KS4 Mathematics

1. KS4 Mathematics D1 Planning and collecting data

2. D1 Planning and collecting data Contents • A D1.2 Types of data • A D1.1 Specifying the problem and planning D1.3 Collecting data • A D1.4 Sampling • A D1.5 The stages of research • A

3. Year 11 pupils with paid jobs don’t do as well in their exams. The first step in planning a statistical enquiry is to decide what problem you want to explore. Formulating a hypothesis This can be done by asking questions that you want your data to answer and by stating a hypothesis. A hypothesis is a statement that you believe to be true but that you have not yet tested. The plural of hypothesis is hypotheses. For example,

4. “Year 11 pupils with paid jobs don’t do as well in their exams.” Forming a hypothesis How could you find out if this statement is true? • Think about: • what data (information) you will need to collect • how you will collect it • which Year 11 pupils this statement covers • how you will ensure the data you collect represents all of these Year 11 pupils • what you will do with the data • what you expect to find.

5. hypothesis– a statement that can be tested Key vocabulary population– the group (often of people) referred to in the hypothesis sample– a selection from the population biased sample– an unfair selection representative sample– a fair selection cross section– a selection that reflects all the subgroups within the population objective data– information that is not affected by people’s opinions

6. subjective data– information that is affected by people’s opinions Key vocabulary primary data– information you collect yourself, by asking people, measuring, carrying out experiments, and so on secondary data– information that has been collected already, that you get from books, the internet, and so on ethical issues– problems to do with confidentiality, personal questions, etc. reliable results– results that will be repeated if the experiment or survey is carried out again with a new sample

7. “The bigger the sample size, the more reliable the results.” Reliable results Do you agree with this statement? Generally, the statement is true so long as the sample is fair and the conditions in which the data was collected normal. For example, suppose we had an experiment into reaction times using a class of ten year olds. This sample is not representative of the population at large and so the results are not reliable if they are applied to the wider population.

8. Using key words

9. “Year 11 pupils with paid jobs don’t do as well in their exams.” Using key words • What decisions will you have to make about the population? • You decide on a sample size of 20. What are the risks of • choosing a small sample? • If you were to do a survey, what questions would you ask? • You decide to ask 20 of your friends. What kind of sample is • this likely to produce? • You need to have a cross section of the population. You • decide to have 10 boys and 10 girls. What other factors do you • need to take into account? • You ask people to estimate the number of hours they spend • on housework as well. What is the problem with this?

10. “People feel stressed when they have exams.” Planning how to test a hypothesis “You get less work done when it is noisy.” “Sleep deprivation affects concentration.” “Coffee can help you revise better.” “The more revision you do, the better your exam results.” • Choose one of these hypotheses and discuss how you would decide on: • the population • the sample size • how you will ensure the sample is representative • what data you will collect and how you will collect it • any problems you might encounter.

11. Improving hypotheses

12. Extending a hypothesis Once you have collected data and drawn conclusions about your hypothesis, you could ask further questions and pursue other lines of enquiry. You will need to plan what these might be beforehand if you are carrying out a survey. For example, “People feel stressed when they have exams.” “You get less work done when it is noisy.” “Sleep deprivation affects concentration.” “Coffee can help you revise better.” “The more revision you do, the better your exam results.” How could you extend these hypotheses? What extra information might it be worth collecting?

13. D1 Planning and collecting data Contents D1.1 Specifying the problem and planning • A • A D1.2 Types of data D1.3 Collecting data • A D1.4 Sampling • A D1.5 The stages of research • A

14. Think about: • the subjectivity of the data “People feel stressed when they have exams.” Measuring stress Kelly decides to ask 30 Year 11 pupils how stressed they are feeling during their mock exams. What are the problems with this approach? • issues of confidentiality • how she will record their answers • whether the data will enable her to decide whether her • hypothesis is correct or not.

15. Circle the most appropriate number for each statement, which refer to the time of your mocks: strongly disagree strongly agree • I am not sleeping well. • I feel anxious. • I feel sick or have stomach problems. • I often get upset or angry. 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 “People feel stressed when they have exams.” Using a scale Kelly decides to use the questionnaire below. What could she do with her results?

16. “People feel stressed when they have exams.” Collecting numerical data Kelly decides to add the numbers circled by each participant to give them a total score. She calls this their “stress score”. These are her results. Does Kelly have enough information to confirm her hypothesis?

17. Kelly gives the same questionnaire to the same participants a month later. Two participants are unable to take part, and their results are removed. She now has two items of data for each participant. Here are the results: Collecting numerical data What could Kelly do with her results?

18. Kelly’s results Sidra’s results Mock After Mock After Mock After Mock After Mock After Mock After 10 7 11 5 9 6 Y N Y Y Y N 14 13 10 10 15 7 Y N Y Y Y Y 8 8 11 12 14 3 N N Y N N N 18 14 12 10 13 1 Y N Y Y Y N 16 12 12 10 15 14 Y N Y Y N Y 14 11 17 12 16 10 N Y Y Y Y N 15 8 18 9 14 8 Y Y N N N N 14 8 15 5 17 9 Y N N N Y Y 16 5 10 2 12 6 N N Y Y N Y 12 2 7 4 11 4 Y Y Y Y N Y Sidra has carried out a survey too. She has asked participants to answer the question “Do you feel stressed?” during and after the mocks. Here are both girls’ results: Numerical and non-numerical data Whose results are better and why?

19. Data can be either: • numerical (quantitative)data • non-numerical (qualitative)data Types of data Examples of quantitative data include: • heights • time • age Examples of qualitative data include: • opinions • favourite subjects • eye colour • gender

20. Which kind of data are each of these? Types of data 1) people’s opinions about third world debt 2) how much sleep you have had each night this month 3) whether people are left handed or right handed 4) the number of full stops in different books 5) how you felt after your last exam 6) how popular your favourite band is among your friends 7) which supermarket people prefer 8) the number of Year 11 pupils who are stressed Now think up your own examples of numerical and non-numerical data.

21. Which of the examples of numerical data given below would need to be rounded off? Measurements • Shoe size • The number of goals in a football match • The temperature of a classroom • The time taken to complete a task • The number of GCSE grade A*s achieved in your school last year • The number of marks gained in a dance exam • The height of a mountain • Numerical data can be either: • continuous • discrete

22. You can have a shoe size of 4 or 4½ but not 4¼ . • Shoe size Discrete data You can score 2 goals but not 2.5. • Number of goals in a • football match • The number of GCSE • grade A*s achieved in • your school last year There could have been 40 or 41 A* grades but not 40.1. You could get 60 but not 60.8 in the exam. • The number of marks • gained in a dance exam Discrete data jumps from one measurement to the next. The measurements in-between have no meaning.

23. Continuous data The temperature could be 21oC, 21.1oC, 21.01oC or …. • The temperature of a • classroom The time could be 57 secs, 57.1 secs, 57.01 secs or …. • The time taken to • complete a task The height could be 300 m, 300.6 m, 300.0006 feet, or ….. • The height of a • mountain Continuous data does not jump from one measurement to the next, but passes smoothly through all the measurements in-between.

24. Discrete or continuous?

25. D1 Planning and collecting data Contents D1.1 Specifying the problem and planning • A D1.2 Types of data • A D1.3 Collecting data • A D1.4 Sampling • A D1.5 The stages of research • A

26. “Year 11 pupils with paid jobs don’t do as well in their exams.” Writing a questionnaire Task 1 You are about to be shown a questionnaire designed to investigate this hypothesis. Discuss how it could be improved. • Think about it from the point of view of • the participants • the researcher collating and analysing the data. Task 2 Write an improved questionnaire.

27. Questionnaire about jobs in Year 11 Name: …………… Form: …………… 1. Do you have a paid job? …………………………………………… 2. If so, how many hours do you do in a week? ……………………. 3. What is your job? …………………………………………………… 4. How long have you had it? ………………………………………… 5. Do your parents make you do any jobs at home? ………………. 6. If so, how many hours do you do in a week? ……………………. 7. How many hours of revision did you do for your mocks? ……… 8. What were your mock results like? ………………………………. 9. Do you think you could have done better if you didn’t have a job? …………………………………………………………………… Writing a questionnaire

28. When writing your own questionnaire, it can be helpful to use the following guidelines: Guidelines for writing a questionnaire • Give participants the option of remaining anonymous. • Record the gender of each participant. • Think about using tick boxes or scales to make the questionnaire easy to fill-in and easy to analyse. • Anticipate problems such as participants working different • hours each week. • Think about whether participants will have the information you • ask for, such as mock results.

29. When writing your own questionnaire, it can be helpful to use the following guidelines: Guidelines for writing a questionnaire • Be specific – you could ask for grades for named subjects. • Don’t ask leading questions. • Only ask relevant questions. • Make sure questions are not ambiguous or misleading. • Carry out a “pilot study” by testing the questionnaire on a few • friends first to see if there are any problems.

30. Gender Hours of work Maths mock Science mock M 3 A B M 1 B A M 6 E C F 2 C B F 0 B A* The table above is an example of part of a data collection sheet. Draw up a data collection sheet for your questionnaire. A data collection sheet is a table, or series of tables, where the data from the questionnaires is collated. Data collection sheets

31. D1 Planning and collecting data Contents D1.1 Specifying the problem and planning • A D1.2 Types of data • A D1.4 Sampling D1.3 Collecting data • A • A D1.5 The stages of research • A

32. Soap wars How are TV viewing figures compiled?

33. Television viewing figures When compiling television viewing figures, it is impractical to find out what everyone in the country is watching at a particular time. Instead, the viewing habits of a sample of households are carefully monitored and this data is then used to compile the figures. To avoid bias, it is important that the sample is representative of all television viewing households across the country. This is done by dividing households into categories and taking samples in proportion to the size of each category. This is an example of a stratified sample.

34. 27 Random sampling People are chosen at random e.g., names are picked from a hat or chosen using a random number generator. Every member of the population has an equal chance of being chosen. Different sampling methods Systematic sampling Members of the population are chosen at regular intervals, such as every 100th person from a telephone directory. Quota sampling You ask a certain number ofpeople from each category. An example would be a survey in the street where you stop asking people over 65 when you have enough results for that age group.

35. Different sampling methods

36. Imagine you are going to investigate the hypothesis Stratified sampling “People who get married at 20 are more likely to divorce than people who get married at 30” Assume the population is all people who married at either 20 or 30 in Great Britain. How would you select a sample? • Relevant factors could be: • the year when they got married • socio-economic class • parents’ marital status

37. The relevant factors are also called variables. Stratified sampling Year of marriage is one variable. We could split this into decades. The different decades are called strata. For example, the 1970s might be one stratum. What are the strata for the variable “parents’ marital status”? The strata for “parent’s marital status” could be: • married • widowed • divorced • single • separated • living with partner

38. If we want the sample to represent a cross section of the population, we need the size of each stratum to be chosen to reflect their proportions in the population. Stratified sampling For example, if 10% of the sample population were married in the 1970s, then the sample needs to contain 10% of people in this strata. The actual number then depends on the number of people in the sample. For example, if there are 3000 people in the sample, 300 of them should be people married in the 1970s.

39. “Pupils from village A are more likely to be late for school than pupils from village B.” Using stratified sampling 40% of pupils in the school are boys and 60% are girls. 30% travel to school by bus and 70% walk. There are 450 pupils in the school. You want to sample 60 pupils from each village. Construct a stratified sample which reflects both the proportions of male and female and methods of getting to school.

40. If there are 60 pupils in each sample then 30% must come by bus and 70% must walk. So in each sample: Using stratified sampling Number of pupils who come by bus = 30% of 60 = 18 Number of pupils who walk = 70% of 60 = 42 Of the 18 who come by bus, 40% are boys and 60% are girls. Number of boys who come by bus = 40% of 18 = 7.2 ≈ 7 Number of girls who come by bus = 60% of 18 = 10.8 ≈ 11 Of the 42 pupils who walk, 40% are boys and 60% are girls. Number of boys who walk = 40% of 42 = 16.8 ≈ 17 Number of girls who walk = 60% of 42 = 25.2 ≈ 25

41. First decide which variables are relevant to your hypothesis. A guide to using stratified sampling • Decide how many strata there are for each variable. • Find out what percentageof the population each of the • different stratamakes up. • Decide on your sample size. • Calculate how many people you need for each of the strata. • To select the people in each of the strata, use another sampling method, for example random sampling.

42. Suppose you have a list of 100 people and want to select 20 of them randomly. This can be done using the random number generator on your calculator. Using a calculator to generate a random sample • Number each person from 0 to 99. • Key 100 into your calculator, followed by the RAN # button. • Press equals. This gives you a number between 0 and 99. • The number may have a decimal. This should be rounded • down (or the decimal ignored). • Press the equals button twenty times, making a note of each • number. • Find the people on your list of 100 people that match your • twenty numbers.

43. Random sampling Evaluating different sampling methods Every member of the population has an equal chance of being chosen, which makes it fair.  It can be very time consuming and usually impractical. Systematic sampling You are unlikely to get a biased sample.  It is notstrictly random: some members of the population cannot be chosen once you have decided where to start on the list.

44. Quota sampling Evaluating different sampling methods This is easier to manage.  It could be biased. For example, if you are only asking people on the street or in a shop, the sample might not represent people that work all day. Stratified sampling It is the best way to reflect the population accurately.  It is time consuming and you have to limit the number of relevant variables to make it practical.

45. D1 Planning and collecting data Contents D1.1 Specifying the problem and planning • A D1.2 Types of data • A D1.5 The stages of research D1.3 Collecting data • A D1.4 Sampling • A • A

46. There are several stages in carrying out a project or piece of research. These include: The stages of research • developing your hypothesis and planning how to test it • collecting data • using graphs and calculations to describe your results • analysing your results; drawing conclusions about whether • your hypothesis has been supported by the data • evaluating and recognizing the limitations of your methods, • and deciding how reliable your conclusions are. • extending your hypothesis and pursuing new lines of enquiry.

47. These stages can be shown by the data collection cycle as follows: 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

48. Your GCSE coursework will be assessed on three strands. Each one is worth the same number of marks. GCSE coursework • Planning and collecting data. • Processing and representing data. • Interpreting and evaluating data. Discuss what each of these strands involves. What kind of evidence will the marker be looking for?

49. Review A key skill in handling data is the correct use of vocabulary. • Give your partner a definition of a word to guess. • See whether you can write a brief definition of each of the sampling methods: random, systematic, quota and sampling. • Choose a hypothesis and write a paragraph about how you • would research it. Compete with your partner to get as • many key words in as possible.