Maths Module 3: Statistics, Exercises of Statistics

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Maths Module 3:

Statistics

Teacher’s Guide

1. Collecting Data

1.1 Qualitative and quantitative data

Chapter Objectives

By the end of this chapter students will be able to:

• Categorise different types of data
• Describe some different data collection methods
• Organise data in a frequency table

Key words

Practice - Answers

Qualitative - Related to things that are described by words not by numbers

Quantitative - Related to things that are described by numbers

Discrete - Has only a fixed set of values. For example, the ages a group of people

Continuous - The opposite of discrete. Can take any numerical value. For example, height

Variable - Something that changes. A quantity which can take on different values

i. a) Discrete b) Discrete c) Continuous d) Discrete e) Continuous f) Discrete

ii. Possible answers:

Discrete variables include: hair colour, eye colour, age in years, gender, number of siblings

Continuous variables include: height, length of arm, leg etc., exact age

iii. a) Qualitative b) Quantitative c) Qualitative d) Quantitative e) Qualitative f) Quantitative g) Qualitative h) Quantitative i) Quantitative

iv. b) Continuous d) Discrete f) Continuous h) Discrete i) Continuous

v. Possible answers:

a) The colour is qualitative, the quantity of petrol that can be held in the tank is quantitative

b) The type of elephant is qualitative, the number of elephants in the herd is quantitative

c) The enthnicity of the person is qualitative, the age in years of the person is quantitative

iii.

Data Advantages Disadvantages Secondary - Cheap to collect

• Easy to collect
  • Data may be old
  • The data may be inaccurate

Primary - You know how it was collected

• Can choose who to collect data from
  • Takes a long time to collect
  • Expensive to collect

iv. If possible divide the students into small groups and tell them to search the internet using www.google.com to fi nd sources of information. Discuss the answers in the following lesson. (Please note that Google itself is not a source but is used to fi nd sources on other websites.)

1.4 Methods for collecting primary data

Key words

Questionnaire - A set of written questions designed to collect data on a subject from people

Interview - A set of written questions designed to collect data on a subject from people

Observation - Collecting data by going to watch a situation

Experiment - A method for collecting data which involves doing tests

1.5 Recording data in tables

Key words

Table - A set of data presented in rows and columns. Choosing one value in the table enables another

connected value to be read

Tally - A simple way of counting things in groups of five using lines

Frequency - How often something which we are studying occurs

Frequency distribution - A table which presents the frequencies of different events we are studying

Class intervals - The groups which we use to organise continuous data

Practice - Answers

i. Possible answers: First question: a) Because it is dif fi cult to de fi ne ‘young’ and ‘old’ b) It would be better to have categories of ages such as ‘10-19’, ‘20-29’ etc. because the catergo- ries given are too general. Second question: a) Hardly anyone is under 1 metre or over 2 metres b) People could either write down their actual height or you could use categories again - ‘1 to 1.2m’ Third question: a) If someone answers ‘no’ then you do not know their real opinion, only that they are not amazing so the information collected is not useful. b) More categories and a more speci fi c question would be better. E.g. ‘What is your opinion of the standard of teaching in your school?’ - Very good, good, fair, poor, very poor. It would also be could to ask for an explanation of the answer, e.g. The teaching is good because........ ii. Ask students to work in pairs to create their questionnaire. The content should focus on what work they would like to do, where they think they will live, choices of family life etc.

After each group has fi nished their questionnaire ask them to swap with another group so that they can give feedback on the quality of the group’s questions.

Finally create a list on the board of the best questions by discussing with the students which ques- tions they like and why.

Think

a) 4 (the students should write 4 in the ‘frequency column’ b) On Sunday 11 students were born c) On Monday and Saturday 7 students were born d) To fi nd this fi gure the students should complete the ‘frequency’ column and then add all the numbers to make 52

2. Analysing Data

2.1 Mean, mode and median

Key words

Average - A number which can be used to represent a set of data

Mean - One kind of average. The mean is calculated by adding up all the values and dividing by the

total number of values

Mode - One kind of average. The mode is the value which occurs most often in a data set

Median - One kind of average. The median is found by ordering the data from smallest to largest and

finding the middle value

Chapter Objectives

By the end of this chapter students will be able to:

• Calculate the mean, mode and median of discrete and continuous data
• Calculate the range and interquartile range of discrete and continuous data
• Draw a scatter diagram from a table of data
• Describe the relationship between two sets of data by reading a scatter diagram

Think

The mean of a set of data is the sum of the values divided by the number of values. The median is the middle value when the data is arranged in order of size. The mode of a set of data is the value which occurs most often.

Practice - Answers

i. a) 34 b) (28 + 29)/2 = 57/2 = 28.5 c) 23.

ii. a) There is no mode because each value occurs only once b) 3,839, c) 4,263,

iii. a) 6,471,

b) twelve million and eighty thousand

iv. a) 9,951,

b) The answer is that there is no mode because each value occurs only once. Explain this to the students if nobody thinks of it themselves

2.3 The quartiles

Think

Key words

quartiles - Numbers which divide a set of data into 4 intervals, each containing 25% of the data

Lower quartile - The number which is one quarter or 25% into the data set when it is arranged in

numerical order

Upper quartile - The number which is three quarters or 75% into the data set when it is arranged in

numerical order

Life expectancy - The number of years a person is predicted (expected) to live based on statistical

analysis of a population

Lower quartile thValue

⎟ ⎠

⎞ ⎜ ⎝

⎛ n +

Median thValue

3 (n + 1 )

Upper quartile thValue

⎟ ⎠

⎞ ⎜ ⎝

⎛ n +

Practice - Answers

In order the populations are: 1,145,000 1,581,082 3,083,000 3,839,000 4,082,000 5,882,000 10,231,

a) Lower quartile = (n + 1)/4 th value = 8/4 = 2nd value = 1,581,

b) Upper quartile = 3(n + 1)/4 th value = 24/4 = 6 th value = 5,882,

Practice - Answers

i. a)

Number of

goals ( x )

Frequency ( f ) fx

Σ f = 31 Σ f x = 56

b) Using the formula the mean = 56/31 = 1.81 goals per game

ii.

Number of

people ( x )

Frequency ( f ) fx

Σ f = 36 Σ f x = 153

The mean = 153/36 = 4.25 people per household

2.6 Range and interquartile range

2.7 Averages from grouped data

Practice - Answers

i.

Number of

goals ( x )

Frequency ( f ) fx

Σ f = 31 Σ f x = 60

a) The range = 7 - 0 = 7 goals

b) There are 31 values.

Lower quartile is the (31 + 1)/4 = 8th value. The 8th value is in the category of 1 goal. The lower quartile is 1 goal.

Upper quartile is the 3(31 + 1)/4 = 24th value. The 24th value is in the category of 3 goals. The upper quartile is 3 goals.

The interquartile range = 3 - 1 = 2 goals ii.

Number of people (x)

Frequency (f) fx

Σf = 36 Σf x = 153

a) The range = 9 - 2 = 7 people

b) There are 36 values.

Lower quartile is the (36 + 1)/4 = 9.25th value. The 9.25th value is the category of 3 people. The lower quartile is 3 people per household.

Upper quartile is the 3(36 + 1)/4 = 24th value. The 27.75th value is the category of 5 people. The upper quartile is 3 people per household.

The interquartile range = 5 - 3 = 2 people per household.

2.8 Scatter diagrams

Practice - Answers

Key words

Scatter diagram - A graph which is used to present statistical data about two variables. The graph

can be used to find relationships between the two variables

Correlation - A measure of the relationship between two sets of data

Positive correlation - If the values in two sets of data increase or decrease at the same time then they

have a positive correlation

Negative correlation - If the value of one set of data decreases as the other increases then the two

sets of data have a negative correlation

i. The answer is quite easy: More drinks are sold when it is hotter because people are hotter!

ii. Yes, there is a relationship. The longer Chandra drives the less distance is remaining.

Think

Practice - Answers (continued)

iii. a)

b) The scatter diagram doesn’t show a relationship between the temperature and the amount of rain

There is a positive correlation between the average daily temperature and the number of cold drinks sold, because as the temperature increases the number of cold drinks sold increases. There is a negative correlation betwen the time spent driving and the distance remaining, because as the time decreases the distance remaining decreases.

3. Presenting Data

3.1 Introduction

Think

Chapter Objectives

By the end of this chapter students will be able to:

• Draw pie charts and bar graphs to present discrete data
• Extract information from pie charts and bar graphs to provide information about data
• Draw histograms and cumulative frequency polygons to present continuous data
• Extract information from histograms and cumulative frequency polygons to provide information

about data

• Calculate the range and interquartile range of data by reading a cumulative frequency polygon

Key words

Diagrams - A picture which is designed to show how something works or how the relationship

between the parts works

Pie charts - A way of showing information using different sized sectors of a circle. The sectors look

like slices of a pie

Bar graph/bar chart - A diagram which uses horizontal or vertical bars of equal width to represent

frequency

Histograms - The name of a type of bar graph which represents grouped continuous data

Cumulative frequency - The number of occurences of something at or before a given point

Cumulative frequency graph - A graph which shows the cumulative frequency plotted against val-

ues of another variable

a) Ask students to make a list. If they can’t think of anything ask them to look around their environ- ment after school. Ask students to explain the diagrams and what was being shown.

b) Discuss students ideas on why we use diagrams to present data. The most obvious answer is that they are easy to look at and understand compared to lists of unorganised data.

3.2 Pie Charts