About Statistics

Book B

TITLE Book Name:

About Statistics Book B: Starting with Statistics 600B 978-1-86968-479-2 2008

Book Code: ISBN 13: Published:

AUTHOR John Thompson

ACKNOWLEDGEMENTS The publisher wishes to acknowledge the work of the following people: Design: Editor:

Glen Honeybone Murray Quartly

PUBLISHER User Friendly Resources New Zealand PO Box 1820 Christchurch Tel: 0508-500-393 Fax: 0508-500-399

Australia PO Box 914 Mascot NSW 2020 Tel: 1800-553-890 Fax: 1800-553-891

United Kingdom Parkside Farm Shortgate Lane Lewes BN8 6DG Tel: 0845-450-7502 Fax: 0845-688-0199

WEBSITE www.userfr.com

E-MAIL info@userfr.com

COPYING NOTICE This is a photocopiable book and permission is given to schools or teachers who buy this resource to make photocopies or transparencies of all pages. The copies must be for internal school use only, and may not be given or sold to other educational institutions or teachers from other institutions.

COPYRIGHT User Friendly Resources, 2008.

User Friendly Resources specialises in publishing educational resources for teachers and students across a wide range of curriculum areas, at both primary and secondary levels. If you wish to know more about our resources, or if you think your resource ideas have publishing potential, please contact us at the above address.

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About Statistics

Book B

Contents STARTING WITH STATISTICS

Introduction

4

Activity 5: Descriptive Statistics Activity Guidelines for Teachers Activity 5A: Getting Started Activity 5B: Creating Data Sets Activity 5C: Cricket Statistics Activity 5D: Some Challenges Activity 5E: Review Questions Self-Assessment Answers

5 6 7 8-9 10-12 13 14 15

Activity 6: Grouped Data Activity Guidelines for Teachers Activity 6A: Constructing a Histogram Activity 6B: Choosing Class Intervals Activity 6C: Creating Frequency Tables Activity 6D: Drawing Histograms Activity 6E: Reading Histograms Activity 6F: Review Questions Self-Assessment Answers

16 17-18 19 20-21 22 23-24 25 26 27

Activity 7: Comparing Data Sets Activity Guidelines for Teachers Activity 7A: Back-to-back Stem-and-Leaf Plots Activity 7B: Box-and-Whiskers Plots Activity 7C: Reading a Box-and-Whisker Display Activity 7D: Comparing Two Types of Display Activity 7E: Review Questions Self-Assessment Answers

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28 29-31 32-34 35 36 37 38 39-41

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About Statistics

Book B

Introduction STARTING WITH STATISTICS It is not always easy for students to analyse statistical data and come to conclusions that are consistent with the information provided. This About Statistics series provides teachers with an accessible resource for guiding students through some of the more difficult steps. The material introduces the basic techniques necessary to carry out statistical analysis and investigation. The data in most of the activities is based on real life situations. It has been collected from newspapers and similar sources. Students will also be able to find data from libraries, or on the internet. Further sources include Yearbooks and other statistical publications. As with the other resources in the About Statistics series, the activities involve students in problem solving and cater to a range of learning styles including cooperative group activities. The activities can also be adapted to provide a programme of independent learning if needed. The study of statistics particularly lends itself to the use of software. As well as assisting students in what have historically been seen as tedious computing tasks, software also has an important role in allowing students to experiment with data. The software and hardware needed is often quite minimal. For most purposes a spreadsheet running on any computer will suffice. Also, data files allow students to investigate real-life data with all its complexity and ambiguity. The major pitfall to avoid with computers is students producing excessive numbers of printed graphs for their data without accompanying interpretations!

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About Statistics

5 F O C U S The calculation of measures of centre: mean and median The calculation of a measure of spread: range The calculation of the quartiles of a data set

Book B

Descriptive Statistics STARTING WITH STATISTICS

Introduction This activity introduces the following descriptive statistics: the calculation median and mean; the calculation of a measure of of measures of centre: med of the quartiles of a data set. spread: range; and the calculation calc Students learn to interpret basic statistics within contexts that are meaningful to them.

Teaching Ideas • • • • • •

Study examples where there is a difference between mean and median. Discuss the different effects of using a mean or median, for example in real estate prices. Construct data with a particular mean or median. Construct data with particular quartile values. Use student generated data to compare with those of samples from the population. Encourage students to use manual methods to compute descriptive statistics as well as a computer or calculator.

Reinforcement • • • • •

Calculating percentiles. Cumulative frequency graphs for calculating medians and quartiles. Relationship between inter-quartile and other measures of spread such as standard deviation of the population. Comparing the mean of a sample to the mean for the whole population. Considering whether the data in a sample has the same inter-quartile range (or some other suitable measure of spread) as the parent population.

Definitions Interquartile The difference between the top of the lower quartile and the bottom of the upper quartile. Percentiles Relative position or rank of each priority score. Cumulative Frequency The number of occurrences at or before a given point. Standard Deviation Measure of spread or variability of the data. © Copyright User Friendly Resources.

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About Statistics

5A F O C U S The calculation culation of measures of centre: mean and median

Book B

Getting Started STARTING WITH STATISTICS

What you need to know There are several basic statistics that are often needed to help understand a set of data. These are listed here. Read their definitions carefully. Maximum: The largest value in a set of data. Minimum: The smallest value in a set of data. Range: The difference between the maximum and minimum. Median: The middle value, when the data is put in numerical order. When there is two middle numbers it is the average of these numbers. This occurs when the data set has an even number of numbers. Mean (or average): The sum of the data, divided by the number of data values. Upper quartile: The median of the higher half of the data values. Lower quartile: The median of the lower half of the data values. Mode: The most frequently occurring data value(s).

The calculation of a measure of spread: range The calculation of the quartiles of a data set

Study the examples below to see how to work these statistics out. In a recent test, Mr James’ class scored the following marks (all marks are percentages). 45, 56, 67, 78, 90, 90, 34, 52, 75 or 34, 45, 52, 56, 67, 75, 78, 90, 90 when written in ascending order. Maximum : 90 Minimum : 34 Range : 90 – 34 = 56 Median : 67 Mean = (45 + 56 + 67 + 78 + 90 + 90 + 34 + 52 + 75) ÷9 = 587 ÷ 9 = 65.2 (to the nearest tenth, or one decimal place) Upper quartile : (45 + 52) ÷ 2 = 48.5 Lower quartile : (78 + 90) ÷ 2 = 84 Inter-quartile range : 84 - 48.5 = 35.5 Mode : 90

What you need to do For each of the following sets of marks calculate the basic statistics on a separate sheet of paper, as shown in the example above:

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1.

45

56

78

41

2.

12.5 6.5 17.5 11

14 9

15.5 0.5 1.5 13.8 16.3 34

6

89

12

13

90

98

12 11.5 16 11.8 0.6 21

58

58

45

9

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About Statistics

5B F O C U S The calculation of measures of centre: mean and median

Book B

Creating Data Sets STARTING WITH STATISTICS

What you need to do Make up your own data set for each of the following. Demonstrate each statistic with working. 1. 4 numbers with a maximum of 10 and a minimum of 2

The calculation of a measure of spread: range The calculation of the quartiles of a data set

2. 5 numbers with a range of 12

3. 7 numbers with a median of 8

4. 6 numbers (all different) with a mean of 10

5. 5 numbers with a median of 12 and a range of 15

6. 6 numbers with a mean of 10 and a median of 8

7. 10 numbers with an upper quartile of 15 and a lower quartile of 9

8. 12 numbers with a mode of 56, a median of 40, a range of 80 and a maximum of 100

Note: It is possible to do this exercise using a spreadsheet that has statistical functions such as: • Average • Mode • Median • Quartile • Max • Min © Copyright User Friendly Resources.

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About Statistics

5C F O C U S The calculation of measures of centre: mean and median The calculation of a measure of spread: range The calculation of the quartiles of a data set

Book B

Cricket Statistics STARTING WITH STATISTICS

What you need to know Different sports have a variety of ways of using statistics to measure how well an individual player or a team is doing. In this activity we look at some of these ways of measuring performance in the game of cricket. Batting For a batter in cricket, statisticians work out a batting average. This is done by adding up all the runs the batter has scored and dividing this by the number of times the batter was out. “NOT outs” are not included in the total count. Read the following example. Peter’s scores were 23, 12, 0, 5 (not out) and 12. His batting average was (23 + 12 + 0 + 5 + 12) ÷ 4 = 13 Bowling A bowling average is worked out by dividing the number of runs scored off the bowler by the number of wickets they took. Read the following example. If a bowler took 7 wickets and had 77 runs scored off their bowling, their average would be 77 ÷ 7 = 11 That is, each wicket they took cost 11 runs on average.

What you need to do 1. Batting At the end of the season there are two players in line for the batting trophy. The prize is for the best batter in the team and is usually given to the player with the highest batting average. The scores of the two players are as follows: Pat 110, 84, 56, 0, 70, 50 (not out), 90, 76, 24 (not out), 20 Chris 20 (not out), 12 (not out), 0 (not out), 30, 25, 10 (not out), 70, 60 (not out) a) Work out the batting average for each player.

b) Who do you think should win the trophy? Give a reason for your answer.

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About Statistics

5C F O C U S The calculation of measures of centre: mean and median The calculation of a measure of spread: range The calculation of the quartiles of a data set

Book B

Cricket Statistics STARTING WITH STATISTICS 2. Bowling Two players were arguing about which of them was the better bowler. They decided that whoever had the lower bowling average in their next match could regard themselves as the best. The next match had two innings. Their bowling figures were as follows. 1st innings

Suli took 2 wickets for 25 runs. Ramesh took 2 wickets for 45 runs.

2nd innings

Suli took 1 wicket for 4 runs Ramesh took 5 wickets for 21 runs.

Overall:

Suli took 3 wickets for 29 runs and Ramesh took 7 wickets for 66 runs.

a) Complete this table for the playerâ€™s bowling averages.

Suli 1st Innings

Ramesh

12.5

2nd Innings

Whole Match

b) Use the table to give a reason why Suli or Ramesh could be regarded as the better player.

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About Statistics

F O C U S The calculation of measures of centre: mean and median The calculation of a measure of spread: range The calculation of the quartiles of a data set

Some Challenges! STARTING WITH STATISTICS

What you need n to d do 1. Here is a bar gra graph aph showing tthe number of visitors to Room 12 during the past week week. Visitors to Room 12 last week Number of Visitors

5D

Book B

6 5 4 3 2 1 0 Mon

Tue

Wed

Thur

Fri

Day

Use this graph to find : a) The number of visitors the class had during the week.

b) The average number of visitors per day.

c) The median number of visitors per day.

d) The mode for the number of visitors per day.

2. To be accepted for Smithfield Music Academy, students need a grade average of 2 or less in their final school exams. Grades are allocated using this table.

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Mark range 80 - 100 65 - 79 50 - 64 40 - 49 0 - 39

Grade 1 2 3 4 5

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About Statistics

5D F O C U S The calculation of measures of centre: mean and median The calculation of a measure of spread: range The calculation of the quartiles of a data set

Book B

Some Challenges! STARTING WITH STATISTICS To find the grade average ave erage the follo following steps are taken. 1. Each mark is converted conve erted to a grade. grad 2. The Th grades d are added dd up. 3 3. The total for the grades is divide divided by 5. Worked Example Mark

Grade

45

4

67

2

52

3

65

2

75

2

Student Sam Ali Jo Kim

Marks 80 65 41 45 67 78 32 59 72 82 67 49

Grade average =

Total of grades Total number of grades

= (4 + 2 + 3 + 2 + 2) ÷5 = 13 ÷ 5 = 2.6 This grade average is not good enough for a student to get into the Music Academy.

Grade Average 75 78 73 67

82 85 95 78

a) Work out the total marks for each student.

b) Work out the grade averages for each of these students.

c) Now write down a list of those students who will not be accepted into the academy.

d) Explain why Kim may feel that the system has treated her unfairly.

e) What system could the Smithfield Music Academy use so that it was fair for all students?

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About Statistics

5D F O C U S The calculation of measures of centre: mean and median

Book B

Some Challenges! STARTING WITH STATISTICS 3. The employees in the steel foundry office were angry when they read a newspaper report of a speech by the owner. In the speech the owner was quoted as saying that the average weekly wage for office staff was $800 per week. The weekly wages for office staff are as follows:

The calculation of a measure of spread: range The calculation of the quartiles of a data set

Cleaner Receptionist Typist Secretary Computer Operator Manager

Average wage

$400.00 $600.00 $600.00 $700.00 $800.00 $2,900.00

=

Total of wages number of staff

a) Calculate the average wage for the office staff.

b) Give a reason why you think the staff felt that the ownerâ€™s speech was misleading.

c) What other statistic could be used to describe the typical weekly wage at the office?

d) What is the value of the statistic used in c)?

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About Statistics

5E F O C U S The calculation of measures of centre: mean and median The calculation of a measure of spread: range The calculation of the quartiles of a data set

Book B

Review Questions STARTING WITH STATISTICS Answer the questions below using the following data. Students at Riverside School are regularly tested on their knowledge of multiplication tables. In each test they are given 10 questions and then marked as follows. • If they get a question right they receive two marks. • If they fail to answer the question they get zero marks. • If they attempt the question but get it wrong they lose one mark. The maximum mark possible in such a test is 20. The lowest mark possible is -10. In his last 8 tests Sam scored : 17 3 -2 6

10

12

11

In her last 7 tests Maria scored : 14 0 5 11 12

8

9

Sam

9

Maria

A. Range

B. Mean

C. Median D. Upper Quartile E. Lower Quartile F. Inter-Quartile Range G. How many marks would Maria need to get in her next test for her average mark over 8 tests to be the same as Sam’s average mark over 8 tests?

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About Statistics

Book B

Self-Assessment STARTING WITH STATISTICS

Use the confidence level scale on the right to assess how confident you feel about each of the activity questions. Tick on the scale for each concept.

CONCEPT

Confidence level scale 1 = I donâ€™t understand it at all. 2 = I would find it difficult. 3 = I could do it, but not easily. 4 = I know how to do this, but would like more practice. 5 = Too easy, give me a challenge!

EXAMPLE

I know how to calculate maximum, minimum and range for a set of data.

Find the range for this set of data.

I know the difference between mean and median.

I know how to calculate upper and lower quartiles for a set of data.

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CONFIDENCE LEVEL 1

2

3

4

5

Explain why the median and the mean for a set of data will usually be different.

1

2

3

4

5

Calculate the inter-quartile range for this set of data.

1

2

3

4

5

14

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About Statistics

Book B

Answers STARTING WITH STATISTICS

Activity 5A

Activity 5D

1. Mean Median Mode Upper quartile Lower quartile Minimum Maximum Range

56.9 57 58 83.5 43 12 98 86

2. Mean Median Mode Upper quartile Lower quartile Minimum Maximum Range

12.3 12 9 16 9 0.5 34 33.5

1. a)17 b) 3.4 c) 4 d) 4 2. a) b) c) d)

Sam 2, Ali 2,4, Jo 2.6, Kim 2.2 Ali, Jo, Kim Sam 343, Ali 353, Jo 331, Kim 343 They had totals equal to or better than Sam, who was accepted. Their average marks were better than 65, which is a grade 2. e) Any reasonable and correct explanation. 3. a) b) c) d)

Activity 5B

$800 All but one employee receives less than $800 Median $475

Review Questions

Students will need to show that their answers satisfy the given conditions.

Activity 5C 1. a) Chris 72.5

Pat 75.7 (1dp)

b) Pat scored more runs and had higher scores. Chris’s average is high only because of the all the “not outs”. Pat should get the trophy. 2. a)

Suli

Ramesh

1st Innings

12.5

22.5

2nd Innings

4

4.2

Whole Match

9.66667

9.42857

Sam

Maria

A. Range

19

14

B. Mean

8.25

8.43 (2dp)

C. Median

9.5

9

D. Upper Quartile

13.25

11.5

E. Lower Quartile

3.75

6.5

F. Inter-Quartile 9.5 Range

5

G. 7

b) Suli has best figures for each innings.

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