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© Boardworks Ltd of 45 KS3 Mathematics D3 Representing and interpreting data

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© Boardworks Ltd of 45 A1A1 A1A1 A1A1 A1A1 A1A1 A1A1 Contents D3 Representing and interpreting data D3.1 Bar charts D3.2 Pie charts D3.3 Frequency diagrams D3.4 Line graphs D3.5 Scatter graphs D3.6 Comparing data

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© Boardworks Ltd of 45 Bar charts for categorical data Bar charts can be used to display categorical or non- numerical data. For example, this bar graph shows how a group of children travel to school.

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© Boardworks Ltd of 45 Bar charts for discrete data Bar charts can be used to display discrete numerical data. For example, this bar graph shows the number of CDs bought by a group of children in a given month.

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© Boardworks Ltd of 45 Bar charts for grouped discrete data Bar charts can be used to display 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.

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© Boardworks Ltd of 45 Bar charts for two sets of data Two or more sets of data can be shown on a bar chart. For example, this bar chart shows favourite subjects for a group of boys and girls.

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© Boardworks Ltd of 45 Bar line graphs Bar line graphs are the same as bar charts except that lines are drawn instead of bars. For example, this bar line graph shows a set of test results. Mental maths test results Mark out of ten Number of pupils

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© Boardworks Ltd of 45 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.

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© Boardworks Ltd of 45 Drawing bar charts Use the data in the frequency table to complete a bar chart showing the number of children absent from school from each year group on a particular day. Year Number of absences

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© Boardworks Ltd of 45 A1A1 A1A1 A1A1 A1A1 A1A1 A1A1 Contents D3 Representing and interpreting data D3.2 Pie charts D3.1 Bar charts D3.3 Frequency diagrams D3.4 Line graphs D3.5 Scatter graphs D3.6 Comparing data

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© Boardworks Ltd of 45 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.

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© Boardworks Ltd of 45 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

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© Boardworks Ltd of 45 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?

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© Boardworks Ltd of 45 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. There are two ways to do this. The first is to work out how many degrees are needed to represent each person or thing in the sample. The second method is to work out what fraction of the total we want to represent and multiply this by 360 degrees.

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© Boardworks Ltd of 45 Drawing pie charts For example, 30 people were asked which newspapers they read regularly. The results were : NewspaperNumber of people The Guardian8 Daily Mirror7 The Times3 The Sun6 Daily Express6

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© Boardworks Ltd of 45 Drawing pie charts Method 1 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: NewspaperNo of peopleWorkingAngle The Guardian8 Daily Mirror7 The Times3 The Sun6 Daily Express6 8 × 12º 96º 7 × 12º 84º 3 × 12º 36º 6 × 12º 72º 6 × 12º 72º Total 30360º

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© Boardworks Ltd of 45 Drawing pie charts Method 2 Write each category as a fraction of the whole and find this fraction of 360º. 8 out of the 30 people in the survey read The Guardian so to work out the size of the sector we calculate 8 30 × 360º = 96º 7 out of the 30 people in the survey read the Daily Mirror so to work out the size of the sector we calculate 7 30 × 360º = 84º

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© Boardworks Ltd of 45 Total AngleWorkingNo of peopleNewspaper 6 Daily Express 6 The Sun 3 The Times 7 Daily Mirror 8 The Guardian Drawing pie charts Method 2 These calculations can be written into the table. 96º 84º 36º 72º 30360º × 360º 8 30 × 360º 7 30 × 360º 3 30 × 360º 6 30 × 360º 6 30

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© Boardworks Ltd of 45 Drawing pie charts Once the angles have been calculated you can draw the pie chart. Start by drawing a circle using compasses. Draw a radius. Measure an angle of 96º from the radius using a protractor and label the sector. 96º The Guardian Measure an angle of 84º from the the last line you drew and label the sector. 84º Daily Mirror Repeat for each sector until the pie chart is complete. 36º The Times 72º The Sun Daily Express

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© Boardworks Ltd of 45 Drawing pie charts Use the data in the frequency table to complete the pie chart showing the favourite colours of a sample of people. No of people Favourite colour Pink Orange Blue Purple Green Total36

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© Boardworks Ltd of 45 Drawing pie charts Use the data in the frequency table to complete the pie chart showing the holiday destinations of a sample of people. Holiday destination No of people UK74 Europe53 America32 Asia11 Other10 Total180

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© Boardworks Ltd of 45 Reading pie charts The following pie chart shows the favourite crisp flavours of 72 children. 35º Smokey bacon 135º Ready salted 50º Cheese and onion 85º 55º Salt and vinegar Prawn cocktail How many children preferred ready salted crisps? The proportion of children who preferred ready salted is: = The number of children who preferred ready salted is: × 72 =27

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© Boardworks Ltd of 45 A1A1 A1A1 A1A1 A1A1 A1A1 A1A1 Contents D3 Representing and interpreting data D3.3 Frequency diagrams D3.2 Pie charts D3.1 Bar charts D3.4 Line graphs D3.5 Scatter graphs D3.6 Comparing data

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© Boardworks Ltd of 45 Frequency diagrams Frequency diagrams are used to display grouped continuous data. For example, this frequency diagram shows the distribution of heights in a group of Year 8 pupils: The divisions between the bars are labelled. Frequency Height (cm) Heights of Year 8 pupils

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© Boardworks Ltd of 45 Drawing frequency diagrams Use the data in the frequency table to complete the frequency diagram showing the time pupils spent watching TV on a particular evening: Time spent (hours) Number of people 0 ≤ h < 14 1 ≤ h < 26 2 ≤ h < 38 3 ≤ h < 45 4 ≤ h < 53 h ≤ 51

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© Boardworks Ltd of 45 Contents D3 Representing and interpreting data A1A1 A1A1 A1A1 A1A1 A1A1 A1A1 D3.4 Line graphs D3.3 Frequency diagrams D3.2 Pie charts D3.1 Bar charts D3.5 Scatter graphs D3.6 Comparing data

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© Boardworks Ltd of 45 Line graphs Line graphs are most often used to show trends over time. For example, this line graph shows the temperature in London, in ºC, over a 12-hour period.

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© Boardworks Ltd of 45 Line graphs This line graph compares the percentage of boys and girls gaining A* to C passes at GCSE in a particular school. What trends are shown by this graph?

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© Boardworks Ltd of 45 Drawing line graphs This data shows the weight of a child taken every birthday. Plot the points on the graph and join them with straight lines. Age (years) Weight (kg)

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© Boardworks Ltd of 45 Contents D3 Representing and interpreting data A1A1 A1A1 A1A1 A1A1 A1A1 A1A1 D3.5 Scatter graphs D3.4 Line graphs D3.3 Frequency diagrams D3.2 Pie charts D3.1 Bar charts D3.6 Comparing data

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© Boardworks Ltd of 45 Scatter graphs We can use scatter graphs to find out if there is any relationship or correlation between two sets of data. Hand span (cm) Foot length (cm)

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© Boardworks Ltd of 45 Scatter graphs and correlation We can use scatter graphs to find out if there is any relationship or correlation between two sets of data. For example, If you revise longer, will you get better marks? Do second-hand car get cheaper with age? Are people with big heads better at maths? Do tall people weigh more than small people? Is more electricity used in cold weather? If there is more rain, will it be colder?

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© Boardworks Ltd of 45 Scatter graphs and correlation When one variable increases as the other variable increases, we have a positive 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. Mass attached to spring (g) Length of spring (cm) The points lie close to an upward sloping line. This is the line of best fit.

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© Boardworks Ltd of 45 Scatter graphs and correlation Sometimes the points in the graph are more scattered. We can still see a trend upwards. This scatter graph shows that there is a weak positive correlation between scores in a maths test and scores in a science test. Maths score Science score The points are scattered above and below a line of best fit.

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© Boardworks Ltd of 45 Scatter graphs and correlation When one variable decreases as the other variable increases, we have a negative correlation. For example, this scatter graph shows that there is a strong negative correlation between rainfall and hours of sunshine. Rainfall (mm) Temperature(°C) The points lie close to a downward sloping line of best fit.

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© Boardworks Ltd of 45 Scatter graphs and correlation Sometimes the points in the graph are more scattered. For example, this scatter graph shows that there is a weak negative correlation between the temperature and the amount of electricity a family used. Electricity used (kWh) Outdoor temperature (ºC) We can still see a trend downwards.

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© Boardworks Ltd of 45 Scatter graphs and correlation Sometimes a scatter graph shows that there is no correlation between two variables. 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. Age (years) Number of hours worked

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© Boardworks Ltd of 45 Plotting scatter graphs This table shows the temperature on 10 days and the number of ice creams a shop sold. Plot the scatter graph. Temperature (°C) Ice creams sold

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© Boardworks Ltd of 45 Plotting scatter graphs We can use scatter graphs to find out if there is any relationship or correlation between two set of data. Hours watching TV Hours doing homework

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© Boardworks Ltd of 45 Contents D3 Representing and interpreting data A1A1 A1A1 A1A1 A1A1 A1A1 A1A1 D3.6 Comparing data D3.5 Scatter graphs D3.4 Line graphs D3.3 Frequency diagrams D3.2 Pie charts D3.1 Bar charts

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© Boardworks Ltd of 45 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. Matt Jamie Who is the better salesman?

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© Boardworks Ltd of 45 Comparing distributions To decide which salesman is best let’s compare the mean number cars sold by each one. Matt Jamie Matt: Mean = = 44 7 = 6.3 (to 1 d.p.) Jamie: Mean = = 50 7 = 7.1 (to 1 d.p.) This tells us that, on average, Jamie sold more cars each day.

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© Boardworks Ltd of 45 Comparing distributions Now let’s compare the range for each salesman. Matt Jamie Matt:Range =8 – 5 = Jamie: 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. 3 Range =12 – 3 =9 We could argue that Jamie is better because he sells more on average, or that Matt is better because he is more consistent.

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© Boardworks Ltd of 45 Comparing the shape of distributions We can compare distributions by looking at the shape of their graphs. This distribution is skewed to the left. This distribution is skewed to the right. This distribution is random. This distribution is symmetrical (or normal).

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© Boardworks Ltd of 45 Comparing the shape of distributions Four groups of pupils sat the same maths test. These graphs show the results. Group A Frequency Group B Frequency Group C Frequency Group D Frequency One of the groups is a top set, one is a middle set, one is a bottom set and one is a mixed ability group. Use the shapes of the distribution to decide which group is which giving reasons for your choice.

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