KS3 Mathematics

1. KS3 Mathematics D3 Representing and interpreting data

2. D3 Representing and interpreting data Contents • A1 D3.2 Pie charts • A1 D3.3 Frequency diagrams • A1 D3.1 Bar charts D3.4 Line graphs • A1 D3.5 Scatter graphs • A1 D3.6 Comparing data • A1

3. 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.

4. 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.

5. Discrete or continuous data

6. Bar charts can be used to display categorical or non-numerical data. Bar charts for categorical data For example, this bar graph shows how a group of children travel to school.

7. Bar charts can be used to display discrete numerical data. Bar charts for discrete data For example, this bar graph shows the number of CDs bought by a group of children in a given month.

8. Bar charts can be used to display grouped discrete data. Bar charts for grouped discrete data For example, this bar graph shows the number of books read by a sample of people over the space of a year.

9. Two or more sets of data can be shown on a bar chart. Bar charts for two sets of data For example, this bar chart shows favourite subjects for a group of boys and girls.

10. Mental maths test results Number of pupils Mark out of ten Bar line graphs are the same as bar charts except that lines are drawn instead of bars. Bar line graphs For example, this bar line graph shows a set of test results.

11. Drawing bar charts When drawing bar chart remember: • Give the bar chart a title. • Use equal intervals on the axes. • Draw bars of equal width. • Leave a gap between each bar. • Label both the axes. • Include a key for the chart if necessary.

12. D3 Representing and interpreting data Contents D3.1 Bar charts • A1 • A1 D3.3 Frequency diagrams • A1 D3.2 Pie charts D3.4 Line graphs • A1 D3.5 Scatter graphs • A1 D3.6 Comparing data • A1

13. Pie charts A pie chart is a circle divided up into sectors which are representative of the data. In a pie chart, each category is shown as a fraction of the circle. For example, in a survey half the people asked drove to work, a quarter walked and a quarter went by bus.

14. Pie charts This pie chart shows the distribution of drinks sold in a cafeteria on a particular day. Altogether 300 drinks were sold. Estimate the number of each type of drink sold. Coffee: 75 Soft drinks: 50 Tea: 175

15. Pie charts These two pie charts compare the proportions of boys and girls in two classes. Dawn says, “There are more girls in Mrs Payne’s class than in Mr Humphry’s class.” Is she right?

16. Drawing pie charts To draw a pie chart you need compasses and a protractor. The first step is to work out the angle needed to represent each category in the pie chart. We need to work out how many degrees are needed to represent each person or thing in the sample.

17. Drawing pie charts For example, 30 people were asked which newspapers they read regularly. The results were :

18. Total Method 1 Drawing pie charts There are 30 people in the survey and 360º in a full pie chart. Each person is therefore represented by 360º ÷ 30 = 12º We can now calculate the angle for each category: 96º 8 × 12º 84º 7 × 12º 36º 3 × 12º 72º 6 × 12º 72º 6 × 12º 30 360º

19. Drawing pie charts Once the angles have been calculated you can draw the pie chart. Start by drawing a circle using compasses. Daily Express The Guardian Draw a radius. 72º Measure an angle of 96º from the radius using a protractor and label the sector. 96º 72º 84º The Sun 36º Daily Mirror Measure an angle of 84º from the the last line you drew and label the sector. The Times Repeat for each sector until the pie chart is complete.

20. Total 36 Drawing pie charts Use the data in the frequency table to complete the pie chart showing the favourite colours of a sample of people. Favourite colour No of people Pink 10 Orange 3 Blue 14 Purple 5 Green 4

21. Holiday destination No of people UK 74 Europe 53 America 32 Asia 11 Other 10 Total 180 Drawing pie charts Use the data in the frequency table to complete the pie chart showing the holiday destinations of a sample of people.

22. Smokey bacon Prawn cocktail 35º 55º Salt and vinegar 135º 85º Ready salted 360 = 50º 72 Cheese and onion Reading pie charts The following pie chart shows the favourite crisp flavours of 72 children. How many children preferred ready salted crisps? How many degrees repesents one child? 5º. The number of children who preferred ready salted is: 135 ÷ 5 = 27

23. D3 Representing and interpreting data Contents D3.1 Bar charts • A1 D3.2 Pie charts • A1 • A1 D3.3 Frequency diagrams D3.4 Line graphs • A1 D3.5 Scatter graphs • A1 D3.6 Comparing data • A1

24. Heights of Year 8 pupils 35 30 25 20 Frequency 15 10 5 0 145 150 155 160 165 170 175 140 Height (cm) Frequency diagrams are used to display grouped continuous data. Frequency diagrams For example, this frequency diagram shows the distribution of heights in a group of Year 8 pupils: The divisions between the bars are labelled.

25. D3 Representing and interpreting data Contents D3.1 Bar charts • A1 D3.2 Pie charts • A1 D3.3 Frequency diagrams • A1 D3.4 Line graphs • A1 D3.5 Scatter graphs • A1 D3.6 Comparing data • A1

26. Line graphs are most often used to show trends over time. Line graphs For example, this line graph shows the temperature in London, in ºC, over a 12-hour period.

27. This line graph compares the percentage of boys and girls gaining A* to C passes at GCSE in a particular school. Line graphs What trends are shown by this graph?

28. D3 Representing and interpreting data Contents D3.1 Bar charts • A1 D3.2 Pie charts • A1 D3.3 Frequency diagrams • A1 D3.5 Scatter graphs D3.4 Line graphs • A1 • A1 D3.6 Comparing data • A1

29. We can use scatter graphs to find out if there is any relationship or correlation between two sets of data. Scatter graphs and correlation For example, • Do tall people weigh more than small people? • If there is more rain, will it be colder? • If you revise longer, will you get better marks? • Do second-hand car get cheaper with age? • Is more electricity used in cold weather? • Are people with big heads better at maths?

30. Length of spring (cm) Mass attached to spring (g) When one variable increases as the other variable increases, we have a positive correlation. Scatter graphs and correlation For example, this scatter graph shows that there is a strong positive correlation between the length of a spring and the mass of an object attached to it. The points lie close to an upward sloping line. This is the line of best fit.

31. Science score Maths score Sometimes the points in the graph are more scattered. We can still see a trend upwards. Scatter graphs and correlation This scatter graph shows that there is a weak positive correlation between scores in a maths test and scores in a science test. The points are scattered above and below a line of best fit.

32. Temperature(°C) Rainfall (mm) When one variable decreases as the other variable increases, we have a negative correlation. Scatter graphs and correlation For example, this scatter graph shows that there is a strong negative correlation between rainfall and hours of sunshine. The points lie close to a downward sloping line of best fit.

33. Outdoor temperature (ºC) Electricity used (kWh) Scatter graphs and correlation Sometimes the points in the graph are more scattered. We can still see a trend downwards. For example, this scatter graph shows that there is a weak negative correlation between the temperature and the amount of electricity a family used.

34. Number of hours worked Age (years) Sometimes a scatter graph shows that there is no correlation between two variables. Scatter graphs and correlation For example, this scatter graph shows that there is a no correlation between a person’s age and the number of hours they work a week. The points are randomly distributed.

35. Temperature (°C) 14 16 20 19 23 21 25 22 18 18 Ice creams sold 10 14 20 22 19 22 30 15 16 19 This table shows the temperature on 10 days and the number of ice creams a shop sold. Plot the scatter graph. Plotting scatter graphs

36. Hours watching TV 2 4 3.5 2 1.5 2.5 3 5 1 0.5 Hours doing homework 2.5 0.5 0.5 2 3 2 1 0 2 3 We can use scatter graphs to find out if there is any relationship or correlation between two set of data. Plotting scatter graphs

37. D3 Representing and interpreting data Contents D3.1 Bar charts • A1 D3.2 Pie charts • A1 D3.3 Frequency diagrams • A1 D3.6 Comparing data D3.4 Line graphs • A1 D3.5 Scatter graphs • A1 • A1

38. Matt 5 7 6 5 7 8 6 Jamie 3 6 4 8 12 9 8 Comparing distributions The distribution of a set of data describes how the data is spread out. Two distributions can be compared using one of the three averages and the range. For example, the number of cars sold by two salesmen each day for a week is shown below. Who is the better salesman?

39. 3 + 6 + 4 + 8 + 12 + 9 + 8 5 + 7 + 6 + 5 + 7 + 8 + 6 7 7 = 44 50 Matt 5 7 6 5 7 8 6 7 7 Jamie 3 6 4 8 12 9 8 = Comparing distributions To decide which salesman is best let’s compare the mean number cars sold by each one. Matt: Mean = = 6.3 (to 1 d.p.) Jamie: Mean = = 7.1 (to 1 d.p.) This tells us that, on average, Jamie sold more cars each day.

40. Matt 5 7 6 5 7 8 6 Jamie 3 6 4 8 12 9 8 Comparing distributions Now let’s compare the range for each salesman. Matt: Range = 8 – 5 = 3 Jamie: Range = 12 – 3 = 9 The range for the number of cars sold each day is smaller for Matt. This means that he is a more consistent or reliable salesman. We could argue that Jamie is better because he sells more on average, or that Matt is better because he is more consistent.