Chapter 8 - Statistics and Probability PDF

8 Statistics and probability Collecting all types of data Statistical calculations, tables, charts and graphs are ABS (Australian Bureau of Statistics): collects essential for interpreting all types of data and are data from samples and also runs...

8 Statistics and probability Collecting all types of data Statistical calculations, tables, charts and graphs are ABS (Australian Bureau of Statistics): collects essential for interpreting all types of data and are data from samples and also runs the Australian widely used, including by governments, hospitals, Census every five years. ABS provides the medical professions, sports clubs, farmers, government with population numbers and data insurance companies and many businesses. on people’s health, education levels, employment, and housing. There are more than one hundred sports played in Australia. Each has a national and state organisation BOM (Bureau of Meteorology): records that keeps a record of results, calculates statistical weather data and forecasts the probabilities measures such as the mean, median and mode, and of rain, storms, wind strengths, temperatures, provides progress graphs and charts. Australia has fires, floods and droughts. This is important sport websites for athletics, basketball, canoeing and information for planning by emergency kayaking, cricket, cycling, diving, football, hockey, services, air traffic control, farmers, and even netball, Paralympics, rowing, rugby, sailing, skiing, holiday-makers! snowboarding, swimming, volleyball and water polo. CSIRO (Commonwealth Scientific and Industrial The following Australian government organisations Research Organisation): provides agricultural obtain and record data and provide statistical data, such as water availability, soil types, land measures and displays. use and crop forecasts. CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  Chapter contents 8A Collecting and classifying data 8B Summarising data numerically 8C Column graphs and dot plots 8D Line graphs 8E Stem-and-leaf plots 8F Sector graphs and divided bar graphs 8G Describing probability 8H Theoretical probability in single-step experiments 8I Experimental probability in single-step experiments NSW Syllabus In this chapter, a student: develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly (MAO-WM-01) classifies and displays data using a variety of graphical representations (MA4-DAT-C-01) analyses simple datasets using measures of centre, range and shape of the data (MA4-DAT-C-02) solves problems involving the probabilities of simple chance experiments (MA4-PRO-C-01) © 2022 NSW Education Standards Authority Online resources A host of additional online resources are included as part of your Interactive Textbook, including HOTmaths content, video demonstrations of all worked examples, auto- marked quizzes and much more. CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  508 Chapter 8 Statistics and probability 8A Collecting and classifying data Learning intentions for this section: To know the meaning of the terms primary source, secondary source, census, sample and observation To be able to classify variables as numerical (discrete or continuous) or categorical To understand that different methods are suitable for collecting different types of data, based on the size and nature of the data Past, present and future learning: These concepts should have been introduced to students in Stage 3 In this section they are revised and extended This topic is revisited and extended in our books for Years 8, 9 and 10 People collect or use data almost every day. Athletes and sports teams look at performance data, customers compare prices at different stores, investors look at daily interest rates, and students compare marks with other students in their class. Companies often collect and analyse data to help produce and promote their products to customers and to make predictions about the future. A doctor records a patient’s medical data to track their Lesson starter: Collecting data recovery. Examples include temperature, which is continuous numerical data, and number of heart beats recorded in a minute which is discrete numerical data. Consider, as a class, the following questions and discuss their implications. Have you or your family ever been surveyed by a telemarketer at home? What did they want? What time did they call? Do you think that telemarketers get accurate data? Why or why not? Why do you think companies collect data this way? If you wanted information about the most popular colour of car sold in Victoria over the course of a year, how could you find out this information? KEY IDEAS In statistics, a variable is something measurable or observable that is expected to change over time or between individual observations. It can be numerical or categorical. Numerical (quantitative) data can be discrete or continuous: – Discrete numerical – data that can only be particular numerical values, e.g. the number of TV sets in a house (could be 0, 1, 2, 3 but not values in between such as 1.3125). – Continuous numerical – data that can take any value in a range. Variables such as heights, weights and temperatures are all continuous. For instance, someone could have a height of 172 cm, 172.4 cm or 172.215 cm (if it can be measured accurately). CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  8A Collecting and classifying data 509 Categorical – data that are not numerical such as colours, gender, brands of cars are all examples of categorical data. In a survey, categorical data come from answers which are given as words (e.g. ‘yellow’ or ‘female’) or ratings (e.g. 1 = dislike, 2 = neutral, 3 = like). Data can be collected from primary or secondary sources. Data from a primary source are firsthand information collected from the original source by the person or organisation needing the data, e.g. a survey an individual student conducts or to answer a question that interests them. Data from a secondary source have been collected, published and possibly summarised by someone else before we use it. Data collected from newspaper articles, textbooks or internet blogs represent secondary source data. Samples and populations When an entire population (e.g. a maths class, all the cars in a parking lot, a company, or a whole country) is surveyed, it is called a census. When a subset of the population is surveyed, it is called a sample. Samples should be randomly selected and large enough to represent the views of the overall population. When we cannot choose which members of the population to survey, and can record only those visible to us (e.g. people posting their political views on a news website), this is called an observation. BUILDING UNDERSTANDING 1 Match each word on the left to its meaning on the right. a sample A only takes on particular numbers within a range b categorical B a complete set of data c discrete numerical C a smaller group taken from the population d primary source D data grouped in categories like ‘blue’, ‘brown’, ‘green’ e continuous numerical E data collected firsthand f population F can take on any number in a range 2 Give an example of: a discrete numerical data b continuous numerical data c categorical data. CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  510 Chapter 8 Statistics and probability Example 1 Classifying variables Classify the following variables as categorical, discrete numerical or continuous numerical. a the Australian state or territory in which a baby is born b the length of a newborn baby SOLUTION EXPLANATION a categorical As the answer is the name of an Australian state or territory (a word, not a number) the data are categorical. b continuous numerical Length is a continuous measurement, so all numbers are theoretically possible. Now you try Classify the following variables as categorical, discrete numerical or continuous numerical. a the number of children a person has b the brand of shoes someone wears Example 2 Collecting data from primary and secondary sources Decide whether a primary source or a secondary source is suitable for collection of data on each of the following and suggest a method for its collection. a a coffee shop wants to know the average number of customers it has each day b a detergent manufacturer wants to know the favourite washing powder or liquid for households in Australia SOLUTION EXPLANATION a primary source by recording daily customer This information is not likely to be available from numbers other sources, so the business will need to collect the data itself, making this a primary source b secondary data source using the results from A market research agency might collect these a market research agency results using a random phone survey. Obtaining a primary source would involve conducting the survey yourself but it is unlikely that the sample will be large enough to be suitable. Now you try Decide whether a primary source or secondary source is suitable for collection of data on each of the following and suggest a method for its collection. a the maximum temperature each year in Australia for the past 100 years b the number of pets owned by everyone in a class at school CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  8A Collecting and classifying data 511 Exercise 8A FLUENCY 1, 2−3(1/2), 4 2−3(1/2), 4 2−3(1/2), 4 Example 1 1 Classify the following as categorical or numerical. a the eye colour of each student in your class b the date of the month each student was born, e.g. the 9th of a month c the weight of each student when they were born d the brands of airplanes landing at Melbourne’s international airport e the temperature of each classroom f the number of students in each classroom period one on Tuesday Example 1 2 Classify the following variables as categorical, continuous numerical or discrete numerical data. a the number of cars in each household b the weights of packages sent by Australia Post on 20 December c the highest temperature of the ocean each day d the favourite brand of chocolate of the teachers at your school e the colours of the cars in the school car park f the brands of cars in the school car park g the number of letters in different words on a page h the number of advertisements in a time period over each of the free-to-air channels i the length of time spent doing this exercise j the arrival times of planes at JFK airport k the daily pollution levels in the Burnley Tunnel on the City Link Freeway l the number of text messages sent by an individual yesterday m the times for the 100 m freestyle event at the world championships over the last 10 years n the number of Blu-ray discs someone owns o the brands of cereals available at the supermarket p marks awarded on a maths test q the star rating on a hotel or motel r the censorship rating on a movie showing at the cinema 3 Is observation or a sample or a census the most appropriate way for a student to collect data on each of the following? a the arrival times of trains at Circular Quay Station during a day b the arrival times of trains at Circular Quay Station over the year c the heights of students in your class d the heights of all Year 7 students in the school e the heights of all Year 7 students in Victoria f the number of plastic water bottles sold in a year g the religions of Australian families h the number of people living in each household in your class i the number of people living in each household in your school j the number of people living in each household in Australia CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  512 Chapter 8 Statistics and probability k the number of native Australian birds found in a suburb l the number of cars travelling past a school between 8 a.m. and 9 a.m. on a school day m the money spent at the canteen by students during a week n the ratings of TV shows Example 2 4 Identify whether a primary or secondary source is suitable for the collection of data on the following. a the number of soft drinks bought by the average Australian family in a week b the age of school leavers in far North Queensland c the number of soft drinks consumed by school age students in a day d the highest level of education by the adults in Australian households e the reading level of students in Year 7 in Australia PROBLEM-SOLVING 5, 6 5, 7−9 8−10 5 Give a reason why someone might have trouble obtaining reliable and representative data using a primary source to find the following. a the temperature of the Indian Ocean over the course of a year b the religions of Australian families c the average income of someone in India d drug use by teenagers within a school e the level of education of different cultural communities within Victoria 6 Secondary sources are already published data that are then used by another party in their own research. Why is the use of this type of data not always reliable? 7 When obtaining primary source data you can survey the population or a sample. a Explain the difference between a ‘population’ and a ‘sample’ when collecting data. b Give an example situation where you should survey a population rather than a sample. c Give an example situation where you should survey a sample rather than a population. 8 A Likert-type scale is for categorical data where items are assigned a number; for example, the answer to a question could be 1 = dislike, 2 = neutral, 3 = like. a Explain why the data collected are categorical even though the answers are given as numbers. b Give examples of a Likert-type scale for the following categorical data. You might need to reorder some of the options. i strongly disagree, somewhat disagree, somewhat agree, strongly agree ii excellent, satisfactory, poor, strong iii never, always, rarely, usually, sometimes iv strongly disagree, neutral, strongly agree, disagree, agree 9 A sample should be representative of the population it reports on. For the following surveys, describe who might be left out and how this might introduce a bias. a a telephone poll with numbers selected from a phone book b a postal questionnaire c door-to-door interviews during the weekdays d a Dolly magazine poll conducted online via social media e a Facebook survey CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  8A Collecting and classifying data 513 10 Another way to collect primary source data is by direct observation. For example, the colour of cars travelling through an intersection (categorical data) is best obtained in this way rather than through a questionnaire. a Give another example of a variable for which data could be collected by observation. b Explain how you could estimate the proportion of black cars parked at a large shopping centre car park without counting every single one. REASONING 11 12, 13 12, 14 11 Television ratings are determined by surveying a sample of the population. a Explain why a sample is taken rather than conducting a census. b What would be a limitation of the survey results if the sample included 50 people nationwide? c If a class census was taken on which (if any) television program students watched from 7.30–8.30 last night, why might the results be different to the official ratings? d Research how many people are sampled by Nielsen Television Audience Measurement in order to get an accurate idea of viewing habits and stick within practical limitations. 12 Australia’s census surveys the entire population every five years. a Why might Australia not conduct a census every year? b Over 40% of all Australians were born overseas or had at least one of their parents born overseas. How does this impact the need to be culturally sensitive when designing and undertaking a census? c The census can be filled out on a paper form or using the internet. Given that the data must be collated in a computer eventually, why does the government still allow paper forms to be used? d Why might a country like India or China conduct their national census every 10 years? 13 When conducting research on First Nations Peoples, the elders of the community are often involved. Explain why the elders are usually involved in the research process. 14 Write a sentence explaining why two different samples taken from the same population can produce different results. How can this problem be minimised? ENRICHMENT: Collecting a sample – – 15 15 a Use a random number generator on your calculator or computer to record the number of times the number 1 to 5 appears (you could even use a die by re-rolling whenever you get a 6) out of 50 trials. Record these data. i Tabulate your results. ii Compare the results of the individuals in the class. iii Explain why differences between different students might occur. b Choose a page at random from a novel or an internet page and count how many times each vowel (A, E, I, O, U) occurs. Assign each vowel the following value A = 1, E = 2, I = 3, O = 4, U = 5 and tabulate your results. i Why are the results different from those in part a? ii How might the results for the vowels vary depending on the webpage or novel chosen? CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  514 Chapter 8 Statistics and probability 8B Summarising data numerically Learning intentions for this section: To understand that numerical data can be analysed and summarised using the mean, median, mode and range To be able to find the range of a set of numerical data To be able to find the mean, median and mode of a set of numerical data Past, present and future learning: These concepts are possibly new to students This topic is revisited and extended in Chapter 8 of our Year 8 book Although sometimes it is important to see a complete set of data, either as a list of numbers or as a graph, it is often useful to summarise the data with a few numbers. For example, instead of listing the height of every Year 7 student in a school, you could summarise this by stating the median height and the difference (in cm) between the tallest and shortest people. Lesson starter: Class summary For each student in the class, find their height (in cm), their age (in years), and how many siblings they have. Which of these three sets of data would you expect to have the largest range? Which of these three sets of data would you expect to have the smallest range? What do you think is the mean height of students in the class? Can you calculate it? The median (i.e. middle) height of school students, of relevant ages, is used to determine suitable dimensions for classroom chairs and desks. KEY IDEAS The range of a set of data is given by: Range = highest number − lowest number. 1 6 7 1 5 range = 7 − 1 = 6 lowest highest Mean, median and mode are three different measures that can be used to summarise a set of data. The word average is used to refer to the mean. These are also called measures of centre or measures of central tendency. CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  8B Summarising data numerically 515 The mean of a set of data is given by: Mean = (sum of all the values) ÷ (total number of values) 1 + 6 + 7 + 1 + 5 = 20 mean = 20 ÷ 5 = 4 The median is the middle value when the values are sorted from lowest to highest. If there are two middle values, then add them together and divide by 2. 1 1 ◯ 5 6 7 1 3 4 9 10 12 1 middle median = 5 middle median = (4 + 9) = 6.5 2 The mode is the most common value. It is the value that occurs most frequently. We also say that it is the value with the highest frequency. There can be more than one mode. ◯ 1 ◯ 1 5 6 7 mode = 1 BUILDING UNDERSTANDING 1 Consider the set of numbers 1, 5, 2, 10, 3. a State the largest number. b State the smallest number. c What is the range? 2 State the range of the following sets of numbers. a 2, 10, 1, 3, 9 b 6, 8, 13, 7, 1 c 0, 6, 3, 9, 1 d 3, 10, 7, 5, 10 3 For the set of numbers 1, 5, 7, 7, 10, find the: a total of the numbers when added b mean c median d mode. These people are lined up in order of height. Whose heights are used to calculate: the range? the median? the mean? CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  516 Chapter 8 Statistics and probability Example 3 Finding the range, mean and mode Consider the ages (in years) of seven people who are surveyed in a shop: 15, 31, 12, 47, 21, 65, 12. a Find the range of values. b Find the mean of this set of data. c Find the mode of this set of data. SOLUTION EXPLANATION a range = 65 − 12 Highest number = 65, lowest number = 12 = 53 The range is the difference. b mean = 203 ÷ 7 Sum of values = 15 + 31 + 12 + 47 + 21 + 65 + 12 = 203 = 29 Number of values = 7 c mode = 12 The most common value is 12. Now you try Consider the test scores of 5 people: 25, 19, 32, 25, 29. a Find the range of values. b Find the mean test score. c Find the mode test score. Example 4 Finding the median Find the median of the following: a 7, 2, 8, 10, 9, 7, 13 b 12, 9, 15, 1, 23, 7 SOLUTION EXPLANATION a Values: 2, 7, 7, 8, 9, 10, 13 Place the numbers in ascending order. (There is just one middle value, because there is an median = 8 odd number of values.) The median is 8, because it is the middle value of the sorted list. b Values: 1, 7, 9, 12, 15, 23 Place the numbers in ascending order and circle the two middle values. (There are 9 + 12 median = _ two middle values because there is an even 2 21 _ numbers of values.) = 2 The median is formed by adding the two 1 (or 10 2 ) = 10.5 _ middle values and dividing by 2. CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  8B Summarising data numerically 517 Now you try Find the median of the following. a 7, 2, 9, 3, 5, 1, 8 b 10, 2, 5, 7, 3, 3, 1, 8 Exercise 8B FLUENCY 1–6(1/2) 3–6(1/2), 7 4–6(1/2), 7, 8 Example 3a 1 Find the range of the following. a 3, 6, 7, 10 b 2, 8, 11, 12, 15 c 4, 1, 8, 2, 9 d 3, 12, 20, 2, 4, 11 Example 3b 2 Find the mean of the following. a 4, 3, 2, 5, 6 b 2, 10, 5, 7 c 2, 2, 7, 1, 4, 2 d 9, 2, 10 Example 3 3 Consider the ages (in years) of nine people who are surveyed at a train station: 18, 37, 61, 24, 7, 74, 51, 28, 24. a Find the range of values. b Find the mean of this set of data. c Find the mode of this set of data. Example 3 4 For each of the following sets of data, calculate the: i range ii mean iii mode. a 1, 7, 1, 2, 4 b 2, 2, 10, 8, 13 c 3, 11, 11, 14, 21 d 25, 25, 20, 37, 25, 24 e 1, 22, 10, 20, 33, 10 f 55, 24, 55, 19, 15, 36 g 114, 84, 83, 81, 39, 12, 84 h 97, 31, 18, 54, 18, 63, 6 Example 4a 5 Find the median of: a 2, 5, 10, 12, 15 b 1, 7, 8, 10, 11 c 3, 1, 5, 2, 9 d 12, 5, 7, 10, 2 e 12, 18, 31, 15, 19, 10, 12 f 17, 63, 4, 13, 97, 82, 56 Example 4b 6 Find the median of: a 3, 8, 10, 14, 16, 19 b 2, 7, 8, 10, 13, 18 c 1, 5, 2, 9, 13, 17 d 5, 2, 3, 11, 7, 15 e 3, 2, 3, 1, 8, 7, 6, 9 f 4, 9, 2, 7, 8, 1, 5, 6 Example 4 7 The median for the data set 5 7 7 10 12 13 17 is 10. What would be the new median if the following number is added to the data set? a 9 b 12 c 20 d 2 CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  518 Chapter 8 Statistics and probability 8 The number of aces that a tennis player serves per match is recorded over eight matches. Match 1 2 3 4 5 6 7 8 Number of aces 11 18 11 17 19 22 23 12 a What is the mean number of aces the player serves per match? Round your answer to 1 decimal place. b What is the median number of aces the player serves per match? c What is the range of this set of data? PROBLEM-SOLVING 9, 10 10, 11 10–12 9 Brent and Ali organise their test marks for a number of topics in Maths, in a table. Test Test Test Test Test Test Test Test Test Test 1 2 3 4 5 6 7 8 9 10 Brent 58 91 91 75 96 60 94 100 96 89 Ali 90 84 82 50 76 67 68 71 85 57 a Which student has the higher mean? b Which student has the higher median? c Which student has the smaller range? d Which student do you think is better at tests? Explain why. 10 Alysha’s tennis coach records how many double faults Alysha has served per match over a number of matches. Her coach presents the results in a table. Number of double faults 0 1 2 3 4 Number of matches with this many double faults 2 3 1 4 2 a In how many matches does Alysha have no double faults? b In how many matches does Alysha have 3 double faults? c How many matches are included in the coach’s study? d What is the total number of double faults scored over the study period? e Calculate the mean of this set of data, correct to 1 decimal place. f What is the range of the data? 11 A soccer goalkeeper recorded the number of saves he makes per game during a season. He presents his records in a table. Number of saves 0 1 2 3 4 5 Number of games 4 3 0 1 2 2 a How many games did he play that season? b What is the mean number of saves this goalkeeper made per game? c What is the most common number of saves that the keeper had to make during a game? 12 The set 1, 2, 5, 5, 5, 8, 10, 12 has a mode of 5 and a mean of 6. a If a set of data has a mode of 5 (and no other modes) and a mean of 6, what is the smallest number of values the set could have? Give an example. b Is it possible to make a data set for which the mode is 5, the mean is 6 and the range is 20? Explain your answer. CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  8B Summarising data numerically 519 REASONING 13 13, 14 14–16 13 Evie surveys all the students in her class to find the distance from their homes to school. One of the students is on exchange from Canada and reports a distance of 16 658 km. Would this very large value have a greater effect on the mean or median distance? Explain your answer. 14 Consider the set of values 1, 3, 5, 10, 10, 13. a Find the mean, median, mode and range. b If each number is increased by 5, state the effect this has on the: i mean ii median iii mode iv range c If each of the original numbers is doubled, state the effect this has on the: i mean ii median iii mode iv range d Is it possible to include extra numbers and keep the same mean, median, mode and range? Try to expand this set to at least 10 numbers, but keep the same values for the mean, median, mode and range. 15 a Two whole numbers are chosen with a mean of 10 and a range of 6. What are the numbers? b Three whole numbers are chosen with a mean of 10 and a range of 2. What are the numbers? c Three whole numbers are chosen with a mean of 10 and a range of 4. Can you determine the numbers? Try to find more than one possibility. 16 Prove that for three consecutive numbers, the mean will equal the median. ENRICHMENT: Mean challenges – – 17, 18 17 a Give an example of a set of numbers with the following properties. i mean = median = mode ii mean > median > mode iii mode > median > mean iv median < mode < mean b If the range of a set of data is 1, is it still possible to find data sets for each of parts i to iv above? 1, _ 18 Find the mean and median of the fractions _ 1, _ 1, _ 1. 2 3 4 5 An important aspect of scientific investigation is collecting data and summarising it numerically. CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  520 Chapter 8 Statistics and probability 8C Column graphs and dot plots Learning intentions for this section: To understand that graphs should include titles and labelled axes, with any numerical values drawn to scale To know what an outlier is To be able to construct and interpret a column graph To be able to construct and interpret a dot plot Past, present and future learning: These concepts should have been introduced to students in Stage 3 In this section they are revised and extended This topic is revisited and extended in Chapter 8 of our Year 8 book Categorical data can be counted and presented as a Favourite colour column graph. Each column’s length indicates the frequency of that category. Column graphs can also be red useful for labelled continuous data (e.g. height of people). green Colour yellow Discrete numerical data can be counted and presented as a dot plot, with the number of dots representing the blue frequency. pink Consider a survey of students who are asked to choose 0 5 10 15 20 their favourite colour from five possibilities, as well Number of students As a column graph (horizontal) as state how many people live in their household. The colours could be shown as a column graph, and the number of people shown in a dot plot. Favourite colour 20 Number of students 15 10 5 0 d n w ue k re ee n llo bl pi gr ye 0 1 2 3 4 5 Colour Household size As a column graph (vertical) Lesson starter: Favourite colours Survey the class to determine the number of people in each student’s family and each student’s favourite colour from the possibilities red, green, yellow, blue and pink. Each student should draw a column graph and a dot plot to represent the results. What are some different ways that the results could be presented into a column graph? (There are more than 200 ways.) CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  8C Column graphs and dot plots 521 KEY IDEAS A dot plot can be used to display discrete numerical data, where each dot represents one datum. A column graph is a way to show data in different categories, and is useful when more than a few items of data are present. Column graphs can be drawn vertically or horizontally. Horizontal column graphs are sometimes also called bar graphs. Graphs should have the following features: An even scale for A title explaining what the numerical axis the graph is about Favourite colour 20 Number of students 15 10 5 0 w n nk ue d llo ee re bl pi gr ye A label on Category labels for each axis Colour any non-numerical data Any numerical axis must be drawn to scale. An outlier is a value that is noticeably distinct from the main cluster of points. main cluster an outlier 0 1 2 3 4 5 6 7 8 9 10 Data represented as a dot plot could be described as symmetrical or skewed (or neither). Negatively skewed Symmetrical Positively skewed 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 mean = median = mode mode < median < mean mean < median < mode CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  522 Chapter 8 Statistics and probability BUILDING UNDERSTANDING Height chart 1 The graph opposite shows the height of four boys. 110 100 Answer true or false to each of the following statements. 90 a Mick is 80 cm tall. 80 Height (cm) b Vince is taller than Tranh. 70 c Peter is the shortest of the four boys. 60 d Tranh is 100 cm tall. 50 40 e Mick is the tallest of the four boys. 30 20 10 0 h ce r k te an ic in Pe M Tr V Child 2 The favourite after-school activity of a number of Year 7 students is After-school activities recorded in the column graph shown opposite. 8 a How many students have chosen television as their favourite activity? 6 Enrolments b How many students have chosen social networking as their 4 favourite activity? c What is the most popular after-school activity for this group 2 of students? d How many students participated in the survey? video games social networking sport television Favourite activity Example 5 Interpreting a dot plot The dot plot on the right represents the results of a survey that Pets at home survey asked some children how many pets they have at home. a Use the graph to state how many children have 2 pets. b How many children participated in the survey? c What is the range of values? d What is the median number of pets? e What is the outlier? f What is the mode? 0 1 2 3 4 5 6 7 8 Number of pets CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  8C Column graphs and dot plots 523 SOLUTION EXPLANATION a 4 children There are 4 dots in the ‘2 pets’ category, so 4 children have 2 pets. b 22 children The total number of dots is 22. c 8−0=8 Range = highest − lowest In this case, highest = 8, lowest = 0. d 1 pet As there are 22 children, the median is the average of the 11th and 12th value. In this case, the 11th and 12th values are both 1. e the child with 8 pets The main cluster of children has between 0 and 3 pets, but the person with 8 pets is significantly outside this cluster. f 1 pet The most common number of pets is 1. Now you try The dot plot shows the number of visits a sample of pet owners Visiting the vet have made to the vet in the past year. a How many people were surveyed? b How many people made more than three visits to the vet in the past year? c What is the range? d What is the median? e What is the outlier? 0 1 2 3 4 5 6 7 f What is the mode? Number of visits Example 6 Constructing a column graph Draw a column graph to represent the following people’s heights. Name Tim Phil Jess Don Nyree Height (cm) 150 120 140 100 130 Continued on next page CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  524 Chapter 8 Statistics and probability SOLUTION EXPLANATION Height chart First decide which scale goes on the vertical 160 axis. 140 120 Maximum height = 150 cm, so axis goes from Height (cm) 100 0 cm to 160 cm (to allow a bit above the highest 80 value). 60 40 Remember to include all the features required, 20 0 including axes labels and a graph title. Tim Phil Jess Don Nyree Name Now you try Draw a column graph to represent the following people’s arm spans. Name Matt Pat Carly Kim Tristan Arm span (cm) 180 190 150 160 170 Exercise 8C FLUENCY 1–6 2–7 2, 3, 5–7 Example 5 1 The dot plot on the right represents the results of a survey that Overseas flights asked some people how many times they had flown overseas. a Use the graph to state how many people have flown overseas once. b How many people participated in the survey? c What is the range of values? d What is the median number of times flown overseas? e What is the outlier? 0 1 2 3 4 5 6 7 f What is the mode? Number of times o/s Example 5 2 In a Year 4 class, the results of a spelling quiz are Spelling quiz results presented as a dot plot. a What is the most common score in the class? b How many students participated in the quiz? c What is the range of scores achieved? d What is the median score? e Identify the outlier. 0 1 2 3 4 5 6 7 8 9 10 Score out of 10 CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  8C Column graphs and dot plots 525 3 From a choice of pink, blue, yellow, green or red, each Favourite colours in Year 7B student of Year 7B chose their favourite colour. The results are graphed on the right. pink Favourite colour a How many students chose yellow? blue b How many students chose blue? c What is the most popular colour? yellow d How many students participated in the class survey? green e Represent these results as a dot plot. red 0 2 4 6 8 10 Number of students 4 Joan has graphed her height at each of her past five birthdays. Joan’s height at different birthdays 180 160 140 120 Height (cm) 100 80 60 40 20 0 8 9 10 11 12 Joan’s age ( years) a How tall was Joan on her 9th birthday? b How much did she grow between her 8th birthday and 9th birthday? c How much did Joan grow between her 8th and 12th birthdays? d How old was Joan when she had her biggest growth spurt? Example 6 5 Draw a column graph to represent each of these students’ heights at their birthdays. a Mitchell b Fatu Age (years) Height (cm) Age (years) Height (cm) 8 120 8 125 9 125 9 132 10 135 10 140 11 140 11 147 12 145 12 150 CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  526 Chapter 8 Statistics and probability 6 Every five years, a company in the city conducts a transport survey of people’s preferred method of getting to work in the mornings. The results are graphed below. Transport methods 70 Number of employees 60 public transport 50 car 40 walk/bicycle 30 20 10 0 1990 1995 2000 2005 2010 2015 Year of survey a Copy the following table into your workbook and complete it, using the graph. 1990 1995 2000 2005 2010 2015 Use public transport 30 Drive a car 60 Walk or cycle 10 b In which year(s) is public transport the most popular option? c In which year(s) are more people walking or cycling to work than driving? d Give a reason why the number of people driving to work has decreased. e What is one other trend that you can see from looking at this graph? 7 a Draw a column graph to show the results of the following survey of the number of male and female puppies sold by a commercial dog breeder. Put time (years) on the horizontal axis. 2010 2011 2012 2013 2014 2015 Number of male 40 42 58 45 30 42 puppies born Number of female 50 40 53 41 26 35 puppies born b During which year(s) were there more female puppies sold than male puppies? c Which year had the fewest number of puppies sold? d Which year had the greatest number of puppies sold? e During the entire period of the survey, were there more male or female puppies sold? PROBLEM-SOLVING 8 8, 9 9, 10 8 The average (mean) income of adults in a particular Average income in a town town is graphed over a 6-year period. 7 Income (× $10 000) 6 a Describe in one sentence what has happened 5 to the income over this period of time. 4 b Estimate what the average income in this town 3 might have been in 2012. 2 c Estimate what the average income might be in 2028 1 if this trend continues. 0 2013 2014 2015 2016 2017 2018 Year CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  8C Column graphs and dot plots 527 9 A survey is conducted of students’ favourite subjects from a choice of Art, Maths, English, History and 30 Science. Someone has attempted to depict the results in a column graph. a What is wrong with the scale on the vertical axis? 25 b Give at least two other problems with this graph. 20 c Redraw the graph with an even scale and appropriate 5 labels. 0 ish y e hs rt or nc d The original graph makes Maths look twice as popular as A at gl ist ie M En Sc H Art, based on the column size. According to the survey, how many times more popular is Maths? e The original graph makes English look three times as popular as Maths. According to the survey, how much more popular is English? f Assume that Music is now added to the survey’s choice of subjects. Five students who had previously chosen History now choose Music, and 16 students who had previously chosen English now choose Music. What is the most popular subject now? 10 Mr Martin and Mrs Stevensson are the two Year 3 teachers Arithmetic quiz scores at a school. For the latest arithmetic quiz, they have plotted their students’ scores on a special dot plot called a parallel Class 3M dot plot, shown opposite. a What is the median score for class 3M? b What is the median score for class 3S? c State the range of scores for each class. d Based on this test, which class has a greater spread of Score 0 1 2 3 4 5 6 7 8 9 10 arithmetic abilities? e If the two classes competed in an arithmetic competition, where each class is allowed only one representative, which class is more likely to win? Justify your answer. Class 3S REASONING 11 11, 12 12, 13 11 At a central city train station, three types of services Train passengers at Urbanville Station No. of passengers run – local, country and interstate. The average 2500 number of passenger departures during each week is 2000 shown in the stacked column graph. 1500 a Approximately how many passenger departures 1000 500 per week were there in 2013? 0 b Approximately how many passenger departures 2013 2014 2015 2016 2017 2018 were there in total during 2018? Year interstate country local c Does this graph suggest that the total number of passenger departures has increased or decreased during the period 2013–2018? d Approximately how many passengers departed from this station in the period 2013–2018? Explain your method clearly and try to get your answer within 10 000 of the actual number. CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  528 Chapter 8 Statistics and probability 12 The mean value of data represented in a dot plot can be found by adding the value represented by each dot, then dividing by the number of dots. a Use this method to find the mean number of children per household, shown in this dot plot of 10 households. b A faster method is to calculate (2 × 1) + (3 × 2) + (4 × 3) + (1 × 4) 0 1 2 3 4 5 and divide by 10. Explain why this also works to find the mean. Number of children 13 Classify the following dot plots as representing symmetrical, positively skewed or negatively skewed data. Find the mean number in each case and compare it to the median. a b 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 c d 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 ENRICHMENT: How many ways? – – 14 14 As well as being able to draw a graph horizontally or vertically, the order of the categories can be changed. For instance, the following three graphs all represent the same data. Animal sightings Number of sightings Animal sightings 8 dingo 6 cougar Animal 4 antelope 2 bear 0 elephant o e nt ar ar op ng ha be ug el di ep co 0 2 4 6 8 t an el Animal Number of sightings How many different column graphs could be used to Animal sightings Number of sightings 8 represent the results of this survey? (Assume that you 6 can only change the order of the columns, and the 4 horizontal or vertical layout.) Try to list the options 2 systematically to help with your count. 0 pe o t ar ar an ng ug lo be h di te ep co an el Animal CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  8D Line graphs 529 8D Line graphs Learning intentions for this section: To understand that a line graph can be used to display continuous numerical data To be able to draw a line graph To be able to use a line graph to estimate values To be able to interpret a travel graph Past, present and future learning: Line graphs should have been introduced to students in Stage 3 In this section they are revised and extended This topic is revisited and extended in Chapter 8 of our Year 8 book A line graph is a connected set of points joined with straight line segments. The variables on both axes should be continuous numerical data. It is often used when a measurement varies over time, in which case time is conventionally listed on the horizontal axis. One advantage of a line graph over a series of disconnected points is that it can often be used to estimate values that are unknown. A business can use line graphs to display data such as expenses, sales and profits, versus time. A line graph makes it easy to visualise trends, and predictions can then be made. Lesson starter: Room temperature As an experiment, the temperature in two rooms is measured hourly over a period of time. The results are graphed below. Room A Room B 30 35 Temperature (°C) Temperature ( °C ) 25 30 20 25 15 20 10 15 5 10 5 0 1 2 3 4 Time (hours) 0 1 2 3 4 Time (hours) Each room has a heater and an air conditioner to control the temperature. At what point do you think these were switched on and off in each room? For each room, what is the approximate temperature 90 minutes after the start of the experiment? What is the proportion of time that room A is hotter than room B? CambridgeMATHS NSW Stage 4 ISBN 978-1-009-40891-2 © Palmer et al. 2024 Cambridge University Press Year 7 Photocopying is restricted under law and this material must not be transferred to another party.  530 Chapter 8 Statistics and probability KEY IDEAS A line graph consists of a series of points joined by straight line segments. Weight The variables on both axes should be continuous numerical data. Time is often shown on the horizontal axis. Time A common type of line graph is a travel graph. 60 Distance (km) Time is shown on the horizontal axis. 50 at rest 40 30 km in 3 hours Distance is shown on the vertical axis. 30 or 10 km/h The slope of the line indicates the rate at 20 which the distance is changing over time 10 30 km in 1 hour or 30 km/h This is called speed. 0 1 2 3 4 5 6 Time (hours) BUILDING UNDERSTANDING Cat’s weight over time 1 The line graph shows the weight of a cat over a 3-month period. It is weighed at the start of each month. State the cat’s weight 6 at the start of: 5 Weight (kg) a January 4 b February 3 c March 2 d April 1 0 Jan Feb Mar Apr Month Lillian’s height 2 The graph shows Lillian’s height over a 10-year period from when she was born. 160 a What was Lillian’s height when she was born? 140 Height (cm) 120 b What was Lillian’s height at the age of 7 years? 100 c At what age did she first reach 130 cm tall? 80 d How much did Lillian grow in