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## KS3 Mathematics

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**KS3 Mathematics**D3 Representing and interpreting data**D3 Representing and interpreting data**Contents D3.2 Pie charts D3.3 Frequency diagrams D3.1 Bar charts D3.4 Line graphs D3.5 Scatter graphs D3.6 Comparing data**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.**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.**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.**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.**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.**Give the bar chart a title.**Use equal intervals on the axes. Label both the axes. Leave a gap between each bar. Drawing bar charts When drawing bar chart remember:**Year**Number of absences 7 74 8 53 9 32 10 11 11 10 Drawing bar charts Use the data in the frequency table to complete a bar chart showing the the number of children absent from school from each year group on a particular day.**D3 Representing and interpreting data**Contents D3.1 Bar charts D3.3 Frequency diagrams D3.2 Pie charts D3.4 Line graphs D3.5 Scatter graphs D3.6 Comparing data**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.**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**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?**Drawing pie charts**To draw a pie chart you need a compass 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.**Drawing pie charts**For example, 30 people were asked which newspapers they read regularly. The results were :**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**8**7 × 360º = × 360º = 30 30 Method 2 Drawing pie charts 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 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 84º**Newspaper**No of people Working Angle 6 3 7 8 6 The Guardian 8 × 360º 30 30 30 30 30 Daily Mirror 7 × 360º The Times 3 × 360º The Sun 6 × 360º Daily Express 6 × 360º Total Method 2 Drawing pie charts These calculations can be written into the table. 96º 84º 36º 72º 72º 30 360º**Drawing pie charts**Once the angles have been calculated you can draw the pie chart. Start by drawing a circle using a compass. The 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 The 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.**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 Red 10 Yellow 3 Blue 14 Green 5 Purple 4**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.**Smokey**bacon Prawn cocktail 35º 55º Salt and vinegar 135º 85º Ready salted 135 = 50º 360 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? The proportion of children who preferred ready salted is: 0.375 The number of children who preferred ready salted is: 0.375 × 72 = 27**D3 Representing and interpreting data**Contents D3.1 Bar charts D3.2 Pie charts D3.3 Frequency diagrams D3.4 Line graphs D3.5 Scatter graphs D3.6 Comparing data**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.**Time spent (hours)**Number of people 0 ≤ h < 1 4 1 ≤ h < 2 6 2 ≤ h < 3 8 3 ≤ h < 4 5 4 ≤ h < 5 3 h ≤ 5 1 Use the data in the frequency table to complete the frequency diagram showing the time pupils spent watching TV on a particular evening: Drawing frequency diagrams**D3 Representing and interpreting data**Contents D3.1 Bar charts D3.2 Pie charts D3.3 Frequency diagrams D3.4 Line graphs D3.5 Scatter graphs D3.6 Comparing data**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.**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?**Age (years)**Weight (kg) 1 9.5 2 12.0 3 14.2 4 16.3 5 18.4 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.**D3 Representing and interpreting data**Contents D3.1 Bar charts D3.2 Pie charts D3.3 Frequency diagrams D3.5 Scatter graphs D3.4 Line graphs D3.6 Comparing data**We can use scatter graphs to find out if there is any**relationship or correlation between two set of data. Scatter graphs Handspan (cm) 18 16 20 15 16 21 19 17 20 18 Foot length (cm) 24 21 28 20 22 30 25 22 27 23**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? 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,**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.**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.**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.**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.**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.**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**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**D3 Representing and interpreting data**Contents D3.1 Bar charts D3.2 Pie charts D3.3 Frequency diagrams D3.6 Comparing data D3.4 Line graphs D3.5 Scatter graphs**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?**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.**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.**This distribution is skewed to the left.**This distribution is symmetrical (or normal). This distribution is skewed to the right. This distribution is random. Comparing the shape of distributions We can comparing distributions by looking at the shape of their graphs.**Group A**Group B Group C Group D Frequency Frequency Frequency Frequency 1-10 11-20 21-30 31-40 41-50 1-10 11-20 21-30 31-40 41-50 1-10 11-20 21-30 31-40 41-50 1-10 11-20 21-30 31-40 41-50 Four groups of pupils sat the same maths test. These graphs show the results. Comparing the shape of distributions 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.