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Transcript of © Boardworks Ltd 2004 1 of 45 KS3 Mathematics D3 Representing and interpreting data.
© Boardworks Ltd 2004 1 of 45
KS3 Mathematics
D3 Representing and interpreting data
© Boardworks Ltd 2004 2 of 45
A1A1
A1
A1
A1
A1
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
© Boardworks Ltd 2004 3 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.
How children travel to school
0
2
4
6
8
10
12
walk train car bicycle bus other
Method of travel
Nu
mb
er
of
ch
ild
ren
© Boardworks Ltd 2004 4 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.
Number of CDs bought in a month
0
5
10
15
20
25
0 1 2 3 4 5
Number of CDs bought
Nu
mb
er
of
ch
ild
ren
© Boardworks Ltd 2004 5 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.
Books read in one year
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
0-3
4-7
8-11
12-15
16-19
20+
Nu
mb
er o
f b
oo
ks
Number of people
© Boardworks Ltd 2004 6 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.
Girls' and boys' favourite subjects
0
1
2
3
4
5
6
7
8
Maths Science English History PE
Favourite subject
Nu
mb
er o
f p
up
ils
Girls
Boys
© Boardworks Ltd 2004 7 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
Nu
mb
er o
f p
up
ils
© Boardworks Ltd 2004 8 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.
© Boardworks Ltd 2004 9 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.
YearNumber of absences
7 74
8 53
9 32
10 11
11 10
© Boardworks Ltd 2004 10 of 45
A1A1
A1
A1
A1
A1
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
© Boardworks Ltd 2004 11 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.
Methods of travel to work
Car
Walk
Bus
© Boardworks Ltd 2004 12 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
Drinks sold in a cafeteria
coffeesoft drinkstea
© Boardworks Ltd 2004 13 of 45
Pie charts
These two pie charts compare the proportions of boys and girls in two classes.
Mr Humphry's class
Number ofboys
Number ofgirls
Mrs Payne's class
Number ofboys
Number ofgirls
Dawn says, “There are more girls in Mrs Payne’s class than in Mr Humphry’s class.” Is she right?
© Boardworks Ltd 2004 14 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.
© Boardworks Ltd 2004 15 of 45
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 6
© Boardworks Ltd 2004 16 of 45
Drawing pie charts
Method 1There 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 6
8 × 12º 96º
7 × 12º 84º
3 × 12º 36º
6 × 12º 72º
6 × 12º 72º
Total 30 360º
© Boardworks Ltd 2004 17 of 45
Drawing pie charts
Method 2Write 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
830 × 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
730 × 360º = 84º
© Boardworks Ltd 2004 18 of 45
Total
AngleWorkingNo of peopleNewspaper
6Daily Express
6The Sun
3The Times
7Daily Mirror
8The Guardian
Drawing pie charts
Method 2These calculations can be written into the table.
96º
84º
36º
72º
72º
30 360º
× 360º830
× 360º730
× 360º330
× 360º630
× 360º630
© Boardworks Ltd 2004 19 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
72º
The Sun
Daily Express
© Boardworks Ltd 2004 20 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
10
3
14
5
4
Favourite colour
Pink
Orange
Blue
Purple
Green
Total 36
© Boardworks Ltd 2004 21 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
UK 74
Europe 53
America 32
Asia 11
Other 10
Total 180
© Boardworks Ltd 2004 22 of 45
Reading pie charts
The following pie chart shows the favourite crisp flavours of 72 children.
35º
Smokeybacon
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:
135360
= 0.375
The number of children who preferred ready salted is:
0.375 × 72 = 27
© Boardworks Ltd 2004 23 of 45
A1A1
A1
A1
A1
A1
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
© Boardworks Ltd 2004 24 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.
Fre
qu
enc
y
Height (cm)
0
5
10
15
20
25
30
35
140 145 150 155 160 165 170 175
Heights of Year 8 pupils
© Boardworks Ltd 2004 25 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 < 1 4
1 ≤ h < 2 6
2 ≤ h < 3 8
3 ≤ h < 4 5
4 ≤ h < 5 3
h ≤ 5 1
© Boardworks Ltd 2004 26 of 45
Contents
D3 Representing and interpreting data
A1A1
A1
A1
A1
A1
D3.4 Line graphs
D3.3 Frequency diagrams
D3.2 Pie charts
D3.1 Bar charts
D3.5 Scatter graphs
D3.6 Comparing data
© Boardworks Ltd 2004 27 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.
Temperature in London
0
2
4
6
8
1012
14
16
18
20
6 am 7 am 8 am 9 am 10 am 11 am 12 pm 1 pm 2 pm 3 pm 4 pm 5 pm 6 pm
Time
Tem
per
atu
re (
ºC)
© Boardworks Ltd 2004 28 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?
Percentage of boys and girls gaining A* to C passes at GCSE
0
10
20
30
40
50
60
70
1998 1999 2000 2001 2002 2003 2004
Girls
Boys
© Boardworks Ltd 2004 29 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)
1 9.5
2 12.0
3 14.2
4 16.3
5 18.4
© Boardworks Ltd 2004 30 of 45
Contents
D3 Representing and interpreting data
A1A1
A1
A1
A1
A1
D3.5 Scatter graphs
D3.4 Line graphs
D3.3 Frequency diagrams
D3.2 Pie charts
D3.1 Bar charts
D3.6 Comparing data
© Boardworks Ltd 2004 31 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)
18
24
16
21
20
28
15
20
16
22
21
30
19
25
17
22
20
27
18
23
© Boardworks Ltd 2004 32 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?
© Boardworks Ltd 2004 33 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)
Leng
th o
f sp
ring
(cm
)
The points lie close to an upward sloping line.
This is the line of best fit.
© Boardworks Ltd 2004 34 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
Sci
ence
sco
re
The points are scattered above and below a line of best fit.
© Boardworks Ltd 2004 35 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)
Tem
pera
ture
(°C
)
The points lie close to a downward sloping line of best fit.
© Boardworks Ltd 2004 36 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)
Out
door
tem
pera
ture
(ºC
)
We can still see a trend downwards.
© Boardworks Ltd 2004 37 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)
Num
ber
of h
ours
wor
ked
© Boardworks Ltd 2004 38 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
14
10
16
14
20
20
19
22
23
19
21
22
25
30
22
15
18
16
18
19
© Boardworks Ltd 2004 39 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
2
2.5
4
0.5
3.5
0.5
2
2
1.5
3
2.5
2
3
1
5
0
1
2
0.5
3
© Boardworks Ltd 2004 40 of 45
Contents
D3 Representing and interpreting data
A1A1
A1
A1
A1
A1
D3.6 Comparing data
D3.5 Scatter graphs
D3.4 Line graphs
D3.3 Frequency diagrams
D3.2 Pie charts
D3.1 Bar charts
© Boardworks Ltd 2004 41 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
5
3
7
6
6
4
5
8
7
12
8
9
6
8
Who is the better salesman?
© Boardworks Ltd 2004 42 of 45
Comparing distributions
To decide which salesman is best let’s compare the mean number cars sold by each one.
Matt
Jamie
5
3
7
6
6
4
5
8
7
12
8
9
6
8
Matt:
Mean =5 + 7 + 6 + 5 + 7 + 8 + 6
7=
447
= 6.3 (to 1 d.p.)
Jamie:
Mean =3 + 6 + 4 + 8 + 12 + 9 + 8
7=
507
= 7.1 (to 1 d.p.)
This tells us that, on average, Jamie sold more cars each day.
© Boardworks Ltd 2004 43 of 45
Comparing distributions
Now let’s compare the range for each salesman.
Matt
Jamie
5
3
7
6
6
4
5
8
7
12
8
9
6
8
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.
© Boardworks Ltd 2004 44 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).
© Boardworks Ltd 2004 45 of 45
Comparing the shape of distributions
Four groups of pupils sat the same maths test. These graphs show the results.
Group A
Fre
quen
cy
1-10 11-20 21-30 31-40 41-50
Group B
1-10 11-20 21-30 31-40 41-50
Fre
quen
cyGroup C
1-10 11-20 21-30 31-40 41-50
Fre
quen
cy
Group D
1-10 11-20 21-30 31-40 41-50
Fre
quen
cy
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.