3 ESO BIL Mathematics

1. 3 ESO BIL Mathematics Statistics

2. D1 Statistics Contents • D1 D1.2 Collecting data • D1 D1.1 Basic concepts D1.3 Organizing data • D1 D1.4 Writing a statistical report • D1

3. The science related to the collection, organization, interpretation and analysis of data is called Statistics. What is the Statistics? A statistic gives us information about a characteristic studied in a set of individuals called population. In order to study the characteristic you can choose a subset of members of the population called sample. Each element of the elements of the population of the sample is called individual. The characteristics we want to study is called statistics variable.

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

5. Suppose, for example, that you wish to investigate the favourite sports of 11 to 15 year-olds. Choosing the sample 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.

6. Qualitative data is data that is non-numerical. For example, • favourite football team, Different kinds of data • eye colour, • birth place. Sometimes qualitative data can contain numbers. For example, • favourite number, • last digit in your telephone number, • most used bus route.

7. Quantitative data is numerical. It can be discrete or continuous. Discrete data can only take certain values. Quantitative data 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.

8. Discrete or continuous data

9. Summary of basic concepts

10. D1 Statistics Contents D1.1 Concepts • D1 • D1 D1.2 Collecting data D1.3 Organizing data • D1 D1.4 Writing a statistical report • D1

11. Data can be collected from a primary source or a secondary source. Deciding on the data 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. In this case you use an observation sheet. Data from a secondary source is data that you have collected from somewhere else including the Internet, reference books or newspapers.

12. Sources of data

13. It is important to design a questionnaire so that: • People will co-operate and answer the questions honestly. Designing a questionnaire • The answers to the questions can be analysed and presented. • The questions are not embarrassing or personal. • The questions, if possible, have a specific answer.

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

16. 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 :

17. 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.

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

19. 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:

20. age gender height (cm) weight (kg) hours of TV watched per week An observation sheet can be used to record data that comes from counting, observing or measuring. Designing an observation sheet 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:

21. For example, in our integrated unit we take the following data to test our physical condition: Designing an observation sheet

22. For example, in our integrated unit we take the following data to test our physical condition: Designing an observation sheet

23. D1 Statistics Contents D1.1 Concepts • D1 D1.2 Collecting data • D1 D1.3 Organizing data • D1 D1.4 Writing a statistical report • D1

24. favourite snack tally frequency crisps fruit nuts sweets 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. Using a tally chart 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. Temperatures tally frequency 24 2 25 2 27 4 33 7 We take the temperatures during one month: Grouping discrete data With these data we can make the frequency table: 34 6 35 4

27. Temperatures tally frequency 24 2 25 2 27 4 33 7 We take the temperatures during one month: Grouping discrete data Can you find the mistake? 34 6 35 4

28. Temperatures tally frequency 24 2 25 2 27 4 33 7 To avoid these kind of mistakes we add another row to calculate the total frequency which must be the same as the number of data. Grouping discrete data 34 5 35 4 Total frequency 24

29. 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. For example, this frequency table shows the favourite take-away meals of a group of pupils: Using a frequency table

30. 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 A group of 20 people were asked how much change they were carrying in their wallets. These were their responses: Grouping discrete data Each amount of money is different and the values cover a large range. This type of data is usually grouped into equal class intervals.

31. 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.

32. 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

33. 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?

34. 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. The class intervals are written using the symbols ≤ and <. What is wrong with the class intervals in this grouped frequency table showing lengths? Grouping continuous data This is an open class interval.

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

36. 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. 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: Using two-way tables

37. In your integrated unit you need first to collect the data of the observations sheet and them organizing them using a frequency table. Integrated unit

38. D1 Statistics Contents D1.1 Concepts • D1 D1.2 Collecting data • D1 D1.4 Processing data D1.3 Organizing data • D1 • D1

39. Processing data • D2 Contents Calculating the mean • D2 Finding the median • D2 Finding the mode Finding the range • D2 Calculating statistics • D2

40. A dice was thrown ten times. These are the results: Finding the mode What was the modal score? 3 is the modal score because it appears most often.

41. 2, 2, 1, 1, 2, 2, 1, 0, 0, 0, 2, 2, 3, 3, 1, 1, 2, 2, 1. 1. The mode or modal value in a set of data is the data value that appears the most often. Finding the mode For example, the number of goals scored by the local football team in the last ten games is: The modal score is 2, and 1. Is it possible to have more than one modal value? Yes Is it possible to have no modal value? Yes

42. The mode is the only average that can be used for categorical or non-numerical data. For example, 30 pupils are asked how they usually travel to school. The results are shown in a frequency table. Finding the mode from a frequency table What is the modal method of travel? 8 Most children travel on foot. Travelling on foot is therefore the modal method of travel.

43. Word length 1 2 3 4 5 6 7 8 9 10 Frequency 3 16 12 16 7 3 11 6 2 1 This frequency table shows the frequency of different length words in a given paragraph of text. Finding the mode from a frequency table 16 16 What was the modal word length? We need to look for the word lengths that occur most frequently. For this data there are two modal word lengths: 2 and 4.

44. 9 8 7 6 Number of pupils 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Marks out of ten This bar chart shows the scores in a science test: Finding the mode from a bar chart What was the modal score? 6 is the modal score because it has the highest bar.

45. This pie chart shows the favourite food of a sample of people: Finding the mode from a pie chart What was the modal food type? The biggest sector of the pie chart is for chocolate, so this is the modal food type.

46. Frequency Time (minutes:seconds) Boys Girls 2:00 ≤ t < 2:15 3 1 2:15 ≤ t < 2:30 7 6 2:30 ≤ t < 2:45 11 10 2:45 ≤ t < 3:00 13 9 3:15 ≤ t < 3:30 8 12 3:30 ≤ t < 3:45 7 10 3:45 ≤ t < 4:00 1 2 This grouped frequency table shows the times 50 girls and 50 boys took to complete one lap around a race track. Finding the modal class for continuous data What is the modal class for the girls? What is the modal class for the boys? What is the modal class for the pupils regardless of whether they are a boy or a girl?

47. Sum of values Mean = Number of values 3 + 6 + 7 + 9 + 9 34 5 5 The mean is the most commonly used average. To calculate the mean of a set of values we add together the values and divide by the total number of values. Calculating the mean For example, the mean of 3, 6, 7, 9 and 9 is = = 6.8

48. 1 3 5 Score 2 4 6 Total 8 6 9 Frequency 11 9 7 Score × Frequency 171 50 The following frequency table shows the scores obtained when a dice is thrown 50 times. What is the mean score? Calculating the mean from a frequency table 50 8 22 18 36 45 42 171 The mean score = = 3.42

49. Sum of Score x Frequency Mean = Sum of frequencies 8 + 22 + 18 + 36 + 45 + 42 171 8 +11 + 6 + 9 + 9 + 7 50 To calculate the mean of a set of values we add together the values and divide by the total number of values. Calculating the mean from a frequency table In the previous example, the mean is = = 3.42

50. A pupil scores 78%, 75% and 82% in three tests. What must she score in the fourth test to get an overall mean of 80%? Problems involving the mean To get a mean of 80% the four marks must add up to 4 × 80% = 320% The three marks that the pupils has so far add up to 78% + 75% + 82% = 235% The mark needed in the fourth test is 320% – 235% = 85%