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**D3 Representing and interpreting data**

KS3 Mathematics The aim of this unit is to teach pupils to: Construct graphs and diagrams to represent data, on paper and using ICT. Interpret diagrams and graphs, and draw inferences. Compare two simple distributions using the range, mode, mean or median. Material in this unit is linked the Framework’s supplement of examples pp262 –275. D3 Representing and interpreting data

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**D3 Representing and interpreting data**

Contents D3 Representing and interpreting data A1 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

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**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. Draw pupils’ attention to the gaps between the bars. The height of each bar represents the frequency for that category. Ask questions based on this data. For example, What is the modal method of transport? How many of the pupils questioned travelled by train? How many children took part in this survey? The chart can be modified by double clicking on it.

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**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. Remind pupils that discrete data is numerical data that can only take certain values. For example, we couldn’t have 2.3 CDs. The chart can be modified by double clicking on it.

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**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. This example shows a horizontal bar chart. The length of each bar represents the frequency for that class interval. Ask pupils if we can tell from this chart how many people didn’t read a book in the previous year. The chart can be modified by double clicking on it.

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**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. Stress that we must include a key when more than one type of data is displayed in the same chart. Ask, What subject did most girls like the best? What subject did most boys like the best? Is it possible to tell if an equal number of boys and girls took part in the survey? The chart can be modified by double clicking on it.

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**Mental maths test results**

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 A bar line graph is more like a bar chart than like a line graph. The heights of the lines represent the frequency. Ask, What is the modal mark out of ten? How many pupils sat the test? (59) Is it possible to work out the median score for this data? (Identify the 30th mark as 6) Is it possible to work out the mean for this data? [We could multiply the frequency for each mark by the mark and add these together to get the total score, (1 × 0) + (2 × 2) + (4 × 3) + (9 × 4) + (8 × 5) + (12 × 6) + (10 × 7) + (7 × 8) + (4 × 9) + (2 × 10) = 346. Dividing this by the total number of pupils gives 346 ÷ 59 = 5.86 to 2 d. p.]

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**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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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 7 74 8 53 9 32 10 11 Start by giving the chart a title and labelling both axes. Then decide on a suitable scale for the vertical axis. For example use each division to represent ten pupils. Number this axis using the pen tool. Ask volunteers to drag each bar to the required frequency. Copy this slide and modify the table to produce more examples if required.

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**D3 Representing and interpreting data**

Contents D3 Representing and interpreting data A1 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

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**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. In a bar chart, the size of each category is compared with each of the others. In a pie chart, each category is compared with the whole. Point out that if the sectors are not labelled we must include a key.

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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 Ask pupils to estimate the fraction shown by each sector. The angle of the sector representing coffee is a right angle. It must therefore represent a quarter of the pie chart. A quarter of 300 is 75. The angle of the sector representing soft drinks looks like a 60º angle or a sixth of the pie chart. A sixth of 300 is 50. The rest of the drinks must be tea. So, using the answers we have already, 300 – 75 – 50 = 175. Explain that if we had a protractor we could find the actual angles and by writing these as fractions of 360º we could calculate the required proportions. Soft drinks: 50 Tea: 175

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Pie charts These two pie charts compare the proportions of boys and girls in two classes. Stress that pie charts compare proportions and not actual amounts. The pie charts in the example show that 2/5 of the pupils in Mr Humphry’s class are girls and 3/5 of the pupils in Mrs Payne's class are girls. We can conclude from this that a higher proportion of the pupils in Mrs Payne’s class are girls. However, unless we are told otherwise we cannot assume that there are the same number of pupils in each class. Suppose, for example that there are 30 pupils in Mr Humphry’s class and 15 pupils in Mrs Payne’s class. 2/5 of 30 is 12, so that would give us 12 girls in Mr Humphry’s class. 3/5 of 15 is 9, so that would give us 9 girls in Mrs Payne’s class. Conclude that we cannot make statement about the actual amounts in a pie chart unless we are told how many the pie chart represents. Dawn says, “There are more girls in Mrs Payne’s class than in Mr Humphry’s class.” Is she right?

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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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Drawing pie charts For example, 30 people were asked which newspapers they read regularly. The results were : Newspaper Number of people The Guardian 8 Daily Mirror 7 The Times 3 The Sun 6 Daily Express

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**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: Newspaper No of people Working Angle The Guardian 8 Daily Mirror 7 The Times 3 The Sun 6 Daily Express 8 × 12º 96º 7 × 12º 84º 3 × 12º 36º Talk through the first method. This method works well if the number of people in the survey (or whatever the pie chart is being used to represent) divides exactly into 360°. Once we know how many degrees represent each person we can multiply this amount by the frequency. Stress that we should check that the angles add up to 360º. (Although, in cases where the angles have been rounded there is the possibility that the angles won’t add up to 360º.) 6 × 12º 72º 6 × 12º 72º Total 30 360º

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**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 In the second method, we find the fraction needed for each category and then find this fraction of 360°. 7 30 × 360º = 84º

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**Drawing pie charts Method 2**

These calculations can be written into the table. Angle Working No of people Newspaper 6 Daily Express The Sun 3 The Times 7 Daily Mirror 8 The Guardian × 360º 8 30 96º × 360º 7 30 84º × 360º 3 30 36º × 360º 6 30 72º × 360º 6 30 72º Total 30 360º

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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. Measure an angle of 96º from the radius using a protractor and label the sector. 72º 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.

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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 Start by working out the total number of people to be represented by the data. There are 36 people altogether so conclude that it would be better to use the total number of degrees representing each person (because 36 divides easily into 360º) to calculate the angle for each sector. 360º ÷ 36 = 10º so each person is represented by a 10º angle. The angle required to represent the number of people who prefer pink is therefore 10 × 10º = 100º. Ask a volunteer to show this angle on one of the sectors in the pie chart and use the pen tool to label it red. Continue until the pie chart is complete. Purple 5 Green 4 Total 36

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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 UK 74 Europe 53 America 32 Asia 11 Other 10 Start by working out the total number of people to be represented by the data. Use this to complete the pie chart. Copy this slide and modify the table to produce more examples if required. Total 180

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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: 135 360 = 0.375 Discuss the method shown on the board. An alternative method would be to divide 360° by 72 to get 5º. That means that every 5º represents one child. 135º therefore represents 135 ÷ 5 = 27 children. This alternative method works well when the total number represented by the pie chart divides easily into 360º. Ask pupils to calculate the number of children that preferred each of the other types of crisp. Encourage them to add the numbers up to check that they add to 72. The number of children who preferred ready salted is: 0.375 × 72 = 27

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**D3 Representing and interpreting data**

Contents D3 Representing and interpreting data A1 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

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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: Frequency Height (cm) 5 10 15 20 25 30 35 140 145 150 155 160 165 170 175 Heights of Year 8 pupils The divisions between the bars are labelled. Point out that a frequency diagram is very similar to a bar chart except that the bars touch each other and the divisions between the bars are labelled. Ask pupils to give the modal class interval for the data on the board. Discuss the type of data that would be shown in a frequency diagram. For example, time taken to run a race, foot length, weights etc. In other words, anything that is measured.

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**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 < 1 4 1 ≤ h < 2 6 2 ≤ h < 3 8 3 ≤ h < 4 5 4 ≤ h < 5 3 h ≤ 5 1 Start by discussing how to label the axes and name the cart. Then ask how to number the vertical axis. Use the pen tool to do this. Drag the first bar to the appropriate height. Ask a volunteer to drag the next bar and continue until the bars are complete. Finally, ask a volunteer to use the pen tool to show how the horizontal axis should be numbered. Copy this slide and modify the table to produce more examples if required.

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**D3 Representing and interpreting data**

Contents D3 Representing and interpreting data A1 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

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**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. Explain that in a line graph points are plotted and then joined together using a ruler. Ask pupils questions based on the line graph shown. For example, What is the temperature at 10 am? What is the range of the recorded temperatures over the 12-hour period? Can we use the graph to work out the exact temperature at 9 am? Can we use the graph to work out the maximum temperature during the day? We can only work out the maximum recorded temperature. Since the temperature was only recorded every two hours, the maximum temperature may have been reached before or after 2 pm. Stress that any value that is taken from the line joining two points is only an estimate of the value at that point.

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Line graphs This line graph compares the percentage of boys and girls gaining A* to C passes at GCSE in a particular school. Stress that when two sets of data are shown on a graph we must label the lines or provide a key. What trends are shown by this graph?

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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) 1 9.5 2 12.0 3 14.2 4 16.3 5 18.4 Start by labelling and numbering the axes. Ask volunteers to plot the points and join them. Ask if it would make sense to join the first point to the origin. Conclude that if we did this we would by saying that when the child was 0 (newborn) it had no weight. We therefore start the line from the point (1, 9.5) because we have not been given the child’s birth weight. Ask pupils if the intermediate points have any significance. Conclude that since the measurements were taken on the child’s birthday, the points are one year apart. This mean that we can predict the child’s age at any point on the line. For example, ask pupils to use the graph to tell you the approximate weight of the child at the age of 3½. Also, ask them to estimate the child age when they weighed 13kg. Copy this slide and modify the table to produce more examples if required.

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**D3 Representing and interpreting data**

Contents D3 Representing and interpreting data A1 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

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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) 18 16 20 15 16 21 19 17 20 18 Foot length (cm) 24 21 28 20 22 30 25 22 27 23 Explain that the table shows the hand span and foot length of ten children and that we can plot the hand span against the foot length for each child. Start by deciding on an appropriate labels and scale for the axes by considering the range for each measurement. Number the axes using the pen tool and ask volunteers to come to the board and plot the points. When the point have been plotted ask pupils if we can deduce anything from this graph. Suggest that we might get a better overall picture if there was more data.

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**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, 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? For each example, discuss the data that would have to be collected and what you would expect the result to be. Is more electricity used in cold weather? Are people with big heads better at maths?

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**Scatter graphs and correlation**

When one variable increases as the other variable increases, we have a positive correlation. Mass attached to spring (g) Length of spring (cm) 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. Establish that this graph shows that the more mass is attached to the spring, the longer it stretches. You may like to point out that if we had a perfect positive correlation, all of the points would lie on the same straight line. It this case there would be a mathematical formula linking the two variables. The length of a spring and the mass attached to it may have a perfect correlation. However, other factors such as errors in measurements, mean that when we conduct an experiment the points do not usually lie on the same straight line. Link: A6 Real-life graphs – plotting graphs The points lie close to an upward sloping line. This is the line of best fit.

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**Scatter graphs and correlation**

Sometimes the points in the graph are more scattered. We can still see a trend upwards. Maths score Science score This scatter graph shows that there is a weak positive correlation between scores in a maths test and scores in a science test. Establish that this graph shows that, in general, the better a pupil is at maths, the better they are at science. The points are scattered above and below a line of best fit.

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**Scatter graphs and correlation**

When one variable decreases as the other variable increases, we have a negative correlation. Rainfall (mm) Temperature(°C) For example, this scatter graph shows that there is a strong negative correlation between rainfall and hours of sunshine. Establish that this graph shows that the more rain there is, the colder it is. In other words, the lower the temperature the greater the rainfall. The points lie close to a downward sloping line of best fit.

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**Scatter graphs and correlation**

Sometimes the points in the graph are more scattered. Electricity used (kWh) Outdoor temperature (ºC) 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. Establish that this graph shows that the colder it is, the more electricity the family uses. Ask pupils to explain why the family might use more electricity on colder days.

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**Scatter graphs and correlation**

Sometimes a scatter graph shows that there is no correlation between two variables. Age (years) Number of hours worked 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. Ask pupils to show a point that could represent a younger person who works long hours. What about a point that could represent an older person who works for only a few hours? A younger person who works for only a few hours? An older person who works long hours? The points are randomly distributed.

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**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 14 10 16 20 19 22 23 21 25 30 15 18 Ask pupils to predict whether or not there will be any correlation shown by the data. Decide on labels and an appropriate scale for the axes by considering the range for each measurement and number the axes using the pen tool. Ask volunteers to come to the board and plot the points from the table on the diagram. Decide if there is any correlation between the values and use the pen tool (set to draw straight lines) to draw a line of best fit.

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**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 2 2.5 4 0.5 3.5 1.5 3 1 5 Ask pupils to predict whether or not there will be any correlation shown by the data. Decide on labels and an appropriate scale for the axes by considering the range for each measurement and number the axes using the pen tool. Ask volunteers to come to the board and plot the points from the table on the diagram. Decide if there is any correlation between the values. Use the pen tool (set to draw straight lines) to draw a line of best fit, if appropriate. Copy this slide and modify the table to produce more examples if required.

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**D3 Representing and interpreting data**

Contents D3 Representing and interpreting data A1 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

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**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 5 3 7 6 4 8 12 9 Ask pupils who they think is the best and why? Who is the better salesman?

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**Comparing distributions**

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

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**Comparing distributions**

Matt Jamie 5 3 7 6 4 8 12 9 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. A small range means that the values are less likely to fluctuate. An employer may prefer a salesman who can be relied upon to sell a minimum number of cars each day. Although, Matt sold 12 cars on one day, he also had two ‘bad days’ where he only sold 3 and 4 cars respectively. Stress to pupils that we could argue that either one of them is better. The important point is to provide a reason for the choice using the data. 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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**Comparing the shape of distributions**

We can compare distributions by looking at the shape of their graphs. This distribution is symmetrical (or normal). This distribution is skewed to the left. This distribution is skewed to the right. This distribution is random. Ask pupils how they think the shapes of these distributions would affect the mean for each one. Ask them if they think one of the other averages would be more appropriate for skewed data.

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**Comparing the shape of distributions**

Four groups of pupils sat the same maths test. These graphs show the results. Group A Frequency 1-10 11-20 21-30 31-40 41-50 Group B 1-10 11-20 21-30 31-40 41-50 Frequency Group C 1-10 11-20 21-30 31-40 41-50 Frequency Group D 1-10 11-20 21-30 31-40 41-50 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. Ask pupils what sort of data is shown in the graphs (grouped discrete data). Discuss how we can use the shape of each graph to establish the ability of each group. Ask pupils what they would expect the mean percentage for each group to be. Group A has a similar number of pupils for each interval. This is most likely to be the mixed ability group. The mean score for this group will be around 50%. Group B has more pupils gaining lower scores. The data is skewed to the left. This is most likely to be the bottom set. The mean score for this group will be around 25%. Group C has more pupils gaining higher scores. The data is skewed to the right. This is most likely to be the top set. The mean score for this group will be around 75%. Group D has more pupils gaining average scores. This is most likely to be the middle set. The mean score for this group will be around 50%. Notice that the mean scores for group A and group D are likely to be very similar although their distributions are very different. They may also have a similar range. This demonstrates that finding the average and the range is sometimes not sufficient to get a clear idea of how the data is distributed. This is why looking at the shape of a distribution is important. Use the shapes of the distribution to decide which group is which giving reasons for your choice.

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