KS3 Mathematics

1. KS3 Mathematics D1 Planning and collecting data

2. D1 Planning and collecting data Contents • D1 D1.2 Collecting data • D1 D1.1 Planning a statistical enquiry D1.3 Organizing data • D1 D1.4 Writing a statistical report • D1

3. We could ask: “Do different types of newspaper use different length words?” The first step in planning a statistical enquiry is to decide what problem you want to explore. Specifying the problem This can be done by asking questions that you want your data to answer and by stating a hypothesis. A hypothesis is a statement of something that you believe to be true but do not have any evidence to support. For example, suppose we wish to investigate the lengths of words used in newspapers.

4. Specifying the problem Related questions could include: “Is there a link between the lengths of the words used and the lengths of the sentences for a particular newspaper?” “Is there a difference between the use of two- and three-letter words?” A possible hypothesis could be: “Tabloid newspapers use shorter words to appeal to a wider audience.”

5. The next step is to decide what data is needed and where it can be collected from. Deciding on the data Data can be collected from a primary source or a secondary source. Data from a primary source is data that you have collected yourself, for example: • From a survey or questionnaire of a group of people. • From an experiment involving observation, counting or measuring. Data from a secondary source is data that you have collected from somewhere else including the Internet, reference books or newspapers.

6. Sources of data

7. When collecting data it is usually impractical to include every member of the group that is being investigated. Choosing the sample A sample is therefore choose to represent the group. How big should a sample be? The sample should be as large as possible. This will depend on the time and resources available. If the sample size is too small, then the results will be unrepresentative.

8. Choosing the sample It is important that the sample is representative of the group that is being investigated. Suppose, for example, that you wish to investigate the favourite sports of 11 to 15 year-olds. Would it be reasonable to question a sample of people outside a football ground following a game? Can you suggest a better sample? You would have to make sure that you ask equal numbers of girls and boys and that the sample is spread out across all age groups in the range.

9. Choosing units If your statistical investigation involves measurement then you must decide what units to use and to what degree of accuracy. Suppose, for example, that you wish to investigate the relationship between age and height. How will you measure age? In weeks? In months? In years and months? In years? How will you measure height? In centimetres? In inches? In metres?

10. Planning a statistical enquiry Once you have decided on • the purpose of the enquiry, • the type of data that will be collected and where it will come from, • the sample size and type, you can start the next stage which is to design a data collection sheet or questionnaire.

11. D1 Planning and collecting data Contents D1.1 Planning a statistical enquiry • D1 • D1 D1.2 Collecting data D1.3 Organizing data • D1 D1.4 Writing a statistical report • D1

12. Data can be collected using a questionnaire or a data collection sheet. Collecting data A questionnaire is used when you wish to ask a sample of people a series of structured questions relevant to your line of enquiry. A data collection sheet or observation sheet is used when recording results involving counting, measuring or observing. It can also be used to collect the answers to a few simple questions. Data can also be collected from secondary sources such as the Internet, newspapers or reference books.

13. 1) Provide an introduction, so that the person filling in the questionnaire knows the purpose of your enquiry. 2) Write questions in a sensible order, putting easier questions first. It is important to design a questionnaire so that: Designing a questionnaire • People will co-operate and answer the questions honestly. • The answers to the questions can be analysed and presented. When designing you own questionnaire you should try to follow these rules:

14. 3) Make sure that questions are not embarrassing or personal. How old are you? Tick one box for your age group. 15-20 21-25 26-30 31 + Designing a questionnaire For example, you need to think carefully about questions asking about age or income. Do not ask : A better question is :

15. 4) If possible, write questions so that they have a specific answer. Did you see the Olympics on TV ? Only the best bits No Sometimes Yes Not much Once a day Designing a questionnaire For example : People could answer :

16. How much of the Olympics coverage did you watch? Tick one box only. None Less than 1 hour a day Between 1 to 2 hours a day More than 2 hours a day A better question would be: Designing a questionnaire Every eventuality has been accounted for and the person answering the question cannot give another choice.

17. How would you rate the leisure facilities available in your local area? Tick one box only. Excellent Good Satisfactory Poor Unsatisfactory Designing a questionnaire A scale can be used when asking for an opinion. For example,

18. 5) Do not ask leading questions. Don’t you agree that football is the best sport? Which one of the following sports do you like the best? football rugby tennis golf cricket boxing Designing a questionnaire For example, this question conveys a particular opinion, A better question is :

19. Would you consider yourself to be: Underweight Average weight Overweight Suggest a better question How much do you weigh? This is too personal, also some people don’t know their weight. A better question would be:

20. Which make of deodorant do you use ? Male: Lynx Adidas Slazenger Other None Female: Sure Impulse Dove Other None Please circle any that apply. Suggest a better question Most people use a deodorant, do you ? This is a leading question and may offend people. A more useful question would be:

21. How many books did you read last month? 0-2 2-4 4-6 How many books did you read last month? Tick one box. 0-2 3-5 6-8 8+ Suggest a better question The intervals given overlap. Also, if a person has read more than 6 books there is nowhere to tick. A better question would be:

22. Trialling a questionnaire Once you have written a questionnaire it is a good idea to try it out on a small sample of people. This is called a pilot survey. Note down their responses and use these to refine any questions that are causing difficulty. Do I use a tick or a cross to show the box I want? There isn’t a box to cover my answer. I don’t want to answer this question because it’s too personal. What does this question mean?

23. age gender height (cm) weight (kg) hours of TV watched per week Designing a data collection sheet A data collection sheet can be used to record data that comes from counting, observing or measuring. It can also be used to record responses to specific questions. For example, to investigate a claim that the amount of TV watched has an impact on weight we can use the following:

24. favourite snack tally frequency crisps fruit nuts sweets Using a tally chart When collecting data that involves counting something we often use a tally chart. For example, this tally chart can be used to record people’s favourite snacks. 13 6 3 8 The tally marks are recorded, as responses are collected, and the frequencies are then filled in.

25. Using a tally chart

26. D1 Planning and collecting data Contents D1.1 Planning a statistical enquiry • D1 D1.2 Collecting data • D1 D1.3 Organizing data • D1 D1.4 Writing a statistical report • D1

27. Categorical data Categorical data is data that is non-numerical. For example, • favourite football team, • eye colour, • birth place. Sometimes categorical data can contain numbers. For example, • favourite number, • last digit in your telephone number, • most used bus route.

28. Numerical data can be discrete or continuous. Discrete and continuous data Discrete data can only take certain values. For example, • shoe sizes, • the number of children in a class, • the number of sweets in a packet. Continuous data comes from measuring and can take any value within a given range. • the weight of a banana, For example, • the time it takes for pupils to get to school, • the height of 13 year-olds.

29. Discrete or continuous data

30. Favourite take-away Frequency Pizza 11 Fish and chips 7 Burgers 8 Indian 5 Chinese 8 Once data has been collected it is often organized into a frequency table. Using a frequency table For example, this frequency table shows the favourite take-away meals of a group of pupils:

31. 34p £1.72 83p £6.36 £4.07 £2.97 £3.53 6p £9.54 34p £1.68 50p 82p £7.54 £1.09 £2.81 £2.43 46p £1.70 £1.29 Grouping discrete data A group of 20 people were asked how much change they were carrying in their wallets. These were their responses: Each amount of money is different and the values cover a large range. This type of data is usually grouped into equal class intervals.

32. For the following data: 34p £1.72 83p £6.36 £4.07 £2.97 £3.53 6p £9.54 34p £1.68 50p 82p £7.54 £1.09 £2.81 £2.43 46p £1.70 £1.29 When choosing class intervals it is important that they include every value without overlapping and are of equal size. Choosing appropriate class intervals We can use class sizes of £1: £0.01 - £1.00, £1.01 - £2.00, £2.01 - £3.00, £3.01 - £4.00, £4.01 - £5.00, Over £5. This is an open class interval.

33. 34p £1.72 83p £6.36 £4.07 £2.97 £3.53 6p £9.54 34p £1.68 50p 82p £7.54 £1.09 £2.81 £2.43 46p £1.70 £1.29 Choosing appropriate class intervals Complete the following frequency table for this data: Amount of money (£) Frequency 0.01 - 1.00 7 1.01 - 2.00 5 2.01 - 3.00 3 3.01 - 4.00 1 4.01 - 5.00 1 Over 5.00 3

34. For the following data: 34p £1.72 83p £6.36 £4.07 £2.97 £3.53 6p £9.54 34p £1.68 50p 82p £7.54 £1.09 £2.81 £2.43 46p £1.70 £1.29 The size of the class intervals depends on the range of the data and the number of intervals required. Choosing appropriate class intervals Explain why class sizes of £5 would be inappropriate. Could we use a class size of 20p?

35. Length (cm) Frequency Length (cm) Frequency 0 ≤ length ≤ 10 0 ≤ length < 10 10 ≤ length ≤ 20 10 ≤ length < 20 20 ≤ length ≤ 30 20 ≤ length < 30 30 ≤ length 30 ≤ length Continuous data is usually grouped into equal class intervals. Grouping continuous data The class intervals are written using the symbols ≤ and <. What is wrong with the class intervals in this grouped frequency table showing lengths? This is an open class interval.

36. Continuous data is usually grouped into equal class intervals. Grouping continuous data What is wrong with the class intervals in this grouped frequency table showing weights?

37. Eat school dinners Eat a packed lunch Eat at home Year 7 35 42 19 Year 8 29 34 22 Year 9 38 32 18 A two-way table can be used to organize two sets of data. Using two-way tables For example, pupils from Years 7, 8 and 9 were asked what they usually did during their lunch break. This two-way table shows the results:

38. D1 Planning and collecting data Contents D1.1 Planning a statistical enquiry • D1 D1.2 Collecting data • D1 D1.4 Writing a statistical report D1.3 Organizing data • D1 • D1

39. The following diagram shows the stages needed to conduct a statistical enquiry. The data collection cycle Specify the problem and plan Interpret and discuss the results Collect the data from a variety of sources Process and display the data

40. The data collection cycle

41. Once you have planned, collected and processed data relevant to a statistical enquiry you will often have to communicate your findings in the form of a report. Writing a statistical report A report should contain the following: • An introduction stating the purpose of the survey and any initial conjectures which you plan to investigate. • A description of what sources were used including a justification of the type and size of any samples used. • Calculations, such as the mean, median and mode, to give an overall picture of the data.

42. Writing a statistical report • Tables or graphs of the results, using ICT as appropriate. (Remember to justify you choice of what is presented). • Problems or ambiguities that arose during the course of the investigation and how you dealt with them. • A summary of the conclusions shown by the data, not forgetting to refer back to your initial hypothesis. Sometimes your data will give results that you did not expect. These will lead to new lines of enquiry which you should investigate if possible.

43. Writing a statistical report Collect the relevant data and write a statistical report investigating one of the following: • The types of sports young people take part in outside of school hours. • How pupils travel to school. • The difference in word lengths used in men’s and woman’s magazines. • Use of mobile phones among teenagers. • The relationship between hand span and foot length.