data Before we collect any data, we need to understand the various types of data we could gather. The diagram below distinguishes the types. Categorical data are observations that ﬁt some qualitative category. Data of this type do not involve numbers or measurements. If there is no order associated with the categories formed by the data, then they are termed nominal data (for example, answers to questions about a student’s hair colour or method of transport used to travel to school). When the data categories are aligned to some qualitative scale, they are termed ordinal data. A response to a question on a scale (for example, strongly disagree to strongly agree) would constitute ordinal data (that is, some order is implied). Numerical data, on the other hand, involve quantitative amounts. Discrete data are responses, observations or records that can take only certain, set values. Examples of discrete data would include, for example, the number of children in a family and the number of marks obtained in a test (even though you can be awarded half marks). Continuous data may take any value within the range of the data. Here, we often ﬁnd data arising as the result of taking measurements (for example, a person’s height, the daily temperature). Data Categorical (qualitative) Numerical (quantitative) Nominal Ordinal Discrete Continuous State whether the following pieces of data are categorical or numerical. a The value of sales recorded at each branch of a fast-food outlet b The breeds of dog that appear at a dog show THINK WRITE a The value of sales at each branch can be measured. a The value of sales are numerical data. b The breeds of dog at a show cannot be measured. b The breeds of dog are categorical data. 1WORKEDExample MQ Maths A Yr 11 - 09 Page 327 Wednesday, July 4, 2001 5:49 PM
4. 4. 328 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Types of data 1 State whether the data collected in each of the following situations would be categorical or numerical. a The number of matches in each box is counted for a large sample of boxes. b The sex of respondents to a questionnaire is recorded as either M or F. c A ﬁsheries inspector records the lengths of 40 cod. d The occurrence of hot, warm, mild and cool weather for each day in January is recorded. e The actual temperature for each day in January is recorded. f Cinema critics are asked to judge a ﬁlm by awarding it a rating from one to ﬁve stars. State whether each of the following records of numerical data is discrete or continuous. a The number of people in each car that passes through a tollgate b The mass of a baby at birth THINK WRITE a The number of people in the car must be a whole number. a Give a written answer. The data are numerical and discrete. b A baby’s mass can be measured to various degrees of accuracy. b Give a written answer. The data are numerical and continuous. 1 2 1 2 2WORKEDExample remember 1. Data can be classiﬁed as either: (a) categorical — the data are in categories, or (b) numerical — the data can be either measured or counted. 2. Numerical data can be either: (a) discrete — the data can take only certain values, usually whole numbers, or (b) continuous — the data can take any value depending on the degree of accuracy. remember 9A WORKED Example 1 MQ Maths A Yr 11 - 09 Page 328 Wednesday, July 4, 2001 5:49 PM
5. 5. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 329 2 State whether the numerical data gathered in each of the following situations are discrete or continuous. a The heights of 60 tomato plants at a plant nursery b The number of jelly beans in each of 50 packets c The time taken for each student in a class of six-year-olds to tie his or her shoelaces d The petrol consumption rate of a large sample of cars e The IQ (intelligence quotient) of each student in a class 3 For each of the following, state if the data are categorical or numerical. If numerical, state if the data are discrete or continuous. a The number of students in each class at your school b The teams people support at a football match c The brands of peanut butter sold at a supermarket d The heights of people in your class e The interest rate charged by each bank f A person’s pulse rate 4 An opinion poll was conducted. A thousand people were given the statement ‘Euthanasia should be legalised’. Each person was offered ﬁve responses: strongly agree, agree, unsure, disagree and strongly disagree. Describe the data type in this example. 5 A teacher marks her students’ work with a grade A, B, C, D, or E. Describe the data type used. 6 A teacher marks his students’ work using a mark out of 100. Describe the data type used. 7 The number of people who are using a particular bus service are counted over a 2-week period. The data formed by this survey would best be described as: A categorical data B numerical and discrete data C numerical and continuous data D quantitative data 8 The graph at right shows the number of days of each weather type for the Gold Coast in January. Describe the data in this example. 9 The graph at right shows a girl’s height each year for 10 years. Describe the data in this example. WORKED Example 2 mmultiple choiceultiple choice 0 Cool W arm H ot M ild 2 4 6 8 10 12 14 NumberofdaysinJanuary Weather 5 7 9 10 11 12 13 14 15 Age Height(cm) 100 120 140 160 180 86 MQ Maths A Yr 11 - 09 Page 329 Wednesday, July 4, 2001 5:49 PM
6. 6. 330 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Collecting data Data can be collected using a variety of techniques. Three common methods are: 1. observation 2. survey 3. experiment Observation Let us return to consider collecting data on bottled water. We could obtain data by visiting outlets selling water and by observing the prices charged. Imagine that on a visit to three stores, a variety of brands, sizes and prices of bottled water were observed. The table below indicates the variation and costs of 1.5-L bottle of water at the three stores. The prices shown are shelf prices for one particular week. Bottled water (1.5 L) Corner store Coles Woolworths Natural spring water \$1.88c Mount Franklin Australian spring water \$1.39 \$1.35 \$1.55 Frantelle Spring water \$1.89c \$1.17 \$1.99c Rain Farm Pure Australian rainwater \$1.14 \$1.09 \$1.19 Peats Ridge Springs Natural spring water \$1.99c \$1.20 First Choice Spring water \$1.09 Evian Natural spring water \$2.66 \$2.67 \$2.39 Brim-Brim Natural spring water \$2.22 Tourquay Natural spring water \$1.19 Savings Still spring water \$1.90c Coles Natural spring water \$1.29 Bells Puriﬁed natural spring water \$1.19 \$1.14 Schweppes Cool Ridge still spring water \$1.70 MQ Maths A Yr 11 - 09 Page 330 Wednesday, July 4, 2001 5:49 PM
8. 8. 332 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Surveys Collecting data by survey is the form that is most frequently used. The survey is admin- istered with the aid of a questionnaire. The degree of success of obtaining meaningful and relevant data from a questionnaire depends largely on the care taken in designing the questions. Methods used to collect data include: 1. personal interviews, where the interviewer usually asks prepared questions, then records the respondents’ replies 2. telephone interviews, where the interview is conducted over the telephone 3. self-administered questionnaires, which are usually mailed to individuals who complete the questionnaire, then return it in a pre-paid envelope or hold it for collection. Reporting your results Write a formal report of this investigation, detailing: 1. Aim 2. Method of obtaining data 3. Table summarising data 4. Your own personal weekly needs 5. Purchases which would satisfy these needs — specify preferred brands, sizes and the total cost. Explain your choice of brand and size. More data collection by observation It is possible to collect a variety of types of data by observation. The following are some suggestions you may wish to investigate. 1 Count the number of vehicles passing a particular point on a road during a given time period. (Road counters across roads typically record this information.) 2 Observe the number of people waiting at a department store for the doors to open the morning before a sale commences and on the morning on which it occurs. 3 Record the wildlife in a park. 4 Take note of the number of early morning joggers during the week compared with the numbers who jog during the weekend. 5 Record the variation in the price of petrol during the week. There are many situations where data are obtained by observation. In some situations, such data provide evidence for development of a scheme, concept or physical resource. This form of data collection is commonly used for planning, marketing and the preparation of reports. inv estigat ioninv estigat ion MQ Maths A Yr 11 - 09 Page 332 Wednesday, July 4, 2001 5:49 PM
13. 13. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 337 Let us return to consider the bottled-water observation discussed earlier. Much vari- ation is evident among the prices of the 1.5-litre bottles of water. An assumption would be that the more expensive bottles contained better-quality water. An experiment could be devised to determine whether consumers can detect differences in the quality of the water samples with a relatively simple taste test. Bottled water taste test The aim of this experiment is to discover whether people can distinguish, by taste, 1. between different samples of bottled water (Test 1) 2. tap water from bottled water (Test 2). This experiment would be best performed as a whole class activity. 1 Prepare three jugs: one containing tap water, one with a cheap variety of bottled water and the third with an expensive variety of bottled water. Provide small cups for each jug. Because the samples may look different (for example, tap water may be slightly coloured), this experiment would be best conducted with the participants blindfolded. 2 For Test 1, blindfolded students will each be given a sip of a cheap variety and a sip of an expensive variety of the bottled water. The test is to see whether the taster can identify which is the cheaper variety and which the dearer. 3 Test 2 is conducted similarly, using bottled water and tap water. 4 Prepare a result sheet (as below). a Record the taste tests. b Are any conclusions obvious at his stage? c Identify any problems with the tests. 5 Retain the results of this experiment for use later, when we will consider ways of displaying data. We must not forget that the results of this taste test could not be published as conclusive evidence of the ability of consumers generally to be able (or unable) to determine the quality of water. The results obtained by your class would need to be conﬁrmed by similar results from tests performed on many other groups. inv estigat ioninv estigat ion Student Test 1 ( or ✗) Test 2 ( or ✗) MQ Maths A Yr 11 - 09 Page 337 Wednesday, July 4, 2001 5:49 PM
14. 14. 338 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Gathering data from the World Wide Web (www) The Web is an excellent resource for data collection. We will spend some time now investigating web sites that provide interesting and relevant data for use later on. 1 A census is conducted on the Australian population every ﬁve years. The Australian Bureau of Statistics (ABS) has published the results of the 1996 census with data available for viewing on their web site. It takes some time for the data from a census to be collated and analysed, so the results of the next census (2001) may not be available until some years afterwards. Browse the site, noting the data available. Take a note of the web address for future reference. 2 The Bureau of Meteorology publishes climatic data for numerous towns in Queensland. Locate its web site and investigate the range of data displayed. Take particular note of the data recorded for the city or town closest to where you live. Record the site address for future use. 3 What is a ‘gallup poll’? This famous poll is named after its founder, the American statistician, George Gallup, who was born in 1901. Another well- known poll is the Morgan Gallup Poll which publishes statistics on a variety of topical issues throughout the world. Search the Web for information on gallup polls. Record your ﬁndings in the form of a poster and present the results of your search to the class. investigat ioninv estigat ion remember 1. Some important methods of collecting data are observations, surveys and experiments. 2. All data collection requires careful planning with regard to the variables recorded in order to maintain the quality of the data collected. 3. A survey may be conducted personally, via the telephone or it may be self- administered. Each of these methods has advantages and disadvantages. 4. Care must be exercised in constructing a questionnaire so that the responses collected are of good quality. 5. The questions may be of an open or closed format. 6. Open questions must eventually be placed in categories before proceeding to the stage of presentation. 7. The data obtained from experiments must be capable of being reproduced before generalisations can be made. 8. In all cases of data collection, the quality and reliability of the data must be examined before processing. remember MQ Maths A Yr 11 - 09 Page 338 Wednesday, July 4, 2001 5:49 PM
15. 15. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 339 Collecting data 1 Explain what you understand by the terms ‘open’ and ‘closed’ questions. Give an example of an open question. Rewrite your question in a closed format. 2 Twenty students were asked their opinions about the cause of congestion at the school’s front gate. Analyse their responses below, suggest categories into which they could be classiﬁed and identify the most commonly stated reasons for the congestion. 1. The cars shouldn’t come up the front driveway. 2. The front entrance is too small. 3. There should be another entrance. 4. The buses are the problem. 5. The students get in the way of the cars. 6. Bike riders should have a separate entrance. 7. The Senior school and the Junior school should start and ﬁnish at different times. 8. The cars block the gate. 9. The bike riders don’t know the road rules. 10. The buses should stop further down the road. 11. Too many students. 12. Parents don’t care where they park. 13. The front gates should be wider. 14. Bike riders should go out the back gate. 15. Kids block the cars. 16. Kids just sit around talking there. 17. There should be a trafﬁc control ofﬁcer there to direct the trafﬁc. 18. Students should not just sit around there. 19. The buses all arrive at the same time. 20. The road is too narrow. 3 Thirty students were asked: ‘Identify one thing in your maths course which you par- ticularly don’t like’. Classify the responses below into appropriate categories, then identify the main reasons. 1. It’s too hard. 2. There’s too much homework. 3. I can’t understand the teacher. 4. It’s boring. 5. The boys are too distracting. 6. The teacher doesn’t like girls. 7. It’s too much work. 8. We get homework every night. 9. I can’t understand it. 10. The boys show off. 11. The boys always get better marks. 12. I can’t concentrate for that length of time. 13. We do something new every lesson. 14. I don’t like working in groups. 15. The teacher expects too much. 16. Our teacher is too strict. 9B WORKED Example 3 MQ Maths A Yr 11 - 09 Page 339 Wednesday, July 4, 2001 5:49 PM
16. 16. 340 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 17. The teacher doesn’t help us enough with our problems. 18. There’s too much work to cover. 19. We’re expected to do assignments. 20. I don’t like doing presentations to the class. 21. Our class is too big. 22. The work is not interesting. 23. I just don’t understand maths. 24. I don’t like the teacher. 25. The teacher expects too much of us. 26. There’s too much work to cover. 27. We’re expected to remember too much. 28. It won’t help me later in life. 29. The teacher picks on me because I don’t understand the work. 30. The course is not relevant. 4 Identify the areas of concern in the following questions, then rewrite each so that the meaning is clear and understandable. (a) How much do you earn? (b) Do you exercise regularly? (c) Is the GST in Australia less than the VAT in England? (d) Do you generally support the causes of murderous terrorists who threaten the lives of peace-loving people? (e) Do you support the Prime Minister’s policy on wildlife preservation? (f) What is your height in inches? (g) Did you buy your sneakers for comfort and quality? (h) You don’t agree with charging more for skim milk (where they’ve taken out the cream) than for full cream milk, do you? (i) Do you agree that we should do more for our ‘diggers’ who risked their lives during the war so that we could be free? 5 Write the following open questions in closed format. (a) What is your age? (b) How much pocket money do you get each week? (c) How do you travel to school? (d) What type of destination do you prefer for a holiday? Data preparation Having considered so far the collection of data by observation, survey and experimental methods, we must now reﬂect on the techniques at our disposal for treating these data. However, before we rush into doing calculations, compiling tables and drawing graphs, we must ﬁrst carefully examine the data for anomalies. We must make a decision with regard to non-compliant responses from a respon- dent. Sometimes a decision is made just to disregard those non-compliant responses and to count the remainder of the compliant responses of the survey from the respon- dent. At other times, the whole survey from that respondent is disregarded. Alterna- tively, we could cope with the dilemma (after the survey has been administered) by providing another category in the question to absorb all those responses which do not slot into the categories given. MQ Maths A Yr 11 - 09 Page 340 Wednesday, July 4, 2001 5:49 PM
18. 18. 342 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d For simplicity, numerical data may be tabulated in groups. Displaying data The most common way of displaying data is by using a graph. Different graphs have different purposes. We will now look brieﬂy at column graphs and sector graphs, then look at histograms, stem plots and boxplots. Column graphs A column graph (or bar graph) is used when we wish to show a quantity. Categories are written on the horizontal axis and frequencies on the vertical axis. A Year 11 class was surveyed on their weekly income. The responses are shown below. \$75 \$115 \$60 \$54 \$88 \$0 \$98 \$102 \$56 \$45 \$83 \$71 \$40 \$37 \$87 \$117 \$43 \$79 \$58 \$89 \$70 \$105 \$99 \$55 Complete the table below. Income Tally Frequency 0–20 21–40 41–60 61–80 81–100 101–120 THINK WRITE Count the number of responses within each category and put a tally mark in the column. Income Tally Frequency 0–20 | 1 21–40 || 2 41–60 |||| || 7 61–80 |||| 4 81–100 |||| | 6 101–120 |||| 4 5WORKEDExample MQ Maths A Yr 11 - 09 Page 342 Wednesday, July 4, 2001 5:49 PM
19. 19. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 343 Sector graphs A sector graph (circle graph, or pie graph) is used when we want the graph to display a comparison of quantities. An angle is drawn at the centre of the circle that is the same fraction of 360° as the fraction of people making each response. The table below shows the results of the survey on favourite sports. Show this information in a column graph. Sport Frequency AFL 6 Basketball 2 Cricket 7 Netball 2 Rugby League 3 Rugby Union 1 Soccer 2 Tennis 1 THINK WRITE Draw the horizontal axis showing each sport. Draw a vertical axis to show frequencies up to 7. Draw the columns all the same width with gaps between. Use a ruler. Label the axes. Give the graph a title. 1 A FL Basketball Cricket N etball RugbyLeague RugbyUnion Soccer Tennis Sport Favourite sports of 24 students Frequency 0 1 2 3 4 5 6 7 2 3 4 5 6 6WORKEDExample MQ Maths A Yr 11 - 09 Page 343 Thursday, July 5, 2001 10:54 AM
20. 20. 344 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d These graphs can also be drawn using a spreadsheet and the charting tool. In our next investigation we shall explore how to enter data into a spreadsheet and how to display the results in a variety of graphical forms. For the table in worked example 6, draw a sector graph. THINK WRITE Calculate each angle as a fraction of 360°. AFL = × 360° Basketball = × 360° = 90° = 30° Cricket = × 360° Netball = × 360° = 105° = 30° Rugby League = × 360° = 45° Rugby Union = × 360° = 15° Soccer = × 360° Tennis = × 360° = 30° = 15° Draw the graph. Label each sector or provide a legend. 1 6 24 ------ 2 24 ------ 7 24 ------ 2 24 ------ 3 24 ------ 1 24 ------ 2 24 ------ 1 24 ------ 2 3 AFL Basketball Cricket Netball Rugby League Rugby Union Soccer Tennis Sport 7WORKEDExample MQ Maths A Yr 11 - 09 Page 344 Wednesday, July 4, 2001 5:49 PM
21. 21. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 345 Organising and displaying data using column and sector graphs 1 A class of students was asked to identify the make of car their family owned. Their responses are shown below. Holden Ford Nissan Mazda Toyota Holden Ford Holden Ford Mitsubishi Toyota Toyota Nissan Holden Holden Ford Toyota Mazda Mazda Toyota Ford Holden Holden Ford Mitsubishi Toyota Holden Ford Ford Toyota Put these results into a table. 2 The results of a spelling test done by 30 students are shown below. 6 7 6 8 4 6 6 7 5 9 5 7 8 10 5 9 7 7 7 6 4 7 8 8 7 8 6 5 9 7 Put these results into a table. 3 The marks scored on a Maths exam, out of 100, by 25 Year 11 students are shown below. 87 44 95 66 78 69 66 92 78 54 60 66 69 66 77 79 66 71 71 83 74 81 69 70 57 Copy and complete the table below. Mark Tally Frequency 40–49 50–59 60–69 70–79 80–89 90–99 remember 1. When data are collected they are usually ﬁrst organised into table form. 2. Data can be easily counted using a tally column and the gatepost method. 3. Sometimes numerical data are better organised into categories. 4. A column graph is drawn when we want to display quantities. 5. A sector graph is drawn when we want to compare quantities. remember 9C WORKED Example 4 E XCEL Spread sheet Frequency tally tables E XCEL Spread sheet Frequency tally tables DIY WORKED Example 5 MQ Maths A Yr 11 - 09 Page 345 Wednesday, July 4, 2001 5:49 PM
22. 22. 346 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 4 The data below show the number of customers that entered a shop each day in a certain month. 114 195 175 163 180 120 204 199 178 216 200 147 168 173 102 150 169 185 173 164 130 119 158 163 141 155 132 143 190 179 200 Choose suitable groupings to tabulate these data. 5 Draw a column graph to display the data from question 1. 6 Draw a sector graph to display the data from question 1. 7 Draw a column graph to display the data from question 2. 8 Draw a column graph to display the data from question 3. 9 Draw a column graph to display the data from question 4. 10 Draw a sector graph to compare the number of people in each category from question 3. For questions 1–4, state if the data are quantitative or qualitative. If they are quantitative, also state whether they are continuous or discrete. 1 Customers in a video shop vote for their favourite movie. 2 Customers in a video shop have records kept on the number of movies they hire each year. 3 The video shop keeps records of the number of times each movie has been hired. 4 The video shop keeps records of the length of each movie. The bar chart at right shows the marital status of respondents to a survey. 5 How many people responded to the survey? 6 What was the most common marital status? 7 How many people were married? 8 How many respondents were either divorced or separated? 9 How many people had been married at some time? 10 Draw a pie/sector graph of the data. EXCE L Spreadshe et Column graphs WORKED Example 6 EXCE L Spreadshe et Column graphs DIY WORKED Example 7 SkillS HEET 9.1 Work SHEET 9.1 1 Frequency Never married Married Divorced Separated Widowed 5 10 15 MQ Maths A Yr 11 - 09 Page 346 Wednesday, July 4, 2001 5:49 PM
23. 23. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 347 Spreadsheets — Displaying numerical data Worked examples 6 and 7 explained the calculations needed to display manually the same numerical data as a column graph and as a sector (pie) graph. In this activity, we look at entering the data into a spreadsheet, then exploring some of the graphical tools available. The instructions provided refer to the Excel spreadsheet. If you are using a different spreadsheet, your teacher will give you the equivalent commands. 1 Enter the heading ‘Sport’ and the categories of sport in Column A as shown above. 2 Enter the heading ‘Frequency’ and the relative frequencies in Column B as shown. • Highlight the sporting classiﬁcations and the frequencies, enter the Chart Wizard and select the Column graph option. Follow the instructions, remembering to label the axes and title the graph, to produce the column graph shown above. • Select the data again and follow through the Chart Wizard to produce the pie graph shown. • Print out a copy of your table and graphs. inv estigat ioninv estigat ion MQ Maths A Yr 11 - 09 Page 347 Wednesday, July 4, 2001 5:49 PM
25. 25. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 349 Notice that: 1. all the questions are of the closed type 2. the response categories for the questions each have a code associated with them (this enables them to be entered into a computer database or spreadsheet for analysis) 3. the ordinal data in questions 15 and 16 are coded. Let us take a small number of responses to question 15 and see how the data might be treated. 1 Open a spreadsheet, head Column A with ‘Respondent’ and enter the respondents’ names as shown below. 2 Head Column B as ‘Code’, then enter the coded responses shown beneath the heading. 3 Columns A and B represent the raw data. We will collate these records on the spreadsheet to the right of these columns. 4 Head Column D with the word ‘Code’, then enter the coded categories 1 to 5 beneath. 5 In Column E, enter the meanings of the coded categories beside their respective code (this just makes the spreadsheet and graphs more meaningful). 6 Head Column F with the word ‘Number’. In this column we are going to count the number of 1’s, 2’s etc. which occur in all the responses in Column B. The formula that enables us to do this is the COUNTIF command. Its format is =COUNTIF(range,criteria). 7 To count the number of 1’s in the range B3 to B14, the formula would be =COUNTIF(B3:B14,1). Enter this formula in Cell F3. MQ Maths A Yr 11 - 09 Page 349 Wednesday, July 4, 2001 5:49 PM
27. 27. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 351 Changing the scale on the vertical axis The following table gives the holdings of ROPE Corporation during 2001. Quarter Holdings in \$’000 000 J–M A–J J–S O–D 200 200 201 202 Secondary roads 1000 2000 3000 4000 5000 6000 7000 8000 9000 10 000 11 000 12 000 13 000 Sealed Paved Formed Unconstructed State highways Urban and sub arterials Main roads Developmental roads Declared roads as at 30 June 1979 Lengthofroad(km) Source: Dept of Mapping Surveying (1980), Queensland resources atlas, 2nd rev. ed. (Courtesy Dept of Lands) MQ Maths A Yr 11 - 09 Page 351 Wednesday, July 4, 2001 5:50 PM
28. 28. 352 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Here is one way of representing this data: But it is not very spectacular, is it? Now look at the following graph. The shareholders would be happier with this one. Omitting certain values If one chose to ignore the second quarter’s value, which shows no increase, then the graph would look even better. Foreshortening the vertical axis Look at the ﬁgures below. Notice in graph (a) that the numbers from 0 to 4000 have been omitted. In graph (b) these numbers have been inserted. The rate of growth of the Queensland Police Force looks far less spectacular in graph (b) than in graph (a). Foreshortening the vertical axis is a very common procedure. It does have the advantage of giving extra detail but it can give the wrong impression about growth rates. 100 200 300 Holdings(\$’000000) Quarter x x x x J-M A-J J-S O-D 200.5 201 201.5 Holdings(\$’000000) Quarter x x x x J-M A-J J-S O-D 200 202 200.5 201 201.5 Holdings(\$’000000) Quarter x x x J-M J-S O-D 200 202 4500 5000 5500 Qldpolicestrength 79 4000 1977 81 83 85 87 x x x x x x x x x x x 1000 2000 3000 Qldpolicestrength 791977 81 83 85 87 x x x x x x x x (a) (b) 4000 5000 x x x Source: Qld Year Book, 1989, p.53 and The Australian Bureau of Statistics. YearYear MQ Maths A Yr 11 - 09 Page 352 Wednesday, July 4, 2001 5:50 PM
29. 29. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 353 Visual impression In this graph, height is the property that gives the true relation, yet the impression of a much greater increase is given by the volume of each money bag. A non-linear scale on an axis or on both axes Consider the following two graphs. Both of these graphs show the same numerical information. But graph (a) has a linear scale on the vertical axis and graph (b) does not. Graph (a) emphasises the ever- increasing rate of growth of pollutants while graph (b) suggests a slower, linear growth. 400 300 200 100 \$m 1990 1995 2000 S S Year Net value of production S 100 200 300 400 500 95 96 97 98 99 Particles/unitarea Year (a) 100 200 300 95 96 97 98 99 Particles/unitarea Year 500 (b) The following data give wages and proﬁts for a certain company. All ﬁgures are in millions of dollars. Continued over page Year 1985 1990 1995 2000 Wages % increase in wages Proﬁts % increase in proﬁts 6 25 1 20 9 50 1·5 50 13 44 2·5 66 20 54 5 100 8WORKEDExample MQ Maths A Yr 11 - 09 Page 353 Wednesday, July 4, 2001 5:50 PM
30. 30. 354 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Now consider the graphs: Consider the following questions. a Do the graphs accurately reﬂect the data? b Which graph would you rather have published if you were: ii an employer dealing with employees requesting pay increases? ii an employee negotiating with an employer for a pay increase? THINK WRITE a Look at the scales on both axes. All scales are linear. a Graphs do represent data accurately. How- ever, quite a different picture of wage and proﬁt increases is painted by graphing with different units on the y-axis. Look at the units on both axes. Graph (a) has y-axis in \$ while graph (b) has y-axis in %. b ii Compare wage increase with proﬁt increases. i The employer wants high proﬁts. b ii The employer would prefer graph (a) because he/she could argue that employees’ wages were increasing at a greater rate than proﬁts. ii Consider again the increases in wages and proﬁts. The employee doesn’t like to see proﬁts increasing at a much greater rate than wages. ii The employee would choose graph (b), arguing that proﬁts were increasing at a great rate while wage increases clearly lagged behind. 4 8 12 16 20 1985 1990 1995 2000 Wagesandprofits(\$m) Year 2 6 10 14 18 Wages Profits (a) 25 50 75 100 1985 1990 1995 2000 Wagesandprofits(%increase) Year Profits Wages (b) 1 2 1 2 1 2 remember To determine whether data in graphical form have been misrepresented, check that: 1. scale on both axes is linear 2. scales on vertical and horizontal axes have not been lengthened or shortened to give a biased impression 3. certain values have not been omitted in the graph 4. picture graphs are drawn to represent a height and not a volume remember MQ Maths A Yr 11 - 09 Page 354 Wednesday, July 4, 2001 5:50 PM
31. 31. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 355 Graphical methods of misrepresenting data 1 This graph shows the dollars spent on health care for 1990, 1994, and 1998. Draw another bar graph that minimises the fall in health care funds. 2 Examine this graph of employment growth. Why is this graph misleading? 3 Examine this graph. a Redraw this graph with the vertical axis showing road fatalities starting at 0. b Does the decrease in road fatalities appear to be as signiﬁcant as the graph suggests? 9D WORKED Example 8 101 102 103 104 1990 1994 1998 Healthcarecost(\$m) 100 105 0 1 2 3 1947 1954 1961 1966 Growth of total employment, 1947–1981 1971 1976 1981 4 5 6 Totalemployment (millions) 400 450 500 550 600 650 1987 1984 1981 1978 1975 1972 1969 Road fatalities, Queensland Source: Qld Year Book, 1989, p. 205 and the Australian Bureau of Statistics. MQ Maths A Yr 11 - 09 Page 355 Wednesday, July 4, 2001 5:50 PM
32. 32. 356 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 4 This graph shows the student-to-teacher ratio in Queensland for the years 1979 to 1987. Apart from the fact that the 0 on the vertical scale has not been shown, this graph is misleading. In 1981 what was the student-to-teacher ratio at: a non-government schools? b government schools? How can you explain this when most classes in the city government schools are 25 or more? Explain fully. 5 You run a company that is listed on the Stock Exchange. During 2002 you have given substantial rises in salary to all your staff. However, proﬁts have not been as spectacular as in 2001. The following table gives the ﬁgures for the mean salary and proﬁts for each quarter. Draw 2 graphs, one showing proﬁts, the other showing salaries, that will show you in the best possible light to your shareholders. 6 You are a manufacturer and your plant is discharging heavy metals into a waterway. Your own chemists do tests every 3 months and the following table gives the results for a period of 2 years. Draw a graph which will show your company in the best light. 1st quarter 2nd quarter 3rd quarter 4th quarter Proﬁts \$’000 000 Salaries \$’000 000 6 4 5·9 5·9 6 6 6·5 7·5 2000 2001 Date Concentration (parts per million) Jan. 7 Apr. 9 July 18 Oct. 25 Jan. 30 Apr. 40 July 49 Oct. 57 15 17 19 21 1987 1985 1983 1981 Student to teacher ratio, Queensland 1979 Rate Non-government Government (a) (a) Break in continuity of series Source: Qld Year Book, 1989, p. 125 and The Australian Bureau of Statistics. MQ Maths A Yr 11 - 09 Page 356 Wednesday, July 4, 2001 5:50 PM
33. 33. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 357 7 This pie graph shows the break-up of workers compensation costs incurred by employers other than the government. a What fraction of the total costs are weekly compensation payouts and statutory lump sum claims? b What angle should be at the centre of this sector? c What angle is at the centre of this sector? d Why has this distortion of angle occurred? Discuss how this might be used to mis- lead the reader. 8 This graph shows how the \$27 that a buyer pays for a CD is distributed among the departments involved in its production and marketing. You are required to ﬁnd out whether or not the graph is misleading, to explain fully your reasoning, and to support any statements that you make. Also, a comment on the shape of the graph and how it could be obtained. b Does your visual impression of the graph support the ﬁgures? The community is constantly bombarded with graphical representations of data. We must be aware that these displays are carefully constructed so that a cursory glance conveys the impression intended by the presenter. On closer inspection, a different impression is often revealed. It is wise to look at the detail in a graph and not to rely simply on the overall visual impact. Break-up of non-government workers compensation costs Common law claims \$143.5m Common law fees and outlays \$19m Total \$202.8m Source: Courier-Mail, 21 September 1991. Weekly compensation payouts and statutory lump sum claims \$40.3m Where your \$27 goes Mechanical royalties \$1.57 Record company profit \$1.54 Advertising \$1.94 Sales tax \$3.27 Production \$3.40 Other recording costs 65c Distribution 56c Record shop \$7.40 Record company administration costs \$1.54 Royalties and costs to artist \$3.86 Record company sales process \$1.27 MQ Maths A Yr 11 - 09 Page 357 Wednesday, July 4, 2001 5:50 PM
35. 35. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 359 Other forms of graphical display We shall now consider three other forms of graphical display of data: histograms, stem plots and boxplots. Displaying data using frequency histograms A frequency histogram is similar to a column graph with the following essential features. 1. Gaps are never left between the columns, except for a half-unit space before the ﬁrst column. 2. If the chart is coloured or shaded then it is done all in one colour. (The columns are essentially all representing different levels of the same thing.) 3. Frequency is always plotted on the vertical axis. 4. For ungrouped data the horizontal scale is marked so that the data labels appear under the centre of each column. For grouped data the horizontal scale is marked so that the class centre of each class appears under the centre of the column. 1 Enter the data as indicated in the spreadsheet above. 2 Graph the data using the Chart Wizard. You should obtain a graph similar to Graph 1. 3 Copy and paste the graph twice within the spreadsheet. 4 Graph 2 gives the impression that the wages are a great deal higher than the proﬁts. This effect was obtained by reducing the length of the horizontal axis. Experiment with shortening the horizontal length and lengthening the vertical axis. 5 In Graph 3 we get the impression that the wages and proﬁts are not very different. This effect was obtained by lengthening the horizontal axis and shortening the vertical axis. Experiment with various combinations. 6 Print out your three graphs and examine their differences. Note that all three graphs have been drawn from the same data using valid scales. A cursory glance leaves us with three different impressions. Clearly, it is important to look carefully at the scales on the axes of graphs. Another method which could be used to change the shape of a graph is to change the scale of the axes. 7 Right click on the axis value, enter the Format axis option, click on the Scale tab, then experiment with changing the scale values on both axes. Techniques such as these are used to create different visual impressions of the same data. 8 Use the data in the table of worked example 8 to create a spreadsheet, then produce two graphs depicting the percentage increase in both wages and proﬁts over the years giving the impression that: a the proﬁts of the company have not grown at the expense of wage increases (the percentage increase in wages is similar to the percentage increase in proﬁts) b the company appears to be exploiting its employees (the percentage increase in proﬁts is greater than that for wages). MQ Maths A Yr 11 - 09 Page 359 Wednesday, July 4, 2001 5:50 PM
36. 36. 360 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d A frequency polygon is a line graph that can be drawn by joining the centres of the tops of each column of the histogram. The polygon starts and ﬁnishes on the horizontal axis a half column width space from the group boundary of the ﬁrst and last groups. The ﬁgure at right shows the frequency polygon drawn on top of the histogram for the previous worked example. It is common practice to draw the histogram and the polygon on the same set of axes. The table below shows the number of people living in each house in a street. Show this information in a frequency histogram. No. of people Frequency 1 1 2 4 3 10 4 15 5 8 THINK WRITE Draw a set of axes with the number of people living in a house on the horizontal axis and frequency on the vertical axis. Draw the graph, leaving half a column width space before the ﬁrst column. 1 1 0 2 4 6 8 10 12 14 16 2 3 4 5 Number of people in a house Frequency 2 9WORKEDExample 1 0 2 4 6 8 10 12 14 16 2 3 4 5 Number of people in a house Frequency The frequency table below shows a class set of marks on an exam. Draw a frequency histogram and polygon on the same set of axes. Mark Class centre Frequency 51–60 55.5 3 61–70 65.5 5 71–80 75.5 12 81–90 85.5 7 91–100 95.5 3 10WORKEDExample MQ Maths A Yr 11 - 09 Page 360 Wednesday, July 4, 2001 5:50 PM
37. 37. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 361 A graphics calculator can also be used to draw histograms. The instructions given below are for the Texas Instruments TI 83 graphics calculator. If you have another var- iety, your teacher will show you the equivalent commands. THINK WRITE Draw a set of axes with the exam mark on the horizontal axis and frequency on the vertical axis. Show the class centres for the exam marks. Draw the columns, leaving a half-column-width space before the ﬁrst column. Draw a line graph to the centre of each column. Make sure the line graph begins and ends on the horizontal axis. 1 55.5 0 2 4 6 8 10 12 65.5 75.5 85.5 95.5 Exam mark Frequency 2 3 4 The marks out of 20 received by 30 students for a book-review assignment are given in the frequency table below. Display these data on a histogram using a graphics calculator. Mark 12 13 14 15 16 17 18 19 20 Frequency 2 7 6 5 4 2 3 0 1 THINK DISPLAY Enter the data. (a) Clear any previous equations. (b) Press and clear any functions. (c) Press , select 1:Edit and press . (d) Enter the marks in L1 and the frequency in L2. Set up the calculator for graphing. (a) Press [STAT PLOT] and select 1:Plot1. Press . (b) Select On and press . (c) Select the type of graph required. The histogram is the third along on the top row. (d) At Xlist type in L1. (e) At Freq type in L2. (f) Press and highlight 9:Zoom Stat; press . (g) If not all of the histogram is shown, press and reset the x- and y-range and step values. 1 Y= STAT ENTER 2 2nd ENTER ENTER ZOOM ENTER WINDOW 11WORKEDExample MQ Maths A Yr 11 - 09 Page 361 Wednesday, July 4, 2001 5:50 PM
38. 38. 362 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Histograms and frequency polygons This exercise can be completed manually, or with the aid of a graphics calculator. 1 A survey is done on young drivers taking the written test for their licence. The number of mistakes each makes is recorded and the results are shown in the frequency distribution table at right. Show this information in a frequency histogram. 2 Students in a class were asked the number of children in their families. The results are shown in the frequency distribution table at right. Show this information in a frequency histogram and polygon. 3 The table below shows the age in years of the members of a surf club. Show this information in a frequency polygon. remember 1. Numerical data can be graphed using histograms and polygons. 2. When drawing histograms always put frequency on the vertical axis and never leave gaps between columns. 3. If the histogram is illustrating ungrouped data, the data labels on the horizontal axis are placed under the centre of each column. 4. If the histogram is illustrating grouped data, the data labels on the horizontal axis (that is, the class centres) are placed under the centre of each column. 5. A frequency polygon is a line graph, which can be drawn by joining the centres of the tops of each column of the histogram. remember 9E GCpr ogram UV statistics WORKED Example 9 No. of mistakes (score) No. of drivers (frequency) 0 5 1 8 2 11 3 4 4 3 5 1 EXCE L Spreadshe et Histograms and frequency polygons No. of children in a family Frequency 0 3 2 5 3 8 4 4 5 2 6 1 Age No. of members 18 3 19 5 20 8 21 13 22 15 23 10 24 8 25 5 MQ Maths A Yr 11 - 09 Page 362 Wednesday, July 4, 2001 5:50 PM
39. 39. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 363 4 The label on a box of matches states that the average contents of a box is 50 matches. Quality control surveyed 50 boxes for the number of matches and the results are shown below. 48 50 50 51 50 49 53 52 48 51 50 50 51 49 48 53 52 50 49 49 49 50 50 51 53 52 54 47 50 49 48 49 47 53 49 52 50 51 50 50 50 48 47 50 51 49 50 49 52 51 a Put this information into a frequency table. b Show the results in a frequency histogram and polygon. 5 The table below shows the length of 71 ﬁsh caught in a competition. Show this information in a frequency histogram and polygon. 6 Sixty people were involved in a psychology experiment. The following frequency table shows the times taken for the 60 people to complete a puzzle for the experiment. a Copy the frequency table and complete the class centre column. b Show the information in a frequency histogram and polygon. Length of ﬁsh (mm) Class centre Frequency 300–309 304.5 9 310–319 314.5 15 320–329 324.5 20 330–339 334.5 12 340–349 344.5 8 350–359 354.5 7 Time taken (seconds) Class centre Frequency 6 to almost 8 1 8 to almost 10 4 10 to almost 12 15 12 to almost 14 18 14 to almost 16 12 16 to almost 18 8 18 to almost 20 2 WORKED Example 10 MQ Maths A Yr 11 - 09 Page 363 Wednesday, July 4, 2001 5:50 PM
40. 40. 364 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Stem-and-leaf plots As an alternative to a frequency table, a stem-and-leaf plot may be used to group and summarise data. A stem is made using the ﬁrst part of each piece of data. The second part of each piece of data forms the leaves. Consider the case below. The following data show the mass (in kg) of 20 possums trapped, weighed then released by a wildlife researcher. 1.8 0.9 0.7 1.4 1.6 2.1 2.7 2.2 1.8 2.3 2.3 1.5 1.1 2.2 3.0 2.5 2.7 3.2 1.9 1.7 The stem is made from the whole number part of the mass and the leaves are the decimal part. The ﬁrst piece of data was 1.8 kg. The stem of this number could be con- sidered to be 1 and the leaf 0.8. The second piece of data was 0.9. It has a stem of 0 and a leaf of 0.9. To compose the stem- and-leaf plot, rule a vertical column of stems then enter the leaf of each piece of data in a neat row beside the appropriate stem. The ﬁrst row of the stem-and-leaf plot records all data from 0.0 to 0.9. The second row records data from 1.0 to 1.9 etc. Attach a key to the plot to show the reader the meaning of each entry. It is convention to assemble the data in order of size, so this stem-and-leaf plot should be written in such a way that the numbers in each row of ‘leafs’ are in ascending order. Key: 0 | 7 = 0.7 kg When preparing a stem-and-leaf plot, it is important to try to keep the numbers in neat vertical columns because a neat plot gives the reader an idea of the distribution of scores. The plot itself looks a bit like a histogram turned on its side. Stem 0 1 2 3 Leaf 7 9 1 4 5 6 7 8 8 9 1 2 2 3 3 5 7 7 0 2 MQ Maths A Yr 11 - 09 Page 364 Wednesday, July 4, 2001 5:50 PM
41. 41. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 365 It is also useful to be able to represent data with a class size of 5. This could be done for the previous stem-and-leaf plot by choosing stems 0*, 1, 1*, 2, 2*, 3 where the class with stem 1 contains all the data from 1.0 to 1.4 and stem 1* contains the data from 1.5 to 1.9 etc. If stems are split in this way it is a good idea to include two entries in the key. The stem-and-leaf plot for the ‘possum’ data would appear as follows: Key: 1 | 1 = 1.1 kg 1* | 5 = 1.5 A stem-and-leaf plot has the following advantages over a frequency distribution table. 1. The plot itself gives a graphical representation of the spread of data. (It is rather like a histogram turned on its side.) 2. All the original data are retained, so there is no loss of accuracy when calculating statistics such as the mean and standard deviation. In a grouped frequency distribution table some generalisations are made when these values are calculated. Stem 0* 1* 1* 2* 2* 3* Leaf 7 9 1 4 5 6 7 8 8 9 1 2 2 3 3 5 7 7 0 2 The information below shows the mass, in kilograms, of twenty 16-year-old boys. 65 45 56 57 58 54 61 72 70 69 61 58 49 52 64 71 66 65 66 60 Show this information in a stem-and-leaf plot. THINK WRITE Make the ‘tens’ the stem and the ‘units’ the leaves. Write a key. Key: 5 | 6 = 56 kg Complete the plot. Note: Complete the plot with the leaves in each row in ascending order. 1 2 3 Stem 4 5 6 7 Leaf 5 9 6 7 8 4 8 2 5 1 9 1 4 6 5 6 0 2 0 1 Stem 4 5 6 7 Leaf 5 9 2 4 6 7 8 8 0 1 1 4 5 5 6 6 9 0 1 2 12WORKEDExample MQ Maths A Yr 11 - 09 Page 365 Wednesday, July 4, 2001 5:50 PM
42. 42. 366 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Since all the original data are recorded on the stem-and-leaf plot and are conveniently arranged in order of size, the plot can be used to locate the upper and lower quartiles and the median. Notes 1. the median is the middle score or the average of the two middle scores 2. the lower quartile is the median of the lower half of the data 3. the upper quartile is the median of the upper half of the data. Using the ‘possum’ mass data as an example: Key: 0 | 7 = 0.7 kg Stem 0 1 2 3 Leaf 7 9 1 4 5 6 7 8 8 9 1 2 2 3 3 5 7 7 0 2 The following data give the length of gestation in days for 24 mothers. Prepare a stem-and-leaf plot of the data using a class size of 5. 280 287 285 276 266 292 288 273 295 279 284 271 292 288 279 281 270 278 281 292 268 282 275 281 THINK WRITE A group size of 5 is required. The smallest piece of data is 266 and the largest is 295 so make the stems: 26*, 27, 27*, 28, 28*, 29, 29*. The key should give a clear indication of the meaning of each entry. Enter the data piece by piece. Enter the leaves in pencil at ﬁrst so that they can be rearranged into order of size. Check that 24 pieces of data have been entered. Now arrange the leaves in order of size. Key: 26* | 6 = 266 days 27 | 0 = 270 days 1 2 3 Stem 26* 27* 27* 28* 28* 29* 29* Leaf 6 8 0 1 3 5 6 8 9 9 0 1 1 1 2 4 5 7 8 8 2 2 2 5 13WORKEDExample MQ Maths A Yr 11 - 09 Page 366 Wednesday, July 4, 2001 5:50 PM
43. 43. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 367 There were 20 records so the median is the average of the 10th and 11th scores. Counting each score as it appeared in the stem-and-leaf plot we can see that the 10th score is the number 1.9 and the 11th score is the number 2.1. Median = = 2.0 kg The median divides the data into halves. The lower quartile is the median of the lower half which has ten scores in it. So the position of the lower quartile is given by the average of the 5th and 6th scores. The 5th score is the number 1.5. The 6th score is the number 1.6. The lower quartile = = 1.55 kg The upper quartile is the median of the upper half which also has ten scores in it. The 5th score in this half is the number 2.3. The 6th score is the number 2.5. The upper quartile = = 2.4 kg The interquartile range is the difference between the upper and lower quartiles. 1.9 2.1+ 2 --------------------- 1.5 1.6+ 2 --------------------- 2.3 2.5+ 2 --------------------- Find the interquartile range of the data presented in the following stem-and-leaf plot. Key: 15 | 7 = 157 kg Stem 15 16 17 18 19 20 Leaf 4 8 8 1 3 3 6 8 0 0 1 4 7 9 9 9 1 2 3 3 5 7 8 8 9 2 7 8 0 2 THINK WRITE There are 30 scores and so the median will be the average of the 15th and 16th scores. Median = Median = 179 kg There are 15 scores in each half and so the lower and upper quartiles will be the 8th score in each half. The lower quartile = 168 kg The upper quartile = 188 kg The interquartile range is the difference between the upper and lower quartiles. Note: Remember to provide appropriate units for the answer. Interquartile range = upper quartile − lower quartile = 188 − 168 = 20 kg 1 179 179+ 2 ------------------------ 2 3 14WORKEDExample MQ Maths A Yr 11 - 09 Page 367 Wednesday, July 4, 2001 5:50 PM
44. 44. 368 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Stem-and-leaf plots 1 The data below give the number of errors made each week by 20 machine operators. Prepare a stem-and-leaf diagram of the data using a stem of 0, 1, 2, etc. 6 15 20 25 28 18 32 43 52 27 17 26 38 31 26 29 32 46 13 20 2 The data below give the time taken (in minutes) for each of 40 runners on a 10 km fun run. Prepare a stem-and-leaf diagram for the data using a class size of 10 minutes. 36 42 52 38 47 59 72 68 57 82 66 75 45 42 55 38 42 46 48 39 42 58 40 41 47 53 68 43 39 48 71 42 50 46 40 52 37 54 48 52 3 The typing speed (in words per minute) of 30 word processors is recorded below. Prepare a stem-and-leaf diagram of the data using a class size of 5. 96 102 92 96 95 102 95 115 110 108 88 86 107 111 107 108 103 121 107 96 124 95 98 102 108 112 120 99 121 130 4 Twenty transistors are tested by applying increasing voltage until they are destroyed. The maximum voltage that each could withstand is recorded below. Prepare a stem- and-leaf plot of the data using a class size of 0.5. 14.8 15.2 13.8 14.0 14.8 15.7 15.5 15.6 14.7 14.3 14.6 15.2 15.9 15.1 14.3 14.6 13.9 14.7 14.5 14.2 5 The stem-and-leaf plot at right gives the exact mass of 24 packets of biscuits. Find the interquartile range of the data. remember When presenting stem-and-leaf plots, observe the following points. 1. Always include a key to assist in the interpretation of the plot. 2. Choose a suitable class size. A class size of 5 is possible by using *notation on class stems. 3. After initially recording each score, rearrange the leaves so that they appear in ascending order. remember 9F WORKED Example 12 WORKED Example 13 WORKED Example 14 Key: 248 | 4 = 248.4 g Stem 248 249 250 251 252 253 Leaf 4 7 8 2 3 6 6 0 0 1 1 6 9 9 1 5 5 5 6 7 1 5 8 0 MQ Maths A Yr 11 - 09 Page 368 Wednesday, July 4, 2001 5:50 PM
45. 45. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 369 6 The time taken (in seconds) for a test vehicle to accelerate from 0 to 100 km/h is recorded during a test of 24 trials. The results are represented by the stem-and-leaf plot at right. a Find the median of the data. b Find the upper and lower quartiles of the data. c Find the interquartile range of the data. Questions 7 to 10 refer to the stem-and-leaf plot below. Key: 12 | 1 = 1210 Key: 12* | 5 = 1250 7 The class size used in the stem-and-leaf plot is: 8 The number of scores that have been recorded is: 9 The median of the data is: 10 The interquartile range of the data is: 11 The maximum hand spans (in cm) of 20 male concert pianists are recorded as follows. 23.6 20.2 22.8 21.4 25.1 24.8 23.2 21.6 20.7 23.6 22.8 24.6 21.8 22.8 23.1 24.6 21.7 24.7 22.2 23.0 a Complete a stem-and-leaf plot to represent the data. b Find the median of the data. c Find the upper and lower quartiles of the data. d Find the interquartile range of the data. Stem 12* 12* 13* 13* 14* 14* Leaf 1 2 4 5 7 7 9 9 0 1 1 2 3 4 4 5 6 6 7 9 9 0 2 3 4 6 7 A 1 B 10 C 33 D 50 A 27 B 33 C 1210 D 1410 A 13.4 B 14 C 1335 D 1340 A 14 B 100 C 1290 D 1390 Key: 7 | 2 = 7.2 s Key: 7* | 6 = 7.6 s Stem 7* 7* 8* 8* 9* 9* Leaf 2 4 4 5 5 7 9 0 0 1 2 4 4 4 5 5 6 8 9 2 2 3 5 7 mmultiple choiceultiple choice mmultiple choiceultiple choice mmultiple choiceultiple choice mmultiple choiceultiple choice MQ Maths A Yr 11 - 09 Page 369 Wednesday, July 4, 2001 5:50 PM
46. 46. 370 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 12 The heights (in cm) of a sample of 30 plants are recorded as follows. 93 88 94 99 91 85 126 107 110 111 98 96 117 101 97 92 101 132 103 82 114 84 96 103 108 115 90 110 126 85 a Complete a stem-and-leaf plot to represent the data. b Find the median of the data. c Find the upper and lower quartiles of the data. d Find the interquartile range of the data. Boxplots (box-and-whisker plots) In drawing a boxplot, ﬁve values are extracted from the data set. These values sum- marise the data and hence attract the name of a ﬁve-number summary. This ﬁve-number summary consists of: • lower extreme — the lowest score in the data set • lower quartile — the middle score of the lower half of the data set • median — the middle score • upper quartile — the middle score of the upper half of the data set • upper extreme — the highest score in the data set. Once a ﬁve-number summary has been developed, it can be graphed using a box-and- whisker plot, a powerful way to display the spread of the data. The box-and-whisker plot consists of a central divided box with attached whiskers. The box spans the interquartile range, the vertical line inside the box marks the median and the whiskers indicate the range. Each section of the boxplot represents one quarter of the scores of the data set. For the set of scores below, develop a ﬁve-number summary. 12 15 46 9 36 85 73 29 64 50 THINK WRITE Re-write the list in ascending order. 9 12 15 29 36 46 50 64 73 85 Write the lowest score. Lower extreme = 9 Calculate the lower quartile. Lower quartile = 15 Calculate the median. Median = = 41 Calculate the upper quartile. Upper quartile = 64 Write the upper extreme. Upper extreme = 85 Five-number summary = 9, 15, 41, 64, 85 1 2 3 4 36 46+ 2 ------------------ 5 6 15WORKEDExample Lower extreme Upper extreme Lower quartile Upper quartile Median of scores 1– 4 of scores 1– 4 of scores 1– 4 of scores 1– 4 MQ Maths A Yr 11 - 09 Page 370 Wednesday, July 4, 2001 5:50 PM
47. 47. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 371 Box-and-whisker plots are always drawn to scale. This can be drawn with the ﬁve- number summary attached as labels: or with a scale presented alongside the box-and-whisker plot. (This representation is preferable.) 4 28 15 21 23 0 5 10 15 20 25 30 Scale The box-and-whisker plot drawn at right shows the marks achieved by students on their end of year exam. a State the median. b Find the interquartile range. c What was the highest mark in the class? THINK WRITE a The mark in the box shows the median (72). a Median = 72 marks b The lower end of the box shows the lower quartile (63). b Lower quartile = 63 marks The upper end of the box shows the upper quartile (77). Upper quartile = 77 marks Subtract the lower quartile from the upper quartile. Interquartile range = 77 − 63 = 14 marks c The top end of the whisker gives the top mark (92). c Top mark = 92 marks 0 10 20 30 40 50 60 Marks achieved 70 80 90 100 1 2 3 16WORKEDExample After analysing the speed (in km/h) of motorists through a particular intersection, the following ﬁve-number summary was developed. The lowest score is 82 km/h. The lower quartile is 84 km/h. The median is 89 km/h. The upper quartile is 95 km/h. The highest score is 114 km/h. Show this information in a box-and-whisker plot. Continued over page 17WORKEDExample MQ Maths A Yr 11 - 09 Page 371 Wednesday, July 4, 2001 5:50 PM
48. 48. 372 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d A graphics calculator can also be used to display data in the form of a boxplot. THINK WRITE Draw a scale from 70 to 120 using 1 cm = 10 km/h. Draw the box from 84 to 95. Mark the median at 89. Draw the whiskers to 82 and 114. 1 70 80 90 Speed in km/h 100 110 120 2 3 4 Use a graphics calculator to draw a boxplot displaying the data below. 3 6 4 8 17 12 9 7 13 13 5 9 7 2 1 7 5 4 2 THINK DISPLAY Clear any previous functions. (a) Press . (b) Press to clear any functions. Enter data. (a) Press [STAT PLOT]. (b) Highlight 4:Plots Off. (c) Press . (d) Enter the list in L1 (press , select 1:Edit... and press to access the screen). Draw the boxplot. (a) Press [STAT PLOT] then select 1:Plot1. (b) Press . (c) Select Plot1, then On, then select the 5th icon along which indicates a boxplot; then at Xlist: enter L1 (use [L1]), and at Freq: enter 1. (d) Press and select 9:ZoomStat. (e) Press and the boxplot will appear. (f) Use the trace key to locate the minimum, maximum and quartile values. 1 Y= CLEAR 1 4 7 9 17 2 2nd ENTER STAT ENTER 3 2nd ENTER 2nd ZOOM ENTER 18WORKEDExample MQ Maths A Yr 11 - 09 Page 372 Wednesday, July 4, 2001 5:50 PM
49. 49. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 373 Exploring boxplots Boxplots provide a powerful tool for comparing sets of data. It is essential, therefore, to understand and interpret correctly the shape of a boxplot. Situation 1 Consider the following situation. Ten randomly chosen students from Class X and Class Y each sit for a test where the highest possible mark is 10. The results of the ten students from the two classes are: Class X: 1 2 3 4 5 6 7 8 9 10 Class Y: 1 2 2 3 3 4 4 5 9 10 Note that the range of the marks is the same in both class samples (1 to 10). In fact, the two highest and two lowest marks are the same in both cases. Determining the ﬁve-number summaries for both class samples shows: Class X 1, 3, 5.5, 8, 10 Class Y 1, 2, 3.5, 5, 10 You should conﬁrm these ﬁgures. If a boxplot was drawn for each set of data and the two plots were shown on the same scale, we would observe the following: In interpreting the differences between the two plots, we must recall the following. 1. The boxplot is divided into four sections and each section represents one quarter of the scores. 2. The median represents a score in the middle of the data. 3. A long ‘box’ or a long ‘whisker’ does not tell us that there are more scores in that section. There is still one quarter of the scores in each whisker and one half of the scores in the whole box. An increase in length indicates that the scores are more spread out. From the boxplots, answer the following questions: 1 The scores in the lower whisker of the Class X sample range from 1 to 3. What is the range of scores for the lower whisker of the Class Y sample? How many scores lie in each of these whiskers? 2 What is the range of scores in the upper whisker of the Class Y plot? How many scores lie in this area? Why is the upper whisker longer than the lower whisker? 3 It is true to say that three-quarters of the Class Y student sample performed more poorly than half the student sample in Class X. Explain how the graph shows this. 4 It is also true to say that more than half the student sample from Class X performed as well as the top quarter of the student sample from Class Y. Explain how the graph indicates this. inv estigat ioninv estigat ion 1 2 3 4 5 6 7 8 9 10 Class Y Class X MQ Maths A Yr 11 - 09 Page 373 Wednesday, July 4, 2001 5:50 PM
50. 50. 374 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d Five-number summaries and boxplots A graphics calculator can be used for many of the following. 1 Write a ﬁve-number summary for the data set below. 15 17 16 8 25 18 20 15 17 14 2 For each of the data sets below, write a ﬁve-number summary. a 23 45 92 80 84 83 43 83 b 2 6 4 2 5 7 1 c 60 75 29 38 69 63 45 20 29 93 8 29 93 3 From the ﬁve-number summary 6, 11, 13, 16, 32 ﬁnd: a the median b the interquartile range c the range. 4 From the ﬁve-number summary 101, 119, 122, 125, 128 ﬁnd: a the median b the interquartile range c the range. Situation 2 Ten randomly chosen students from Class Z sat for the same test with the following results. Class Z: 1 2 5 5 6 7 8 8 9 10 5 Determine the ﬁve-number summary for the Class Z sample. 6 Draw parallel boxplots of the three class samples, using the same horizontal scale. 7 Write a report comparing the performances of the three classes on the test. You must refer to speciﬁc data from the boxplots to support your conclusions. remember 1. A ﬁve-number summary is a summary set for a distribution. 2. The ﬁve numbers used in a ﬁve-number summary are the lower extreme, lower quartile, median, upper quartile and upper extreme. 3. A box-and-whisker plot can be used to graph a ﬁve-number summary. 4. The box is used to show the interquartile range and the median is marked with a line in the box. The whiskers then extend to show the range of the data set. remember 9G WORKED Example 15 EXCE L Spreadshe et Interquartile range WORKED Example 16 MQ Maths A Yr 11 - 09 Page 374 Wednesday, July 4, 2001 5:50 PM
51. 51. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 375 5 A ﬁve-number summary is given below. Lower extreme = 39.2 Upper quartile = 52.3 Lower quartile = 46.5 Upper extreme = 57.8 Median = 49.0 Draw a box-and-whisker plot of the data. 6 The box-and-whisker plot at right shows the distribution of ﬁnal points scored by a football team over a season’s roster. a What was the team’s greatest points score? b What was the team’s smallest points score? c What was the team’s median points score? d What was the range of points scored? e What was the interquartile range of points scored? 7 The box-and-whisker plot at right shows the distribution of data formed by counting the number of honey bears in each of a large sample of packs. In any pack, what was: a the largest number of honey bears? b the smallest number of honey bears? c the median number of honey bears? d the range of numbers of honey bears? e the interquartile range of honey bears? Questions 8, 9 and 10 refer to the box-and-whisker plot drawn below. 8 The median of the data is: A 20 B 23 C 35 D 31 WORKED Example 17 E XCEL Spread sheet Boxplots 50 70 90 110 130 150 Points 30 35 40 45 50 55 60 Scale 5 10 15 20 25 30 Scale mmultiple choiceultiple choice MQ Maths A Yr 11 - 09 Page 375 Wednesday, July 4, 2001 5:50 PM
52. 52. 376 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 9 The interquartile range of the data is: 10 Which of the following is not true of the data represented by the box-and-whisker plot? A One-quarter of the scores is between 5 and 20. B One-half of the scores is between 20 and 25. C The lowest quarter of the data is spread over a wide range. D Most of the data are contained between the scores of 5 and 20. 11 The number of sales made each day by a salesperson is recorded over a fortnight: 25, 31, 28, 43, 37, 43, 22, 45, 48, 33 a Write a ﬁve-number summary of the data. b Draw a box-and-whisker plot of the data. 12 The data below show monthly rainfall in millimetres. a Provide a ﬁve-number summary of the data. b Draw a box-and-whisker plot of the data. A 23 B 26 C 5 D 20 to 25 Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. 10 12 21 23 39 22 15 11 22 37 45 30 Drawing conclusions from the bottled-water investigations Having considered ways of collecting, organising and displaying data, it is now appropriate that you should draw together all your results from your investigations on bottled water and form them into a report. You have collected data through: 1. observation 2. survey and 3. taste testing. 1 Consider each activity separately. For each: a deﬁne the AIM of the investigation b outline the PROCEDURE undertaken c present a table of your RESULTS (preferably on a spreadsheet) d represent your data in an appropriate GRAPHICAL FORM (preferably using a computer or graphics calculator) e summarise and draw some CONCLUSIONS from your investigation. 2 Present your whole investigation as a report to the class. mmultiple choiceultiple choice mmultiple choiceultiple choice GCpr ogram UV statistics Work SHEET 9.2 inv estigat ioninv estigat ion MQ Maths A Yr 11 - 09 Page 376 Wednesday, July 4, 2001 5:50 PM
53. 53. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 377 Classiﬁcation of data • Data can be classiﬁed as being categorical or numerical. • Categorical data are data that are non-numerical. For example, a survey of car types is not numerical. • Numerical data are data that can be either counted or measured. For example a survey of the daily temperature is numerical. • Numerical data can be either discrete or continuous. • Discrete data can take only certain values such as whole numbers. • Continuous data can take any value within a certain range. Collecting data • Data can be collected by observation, survey or experiment. • Observations can be made on categorical or numerical data. • Surveys can be conducted through personal interview, telephone interview, or self- administered questionnaire. • Questionnaires require careful construction, particularly with question construction and classiﬁcation. • Experiments must be capable of being reproduced for conclusions to be drawn. • In all cases of collecting data, the quality of the data must be investigated before any processing occurs. Organising and displaying data • Data are usually organised by ﬁrst collating the records in the form of a table using grouped or ungrouped data. • Data are generally displayed graphically. Types of graphs include column, pie, histogram, stem plots or boxplots. Misrepresenting data graphically • View graphical presentations carefully to detect signs of bias in impressions. • Check that scales on both axes are linear and not lengthened or shortened. • Check that picture graphs are not represented as volumes. Histograms and frequency polygons • Numerical data are best displayed by a frequency histogram and polygon. • A frequency histogram is a column graph that is drawn with a 0.5-unit (half column) space before the ﬁrst column and no other spaces between the columns. • A frequency polygon is drawn as a line graph from the corner of the axes to the centre of each column. Stem-and-leaf plot • A stem-and-leaf plot is like a histogram on its side. • The ﬁrst part of the data forms the stem. • The last part of the data forms the leaves. • The data in the leaves must be in ascending order. Boxplots (box-and-whisker plots) • A ﬁve-number summary of a data set is the lower extreme, lower quartile, median, upper quartile and upper extreme. • A ﬁve-number summary can be graphed using a box-and-whisker plot. • A box-and-whisker plot shows the spread of a data set on a scale. summary MQ Maths A Yr 11 - 09 Page 377 Wednesday, July 4, 2001 5:50 PM
54. 54. 378 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 1 State whether each of the following data types is categorical or numerical. a The television program that people watch at 7:00 pm b The number of pets in each household c The amount of water consumed by athletes in a marathon run d The average distance that students live from school e The mode of transport used between home and school 2 For each of the numerical data types below, determine if the data are discrete or continuous. a The dress sizes of Year 11 girls b The volume of backyard swimming pools c The amount of water used in households d The number of viewers of a particular television program e The amount of time Year 11 students spent studying 3 Identify the areas of concern with the following questions, then rewrite each so that the meaning is clear and understandable. a Do you think that the school holidays are too long at Christmas time and that they should be spread evenly over the year? b Have you never said that you don’t like chocolate? c Do you support the building of a fence around the duck pond to protect the poor defenceless ducklings from the savage dogs in the area? d Did you know that the BSSSS administer the QCS test? e You support the police in their endeavour to reduce the road toll, don’t you? 4 A survey is conducted on the number of people living in each household in a street. The results are shown below. 1 4 5 2 2 3 4 6 1 2 5 6 4 4 6 3 2 3 5 1 3 4 3 3 4 2 2 Put these results into a table. 5 A group of Year 11 students was asked to state the number of CDs that they had purchased in the last year. The results are shown below. 12 1 13 20 5 22 35 12 17 20 9 5 11 0 14 25 3 8 10 9 12 6 18 7 10 9 6 23 14 19 Put the results into a table using the categories 0–4, 5–9, 10–14 etc. 6 Draw a column and a sector graph to represent the results to question 4. 7 Draw a column and a sector graph to represent the results to question 5. CHAPTER review 9A 9A 9B 9C 9C 9C 9C MQ Maths A Yr 11 - 09 Page 378 Wednesday, July 4, 2001 5:50 PM
55. 55. C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 379 8 This table shows the number of students in each year level from Years 8 to 12. Draw two separate graphs to illustrate the following: a The principal of the school claims a high retention rate of students in Years 11 and 12 (that is, most of the students from Year 10 continue on to complete Years 11 and 12). b The parents claim that the retention rate of students in Years 11 and 12 is low (that is, a large number of students leave at the end of Year 10). 9 The table below shows the number of sales made each day over a month in a car yard. Show this information in a frequency histogram and polygon. 10 The frequency table below shows the crowds at football matches for a team over a season. 11 Display the following scores in a stem-and-leaf plot. 45 21 38 46 42 41 42 49 35 29 24 28 36 21 38 45 44 40 29 28 35 35 33 38 40 41 48 39 34 38 45 28 23 29 30 40 12 Use the stem-and-leaf plot drawn in the previous question to ﬁnd: a the range b the median c the interquartile range. Year No. of students 8 200 9 189 10 175 11 133 12 124 Number of sales Frequency 0 2 1 5 2 12 3 6 4 2 5 0 6 1 Class Class centre Frequency 5000–9999 1 10 000–14 999 5 15 000–19 999 9 20 000–24 999 3 25 000–29 999 2 30 000–34 999 2 9D 9E a Copy the frequency table and complete the class centre column. b Show the information in a frequency histogram and polygon. 9E 9F 9F MQ Maths A Yr 11 - 09 Page 379 Wednesday, July 4, 2001 5:50 PM
56. 56. 380 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d 13 For the data set below, give a ﬁve-number summary. 24 53 91 57 29 69 29 15 84 6 14 For the box-and-whisker plot drawn at right: a state the median b calculate the range c calculate the interquartile range. 15 The number of babies born each day at a hospital over a year is tabulated and the ﬁve- number summary is given below. Lower extreme = 1 Upper quartile = 16 Lower quartile = 8 Upper extreme = 18 Median = 14 Show this information in a box-and-whisker plot. The stem-and-leaf plot below refers to questions 16 to 20. It represents the number of typing errors recorded by a class of students in 1 page of typing. 16 How many students are in the class? 17 What is the median number of errors? 18 State the value of the lower quartile. 19 Determine the interquartile range. 20 Draw a boxplot of the data. The box-and-whisker plots at right refer to questions 21 to 25. They show the sales of two different brands of washing powder at a supermarket each day. 21 State the range for Brand A. 22 State the interquartile range for Brand A. 23 State the range for Brand B. 24 State the interquartile range for Brand B. 25 Describe the spread of the sales for each brand of washing powder. Key: 1 | 2 = 12 1* | 5 = 15 Stem 0* 0* 1* 1* 2* Leaf 0 1 4 6 7 8 9 0 0 1 1 2 3 3 4 5 6 8 9 3 9G 9G 0 5 10 15 20 25 30 35 40 45 50 55 60 9G 10 15 20 25 30 350 5 40 Scale Brand A Brand B 45 50 testtest CHAPTER yyourselfourself testyyourselfourself 9 MQ Maths A Yr 11 - 09 Page 380 Wednesday, July 4, 2001 5:50 PM