O Level
Statistics
Dean James Chalmers
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CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, SĂŁo Paulo, New Delhi
Cambridge University Press c/o Cambridge University Press India Pvt. Ltd. Cambridge House 4381/4, Ansari Road, Daryaganj New Delhi 110002 India
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Š Cambridge University Press, 2009
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Printed at
ISBN: 978-0-521-16954-7
paperback
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Contents Introduction
v
1. Representation of Data
1
2. Basic Probability
25
3. Sampling
31
4. Frequency Distributions
39
5. Measures of Central Tendency
67
6. Weighted Averages
88
7. Measures of Dispersion
104
8. Scaling
126
9. Probability and Expectation
132
10. Correlation and Scatter
165
11. Time Series
177
12. Linear Interpolation
194
13. Essential Tools
203
Answers to the exercises
209
Index
223
Introduction Statistics is a practical subject that concerns the collection, recording, representation and analysis of data. Look in your local or national newspaper, listen to the radio or watch television, and you will find that statistical information on many aspects of everyday life is frequently being presented to you in a variety of forms. Once data have been analysed, it is possible to predict what the future may hold in store by looking for trends; appropriate action can then be taken. The aim of this book is to serve as a basic introduction to the study of Statistics and Probability, enabling students to gain a sound knowledge and understanding of the elementary ideas, methods and terminology used in the subject. Students who are studying the Cambridge (CIE) O Level Statistics 4040 syllabus will find that this book covers the entire course. It will also be of invaluable use to those studying Statistics and/or Probability on any other syllabus at a similar or higher level. Although teachers and students are at liberty to study the topics in any order, the chapters in this book have been constructed and arranged in such a way that the entire syllabus can be covered by working through chapters 1 to 12 in sequence. Chapter 13 covers three additional topics that can be used as and when needed. The chart below indicates where each of the 11 sections of the Cambridge (CIE) O Level Statistics 4040 syllabus can be found in this book. Location of each section of the syllabus: Syllabus Section Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
1
2
3
4
5
a
6 b
c
7
8
9
10
11
•
•
• • •
•
• •
• • •
• • • • •
•
• •
• •
Note: Syllabus section 6. a Measures of central tendency b Measures of dispersion c Linear transformation of data
• •
Representation of Data 1
Chapter
1
Representation of Data Tabular Representation of Data Statistical information or data should be displayed in a clear and easy-to-understand way. One method of doing this is to enter the data into a table – data can be tabulated. The columns and rows of a table should have headings, and totals should be included where appropriate. ! Numbers and proportions (percentages, fractions and ratios) are easy to compare if the data are tabulated. Example. The table shows the results of a geography test that was taken by a group of 45 students. From the table we find that: 26 boys took the test; 2 girls failed the test; Pass Fail Total 39 students passed the test; 17 19 of the girls Boys 22 4 26 passed the test; 13 or 33.3% of those who Girls 17 2 19 Total
39
6
45
failed were girls, and the ratio of passes to failures for boys was 11:2
(a) What fraction of the students passed the test? (b) What fraction of those who passed were boys? (c) Express the number that passed to the number that failed as a simple ratio. (d) What percentage of the boys failed? Exercise 1A 1. The table below gives information about 100 students who wrote an examination in physics. (i) How many students passed? (ii) What fraction of the students failed? Pass Fail (iii) What proportion of the girls passed? Boys 45 15 (iv) Calculate (a) the percentage pass rate for boys, Girls 28 12 (b) the percentage pass rate for girls. (v) Which group performed better, boys or girls?
2 Statistics
2. A woman sells red and yellow T-shirts at a market. She has small, medium and large in each colour. There are 120 T-shirts altogether, and 75 of them are red. She has 20 small red T-shirts, 14 medium yellow T-shirts, and 48 large T-shirts. The ratio of large-red T-shirts to large-yellow T-shirts is 2:1. (i) Copy and complete this table: Small Red Yellow Total
3.
4.
5.
6.
Medium
Large
20
Total 75
14 48
120
(ii) What fraction of the T-shirts is (a) yellow, (b) medium? (iii) What proportion of the red T-shirts is large? (iv) What percentage of the small T-shirts is yellow? [Answer to 1 decimal place] A girl was selling two types of canned drinks on sports day: these were Cherry-Fizz and Orange-Wizz. She had both types of drink in small and medium bottles, and there were 15 bottles of each type of drink. She had 13 small bottles altogether, and 7 of the medium bottles contained Orange-Wizz. (i) Tabulate the data with the numbers of each of the four items. (ii) What fraction of the bottles were small Cherry-Fizzes? (iii) How many of the bottles were neither small nor contained Orange-Wizz? At lunchtime 100 students were each asked to choose one main course from either stew or hot pot. The students were also asked to choose one dessert from either ice cream or tart: 24 students chose both ice cream and hot pot. Altogether 58 students chose stew, and 44 chose tart. All students chose one main course and one dessert. (i) Illustrate the data in a fully labelled table. (ii) What fraction of the students chose both hot pot and tart? A boy has 60 discs: some are black and some are white. The discs are either plastic or wooden. Forty-percent of the discs are black, and 95 of the white discs are plastic. Of the wooden discs, there is one less black than white. (i) Draw up a table, with headings, showing the numbers of the four types of disc. (ii) What proportion of the discs is not a black plastic disc? Joan wants to tabulate data showing the numbers of students that passed or failed each of the three final papers in geography at her school in 2007 and in 2008. Altogether 322 students sat for each of the three papers in 2007, and that was 23 less than in 2008.
Representation of Data 3
The numbers passing papers 1, 2 and 3 in 2007 were 310, 303 and 305, respectively. Fifteen more students failed paper 2 in 2008 than in 2007; equal numbers passed paper 1 in both years, and three times as many students failed paper 3 in 2008 than in 2007. (i) Tabulate the data. (ii) What is the greatest possible number of students that failed all three papers in 2008? 7. A survey gave the following data on the numbers of adults and the numbers of employed adults in each of the households in a particular street. o
N adults o
N employed adults 0 1 2 3
1 6 4 -
2 3 5 5 -
3 0 2 4 1
(i) In how many households was there more than one adult? (ii) In how many households were all the adults employed? (iii) How many households had just one unemployed adult? (iv) How many employed adults were there in all these households together? (v) Explain why there are no values entered into three of the boxes in the table above. 8. The 124 employees at a company are classified by gender, employment status and skill. The diagram below shows, for example, that there are 25 skilled full-time females, and 6 unskilled part-time males. Females Males 9 Full-time
Part-time
Skilled 25
35
10
20
15
4 6 Find (i) the number of skilled employees, (ii) the number of male employees, (iii) the number of unskilled full-time female employees, (iv) what fraction of the full-time employees are skilled, (v) what percentage of the part-time employees are male, (vi) what proportion of the skilled males work part-time, (vii) what percentage of the unskilled part-timers are female.
4 Statistics
Pictorial Representation of Data A variety of diagrams can be used to show data. They impart information 'at-a-glance' and usually appeal to people more than lists or tables of numbers. The statistical diagrams that are generally used to illustrate ungrouped data include pictograms, bar charts, line graphs and pie charts.
Pictogram Data are illustrated by the use of symbols with a key to indicate what each one represents. Care must be taken when drawing symbols: if two symbols are different in shape or size, they will represent neither the same item nor the same number of items. Symbols with simple shapes are recommended, as it may sometimes be necessary to draw fractions of them.
Example. Three farmers have planted maize and sorghum on their land. The pictogram below shows the number of hectares that each farmer has used for each crop. Farmer David
Represents 10 hectares of maize.
Samuel
Bonolo
Represents 12 hectares of sorghum.
Bonolo planted 65 hectares of maize and David planted the largest area of maize. 48 = 13 of the total area used for Samuel planted 48 hectares of sorghum, which was 144 sorghum.
(a) How many fewer hectares of sorghum did David plant than Bonolo? (b) What fraction of the total area used for sorghum was planted by David? (c) What percentage of all the land used by the three farmers was planted with maize?
Representation of Data 5
Exercise 1B 1. Forty-five students were asked which sports they play. All students play at least one sport. The data were collected, and illustrated in the pictogram below. Sport
Represents two students
Softball Football Volleyball Table tennis Badminton Tennis (i) Find (a) the most popular sport, (b) the least popular sport, (c) what fraction plays volleyball, (d) what percentage plays tennis. (ii) Explain why there are more than 45 students represented in the pictogram. (iii) Express the number that plays softball to the number that does not play table tennis, as a simple ratio. 2. A farmer grows five different types of vegetable on his farm. The area used for growing each is given. (i) Draw one symbol that would be suitable for showing Vegetable Area (m2) these data in a pictogram, and state what your symbol would represent. Cabbage 32 (ii) What is the total area used for growing vegetables? Carrot 64 (iii) What proportion of the total area is used for onions? Onion 48 (iv) What percentage, to 1 decimal place, is used for Tomato 80 tomatoes? Potato 40 (v) Draw the pictogram, using the symbol chosen in (i).
Bar Chart Bars or columns of equal width are drawn to the heights of the frequencies. Equal-width gaps may be left between bars. It is preferable to leave all bars unshaded, or to shade them all in the same way; avoid using lots of bright colours or random patterns, as this may give a false impression of the relative size of the bars.
6 Statistics
Example. A child asked each of a group of adults "What is your favourite drink?" The adults’ responses were: 10 tea, 7 coffee, 5 juice, 4 milk, 3 water and 1 lemonade. o
N adults ( frequency) Key: T C J M W L
= = = = = =
T C J M W L
•
•
tea coffee juice milk water lemonade
(a) How many adults were in the group? (b) What fraction of the adults prefers coffee? (c) What percentage of the adults does not prefer milk? (d) How many more adults prefer drinks that are usually taken hot than drinks that are usually taken cold?
Favourite drink
o
N adults (frequency)
As we are used to writing words horizontally, the axes of the bar o chart may be rotated by 90 . There are several other types of bar chart.
Change Chart This is a type of bar chart that shows changes in the values of various quantities over a period of time. Example. The numbers and types of crime reported to city police in November and December 2008 were: Crime
Robbery
Burglary
Assault
Vandalism
Rape
November
6
10
26
7
2
December
3
18
24
7
6
Change
–3
+8
–2
0
+4
Month
Representation of Data 7
The changes in the numbers of different crimes are illustrated in the following change chart. Notice that the scales showing the changes are written top and bottom, and that the bars are drawn horizontally for convenience. The largest increase in the number of crimes (+8) was in burglary, which was an increase of 8 10
100%
80%
– 10
0
+10
0
+10
Robbery Burglary Assault Vandalism
The largest proportional increase was in rape, which was an increase of 4 over the original number 2. Rapes increased by 42 100% = 200%
Rape – 10
Exercise 1C 1. Categories of road traffic accidents attended by village police in November and December 2008 are given. Category Intoxication Careless driving Defective vehicle Hit a beast Month November
8
4
8
16
December
14
8
8
11
(i) Illustrate the data in a change chart. (ii) From November to December, which category shows (a) the greatest change, (b) the greatest proportional change? 2. The bar chart below shows the methods of transport used by some primary school students when travelling to school. (i) Name the least commonly used method of transport. (ii) How many more students walk than use the bus? (iii) Express the number of students who use the bus to the number of students who cycle, as a simple ratio. (iv) What proportion of the students travels in a car? (v) What fraction of the students Walk Bus Car Bicycle does not walk to school?
8 Statistics
3. The owner of a shop recorded the mass of potatoes sold on each day that his shop was open last week. Day Mass (kg)
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
32
16
24
18
40
36
(i) Illustrate the data in a bar chart. (ii) What was the total mass of potatoes sold last week? (iii) Between which two consecutive days did the largest change in mass of potatoes sold occur? (iv) Is it necessarily true that more potatoes were sold on Thursday than on Tuesday? Explain your answer. 4. The bar chart below illustrates the number of goals scored by a hockey team in its 32 games last season.
8
(i) In how many games did the team score exactly 2 goals? (ii) What was the most frequently scored number of goals? (iii) In what percentage of the games did the team fail to score? (iv) How many goals did the team score altogether in its 32 games last season? (v) In how many games did the team score at least 4 goals?
Comparative Bar Chart These are bar charts that are used to compare two or more sets of related data. One special type of comparative bar chart is a dual bar chart. ! A dual bar chart can be used to compare two sets of related data. It is made by drawing pairs of bars, with one bar in each pair for each set of data. Equal-width gaps may be left between pairs of bars, and all bars should be of equal width. A key, in which different shading styles or colours can be used, is needed to distinguish each set of data.
Representation of Data 9
Example. Last month two travel companies, TourWell and TravelSafe, organised holidays for customers in four destinations. The numbers of each are given in the table. Destination
France
U.S.A.
China
South Africa
TourWell
11
9
5
7
TravelSafe
9
12
3
4
Company
The dual bar chart below illustrates the data.
o
N customers ( f )
12 8 4 0
France U.S.A. China S. Africa
Destination
(a) How many of these customers' holidays were organised by TourWell last month? (b) How many customers altogether went to South Africa? (c) Which was the least popular of the four destinations? (d) Express the total number of TourWell customers to TravelSafe customers as a simple ratio. ! The data in the example above could be illustrated in a different type of comparative bar chart, with two groups of four bars – one group of four bars for each travel company.
Exercise 1D 1. The results of a survey in two classes on students' favourite flavours in potato chips are shown. Potato chip flavour Class A Class B (i) Illustrate these data in a clearly labelled dual bar chart showing one pair of bars Smokey bacon 6 14 for each of the 5 flavours. Barbecue 8 6 (ii) What proportion of the students from class A prefers barbecue flavour chips? Cheese & onion 1 0 (iii) What fraction of the students that Salt & vinegar 11 8 prefers hot chilli flavour is from class Hot chilli 4 2 B?
10 Statistics
2. The numbers of patients admitted with malaria into three hospitals over a six-week period are shown. Hospital A Hospital B Hospital C
(i) Tabulate the data shown in the comparative bar chart above. (ii) Use the bar chart and your table to find (a) how many patients were admitted with malaria to the three hospitals altogether during these 6 weeks, (b) during which week the greatest number of patients was admitted into the three hospitals altogether, (c) between which two consecutive weeks was the greatest change in the number of patients admitted to hospital C. (iii) Express the total number of patients admitted to hospitals A, B and C as a simple ratio. (iv) What percentage of the total number of patients admitted during this period was admitted to hospital B? 3. A company has recorded the numbers of its full-time and part-time employees for the years 2006 to 2008. (i) Illustrate the data in a dual bar chart. (ii) Express, in simple form, the ratio Year of the two types of employee in each year. Type of employee 2006 2007 2008 (iii) In which year was the highest Full-time 78 77 95 proportion of employees fullPart-time 26 22 19 time? (iv) During the entire three-year period, what percentage of the employees has been part-time?
Representation of Data 11
4. Sixty men and eighty women applied for a job with an international aid organization. The advertisement for the job stipulated that applicants must be fluent in at least one of Xhosa or Zulu. It was discovered that 55% of the male applicants and 70% of the female applicants were fluent in Xhosa and that 60% of the male applicants and 55% of the female applicants were fluent in Zulu. (i) Calculate the number of female applicants that were fluent in Xhosa and fluent in Zulu. (ii) Calculate the number of male applicants that were fluent in Zulu only. (iii) Display the data in a dual bar chart with three pairs of bars showing the numbers of males and females who were fluent in Xhosa only, fluent in Zulu only and fluent in both languages.
Sectional/Composite Bar Chart Each bar represents a total and is divided into sections to show how the total is made up. Example. From her monthly salary of £800 a woman saves £250. She spends £250 on accommodation, £150 on food, £100 on transport and £50 on clothes. The sectional bar chart illustrates the data. Amount spent (Pounds)
The woman's sister is paid a monthly salary of £1000. Of this she saves £300; she spends £300 on accommodation, £200 on food, £150 on transport and £50 on clothes. (a) Construct another sectional bar chart to show how the woman's sister spends her monthly salary. (b) Who saves the largest amount of money each month? (c) Who saves the largest proportion of her salary each month?
12 Statistics
Exercise 1E 1. The sectional bar chart below shows the number of men, women and children living in Hill Street. 0
10
20
o
40 N people (f)
30
Hill Street
Men
Women
(i) Copy and complete the table. o N men
Children o
N women
o
N children
(ii) Find (a) the fraction of people living in Hill Street that are men, (b) the percentage of people living in Hill Street that are women. (iii) The number of men, women and children living in Valley Road are given in the table. o o o N men N women N children 15 11 34 (a) draw a sectional bar chart to illustrate the data for the people living in Valley Road, (b) find the percentage of people living in Valley Road that are women. (iv) Which has the highest proportion of children, Hill Street or Valley Road? 2. The number of cars, trucks and motorbikes serviced at Mike's garage and Jomo's garage last month are shown in the sectional bar chart below.
N vehicles ( f )
Trucks
o
Cars Motorbikes
Mike
Jomo
Garage Owner
Representation of Data 13
(i) Tabulate the data given in the sectional bar chart on the previous page. (ii) Showing your working, find which garage did the highest proportion of its servicing on motorbikes. The cost of servicing each of the different types of vehicle at each of the two garages is given: Vehicle
Mike’s ($ per vehicle)
Jomo’s ($ per vehicle)
Motorbike
40
50
Car
130
120
Truck
230
260
(iii) Calculate the total amount spent on servicing vehicles at each of the two garages last month. (iv) Express the amount spent on servicing vehicles at Jomo's as a percentage of the total amount spent on servicing vehicles at the two garages together. 3. The numbers and status of employees at a clothing store between 2006 and 2008 are shown.
o
N employees ( f )
Managerial Manual Clerical
(i) How many employees were there altogether in 2007? (ii) How many manual employees were there in 2006? (iii) How many fewer manual employees were there in 2008 than in 2006? (iv) What percentage of those employed in 2006 was clerical? (v) Describe the proportional change in management between 2006 and 2008.
4. There are 40 students in form 3A and 24 are boys. There are 32 students in form 3B and 12 are girls. (i) On the same diagram, construct two sectional bars with one bar for each gender. (ii) In which class are there more boys? (iii) Which class has the highest percentage of boys? (iv) Express the number of girls in 3A to the number of girls in 3B as a simple ratio, (v) Which class has the highest proportion of girls? (vi) Illustrate the data in a dual bar chart showing the number of boys and girls in each of the two classes.
14 Statistics
Sectional Percentage Bar Chart Each bar represents a total and is drawn to a height of 100%. The bars are divided into sections, and each represents a percentage of a particular total. Example. A farmer used her land to grow potatoes and wheat in 2007 and in 2008. The sectional percentage bar chart below shows the percentage of each year's production, by mass.
Wheat Potatoes
The data shown in the chart are tabulated:
2007
2008
Crop
2007
2008
Wheat
40%
30%
Potatoes
60%
70%
Total
100%
100%
The total mass represented by each bar is not indicated in the chart or in the table. If the sections of the chart are labelled as shown, then in terms of quantities (masses), (a) P is definitely greater than Q, (b) X is definitely not less than Y, (c) X appears to be greater than P, but it may not be, (d) Q does not appear to be less than Y, but it may be. In the context of the data given by the chart and the table, the four statements above mean: (a) It is necessarily true that a greater mass of potatoes was produced than wheat in 2007. (b) It is necessarily false that a smaller mass of potatoes was produced than wheat in 2008. (c) It is not necessarily true that a greater mass of potatoes was produced in 2008 than in 2007. (d) It is not necessarily false that a smaller mass of wheat was produced in 2007 than in 2008.
Representation of Data 15
! Necessarily true or necessarily false statements (comparing two quantities given as percentages) can be made only if the total quantities are known, or if the two percentages are percentages of the same total. If the total production in 2007 was 700 tonnes and the total production in 2008 was 1000 tonnes, then (a) P is greater than Q [420t and 280t] – as it appears. (b) X is not less than Y [700t and 300t] – as it appears. (c) X is greater than P [700t and 420t] – as it appears. (d) Q is less than Y [280t and 300t] – not as it appears.
Exercise 1F 1. The table below gives the numbers of boys and girls who passed or failed a history examination last term.
Boys Girls
Pass 156 210
Fail 84 90
(i) Find the percentage of (a) girls that failed, (b) boys that passed, (c) students that passed. (ii) On the same diagram, illustrate the data with three sectional percentage bars: one for boys, one for girls, and one for students.
2. Mrs Mafela owns three filling stations. The volume of petrol, diesel and paraffin sold, in litres, at each of the stations last week is shown. Filling station
Volume of petrol (l)
Volume of diesel (l)
Volume of paraffin (l)
North
21000
12000
1500
South
18000
16500
6000
West
11000
11500
2500
(i) (a) Calculate the total volume of each type of fuel that was sold. (b) What percentage of the petrol was sold at the North filling station? (c) Draw a sectional percentage bar chart with one bar for each type of fuel. (ii) (a) Calculate the total volume of fuel sold at each of the three filling stations. (b) What percentage of the fuel sold at the South filling station was diesel? (c) Draw a sectional percentage bar chart with one bar for each filling station.
16 Statistics
3. The percentages of students in forms 1, 2 and 3 at Castle junior school are shown in the chart below.
(i) If there are 320 students at Castle junior school, find the number of students in each of the form groups. (ii) Bishop junior school has 164 form 1s, 96 form 2s, and half of all the students are in form 3. (a) How many students are there altogether in Bishop junior school? (b) What percentage of the students is in form 1? (c) Draw a sectional percentage bar chart to show the data for Bishop junior school. 4. The sectional percentage bar chart below shows the proportion of total profit made from the sales of food and hardware at a general dealer's in March and in April. Percentage of total profit 0
50
100
March April Month
Food
Hardware
Indicate whether each of the statements below is necessarily true, not necessarily true, necessarily false or not necessarily false. A: More profit was made from food than from hardware in April B: Less profit was made from Hardware than from food in March C: Less profit was made from food in April than in March D: More profit was made from hardware in March than in April 5. A man and a woman made separate journeys between Moscow and their hometowns in Russia. The man took a train 45% of the distance, and the woman took a train 65% of the distance.
Representation of Data 17
Both of their journeys were completed by car. Indicate whether each of the statements given below is necessarily true, not necessarily true, necessarily false or not necessarily false. A: The man travelled further than the woman by train. B: The woman travelled further by train than by car. C: The man travelled a shorter distance by car than by train. D: The woman travelled a shorter distance by car than the man. 6. The trainees at a Vocational Training Centre take a combination of two trades. Each trainee chooses a combination of two trades from carpentry, bricklaying and mechanics. The chart below shows the percentages of first- and second-year trainees taking each of the combinations. Percentage of trainees 0
50
100
First
Second
Year-group
Carpentry & Bricklaying
Mechanics & Carpentry
Bricklaying & Mechanics
There are 150 first-year trainees and 180 second-year trainees altogether. (i) Tabulate the number of trainees in each year group that takes each of the three combinations. (ii) Use the numbers from your table in (i) to find the number of trainees in each year group that takes each of the three trades. Show these data in a table. (iii) Display the data from your table in (ii) in the form of a dual bar chart. (iv) State two advantages that the dual bar chart has over the original sectional percentage bar chart.
18 Statistics
Line Graph There are two types of line graph that can be used to illustrate data. Vertical Line Graph
This type of line graph is essentially a bar chart with pencil-thin bars.
Normal Line Graph Points are plotted and joined by ruled lines. Readings must not be taken from these lines, as they are only drawn to show the changes in frequency. The line graph can be 'closed' with zero frequencies at values equally spaced on each side of the data. In both types of line graph, values of the variable should be marked clearly along the horizontal axis. Example 1. The numbers of O level passes obtained by a group of school leavers are given. o
N O level passes
0
1
2
3
4
5
6
7
8
9
10
11
N school leavers ( f ) 2
5
7
8
10
20
16
13
11
6
8
11
o
A vertical line graph can be used to illustrate the data.
10
o
N school leavers ( f )
20
o
0
0 1 2 3 4 5 6 7 8 9 10 11
N O level passes
Representation of Data 19
(a) (b) (c) (d) (e)
How many school leavers are represented in the line graph? How many O level passes did these school leavers obtain altogether? What proportion of the school leavers obtained just 4 O level passes? What percentage of the school leavers obtained less than 3 O level passes? Of those that have more than 5 passes, what fraction has exactly 7 passes?
Example 2. A bag full of £10 notes was distributed amongst 76 old men. The amounts of money that they received are shown in the table and in the line graph below. Amount of money (£) o
N old men ( f )
20
30
40
50
12
32
24
8
32 The point on the line at (45, 16) does not indicate that there were 16 old men who each received £45.
24 16
o
N old men (f )
The line graph can be closed at £10 and at £60 (both with zero frequency), showing that none of the old men received £10, and that none of the old men received £60.
8 0
10
20
30
40
50
60 Amount of money (£)
Exercise 1G 1. The owner of a small shop recorded the number of loaves of bread that she sold each day last week. Day
Mon
Tue
Wed
Thu
Fri
Sat
Sun
o
6
5
3
7
11
10
8
N loaves
Illustrate these data in a vertical line graph.
20 Statistics
2. The line graph below shows the numbers of absences at a primary school for the first 8 weeks of last term.
10
o
N absences ( f )
20
0
1
2
3
4
5
6
7
8
Week
(i) Find the number of absences in week 3. (ii) During how many weeks were there more than 8 absences? (iii) How many absences were there altogether in this 8-week period? (iv) Between which two consecutive weeks was the greatest change in the number of absences? (v) The diagram shows that there were 5 absences in week 8. In fact, only one student was absent during that week. Explain how this could be the case. 3. On each day in April, four young men went into the forest to collect mushrooms. At the end of each day they sold what they had collected to a man who paid them !25 per bag. The table below shows on how many days the young men received various amounts of money. Amount (!) o
N days ( f )
100
125
150
175
200
225
2
4
7
9
5
X
(i) State the value of X. (ii) Illustrate the data in (a) a vertical line graph, showing the amounts of money received, (b) a normal line graph, showing the numbers of bags sold.
Pie Chart A pie chart shows how the whole of something is divided up. A full circle is divided into sectors where each sector represents a part of the whole. ! Each sector should be labelled with the title of the part that it represents. Each sector angle and each sector area must be proportional to the size of the part that it represents.
Representation of Data 21
It is preferable that sector angles are not written inside the chart: they should only be shown in calculations. The percentages of the total that the sectors represent can be written onto or around the outside of the chart. Example. The numbers of students using each of four methods of transport to travel to school are shown. Method of transport o
N students ( f )
Walking
Bus
Bicycle
Car
276
159
75
32
Total = 542
Sector angles are given to the nearest degree and percentages to 3 sf. Angles
Percentages
o 276 Walking : 542 × 360 = 183
276 542 ×100 = 50.9%
o 159 542 × 360 = 106
159 542 ×100 = 29.3%
Bus :
75 Bicycle : 542 × 360 = 50
o 32 542 × 360 = 21
Bicycle 13.8%
Bus 29.3%
75 542 ×100 = 13.8% 32 542 ×100 = 5.90%
Car 5.90%
Car :
o
Walking 50.9%
Exercise 1H 1. There were 36 women, 21 men and 3 children in attendance at a community meeting. (i) Calculate the sector angles required to construct a pie chart to illustrate these data. (ii) Construct and label the pie chart using a radius of 4 cm. Include percentages on the chart.
22 Statistics
2. Morgan's music collection consists of 34 records, 51 cassette tapes, 62 compact discs and 33 mini discs. Construct and label a pie chart using a radius of 4.5 cm. Include percentages on the chart. 3. The pie chart below illustrates the proportion of votes obtained by the Labour, S.N.P and Conservative parties at a by-election held in Scotland. Cons.
Labour S.N.P
o
o
o
The sector angles used are: Labour 158.4 ; S.N.P 151.2 and Conservative 50.4 . (i) What percentage of the votes was won by the party with the second highest number of votes? (ii) If the Conservative party obtained 4116 votes, find (a) the total number of votes cast, (b) by how many votes the Labour party beat the S.N.P. in the by-election. 4. A commerce examination was written by a group of students for which 6 different grades were awarded. The percentage of students obtaining each grade is given: grades A to E are passes and F is a failure. Grades
A
B
C
D
E
F
Percentage of students
5
15
45
17.5
12.5
5
(i) Calculate the six sector angles needed to construct a pie chart. (ii) Construct the pie chart using a radius of 4 cm. (iii) Given that 36 students obtained grade B, find (a) the number of students that obtained grade C, (b) the number of students that failed the examination, (c) the number of students that wrote the examination.
Comparative Pie Chart Sets of related data can be shown and compared using comparative pie charts. ! The areas of the comparative pie charts must be proportional to the totals that they represent, which means that their areas must be in the same ratio as the totals that they represent.
Representation of Data 23 2
Area of a circle = !r , where r is the radius of the circle. If two pie charts with radii r and R are drawn to represent totals t and T, respectively: 2
2
Ratio of areas is !r : !R , which simplifies to r 2 : R2 ………. [1] Ratio of the two totals is …………………….. t : T ……….. [2] Ratios [1] and [2] must be equal, therefore r 2 : R2 = t : T 2 2 Cross-multiplying gives tR = Tr R =
Tr t
2
or R
=
r
T t
Example 1. A pie chart of radius 4 cm is drawn to represent 60 vehicles. What is the radius that should be used to represent 135 vehicles in a comparative pie chart? 2 2 Ratio of areas is 4 : R Ratio of totals is 60 : 135 and these ratios must be equal, so 42 : R 2 = 60 :135 giving 60 × R 2 = 135 × 4 2 2 ×4 R 2 = 13560
×4 R = 13560
2
R = 6 cm The correct radius for the comparative pie chart to represent 135 vehicles is 6 cm. Example 2. A crowd of 6400 that attended a football match on Saturday was represented in a pie chart of radius 5 cm. There was another match on Sunday and the crowd was represented in a comparative pie chart of radius 6.5 cm. What was the size of the crowd on Sunday? Let the size of the crowd on Sunday be T, then the two equal ratios are 2 e 5 : 6.52 and 6400 : T 52 : 6.52
6400 : T 5 × T = 6400× 6.52 2
2 T = 6400 ×2 6.5
5
T = 10 816
The size of the crowd on Sunday was 10 816
24 Statistics
Exercise 1I 1. At Deepdale School, 400 students sat an English language examination. A pie chart was drawn to illustrate the results of these students using a radius of 6 cm. At Shallowvale School, 576 students sat the same examination. Find the correct radius for a comparative pie chart to illustrate the results of the Shallowvale students. 2. A man surveyed 350 shoppers at CutPrice supermarket and 686 shoppers at BargainBin supermarket. The 350 shoppers at CutPrice were represented in a pie chart of radius 7 cm. What is the correct radius to be used in a comparative pie chart to represent the shoppers at BargainBin? 3. A census was taken on the populations of the villages of Gobolale and Semojango. A pie chart of radius 8 cm was used to represent Gobolale's population of 6800, and a comparative pie chart with a radius of 10 cm was used to represent the population of Semojango. What was the population of Semojango? 4. Kay earns ÂŁ27 000 per annum. She has drawn a pie chart of radius 6.3 cm to show how she spends her salary, and her partner Joshua has drawn a comparative pie chart with radius 5.67 cm to show how he spends his salary. (i) Calculate Joshua's annual salary. (ii) Calculate, correct to 3 significant figures, the correct radius for another comparative pie chart that could be drawn to show how Kay and Joshua spend their combined salaries. 5. The number of employees at a clothing manufacturer's in 2008 was 19% less than in 2007. A pie chart of radius 12.5 cm was drawn to represent the employees in 2007. (i) Calculate the radius of a comparative pie chart that could be drawn to represent the employees in 2008. (ii) If there were 486 employees in 2008, how many employees were there in 2007? 6. The table gives the number of males and the number of females who applied to join the Armed Forces at a recruitment office in 1998 and in 2008. A pie chart drawn to represent those applying in 2008 has a sector area for females of 62 cm2. 1998 2008 Using ! = 3.142, find Males 423 424 (i) The area of the pie chart used for 2008. (ii) The radius of the pie chart used for 2008. Females 137 189 (iii) The correct radius to be used for a comparative pie chart to represent those who applied in 1998. (iv) The area of the sector for the males that applied in 1998.