Post on 30-Mar-2016
description
transcript
© Boardworks Ltd 2004 1 of 45
Mathematics
Representing and interpreting data
© Boardworks Ltd 2004 2 of 45
A1A1
A1
A1
A1
A1
Contents
Representing and interpreting data
1 Bar charts
2 Pie charts
3 Frequency diagrams
4 Line graphs
5 Scatter graphs
6 Comparing data
© Boardworks Ltd 2004 3 of 45
Categorical data
Categorical data is data that is non-numerical.
For example,
Sometimes categorical data can contain numbers.
For example,
favourite football team, eye colour, birth place.
favourite number, last digit in your telephone number, most used bus route.
© Boardworks Ltd 2004 4 of 45
Discrete and continuous data
Discrete data can only take certain values.
Continuous data comes from measuring and can take any value within a given range.
Numerical data can be discrete or continuous.
For example,
For example,
shoe sizes, the number of children in a class, the number of sweets in a packet.
the weight of a banana, the time it takes for pupils to get to school, the height of 13 year-olds.
© Boardworks Ltd 2004 5 of 45
Discrete or continuous data
© Boardworks Ltd 2004 6 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
Num
ber o
f chi
ldre
n
© Boardworks Ltd 2004 7 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
Num
ber o
f chi
ldre
n
© Boardworks Ltd 2004 8 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+
Num
ber o
f boo
ks
Number of people
© Boardworks Ltd 2004 9 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
Num
ber o
f pup
ils
GirlsBoys
© Boardworks Ltd 2004 10 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
Num
ber o
f pup
ils
© Boardworks Ltd 2004 11 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 12 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 13 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
CarWalkBus
© Boardworks Ltd 2004 14 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 15 of 45
Pie charts
These two pie charts compare the proportions of boys and girls in two classes.
Mr Humphry's class
Number ofboysNumber ofgirls
Mrs Payne's class
Number ofboysNumber ofgirls
Dawn says, “There are more girls in Mrs Payne’s class than in Mr Humphry’s class.” Is she right?
© Boardworks Ltd 2004 16 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.
We need to work out how many degrees are needed to represent each person or thing in the sample.
© Boardworks Ltd 2004 17 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 18 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 = 12ºWe can now calculate the angle for each category:
Newspaper No of people Working AngleThe Guardian 8Daily Mirror 7The Times 3The Sun 6Daily Express 6
8 × 12º 96º7 × 12º 84º3 × 12º 36º6 × 12º 72º6 × 12º 72º
Total 30 360º
© 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 salted50º
Cheese and
onion
85º
55ºSalt and vinegar
Prawn cocktail
How many children preferred ready salted crisps?
How many degrees repesents one child?
360 72 = 5º.
The number of children who preferred ready salted is:
135 ÷ 5 = 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.Fr
eque
ncy
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
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 26 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
02468
101214161820
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
pera
ture
(ºC
)
© Boardworks Ltd 2004 27 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
GirlsBoys
© Boardworks Ltd 2004 28 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 29 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 30 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 spr
ing
(cm
)
The points lie close to an upward sloping line.
This is the line of best fit.
© Boardworks Ltd 2004 31 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 32 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 33 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 34 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 o
f hou
rs w
orke
d
© Boardworks Ltd 2004 35 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 36 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 37 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 38 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 39 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 + 67 = 44
7 = 6.3 (to 1 d.p.)
Jamie:
Mean = 3 + 6 + 4 + 8 + 12 + 9 + 87 = 50
7 = 7.1 (to 1 d.p.)
This tells us that, on average, Jamie sold more cars each day.
© Boardworks Ltd 2004 40 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.
3Range = 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.