Representation and Interpretation of Data

Representation and
Interpretation
of Data
Doreen Drews
1
Glossary
Data collection sheet: A way of collecting and organising data, a complex example being a
questionnaire.
‘Frequency’: The number of times an event occurs.
Tables and lists: With headings, columns and rows.
Block graphs: A form of pictorial representation where the data is represented by blocks which
form columns. Used with discrete data.
Bar charts: A form of pictorial representation where the data is represented by bar columns.
The height (length) of each bar indicates its value. Can be used with discrete or continuous
data.
Bar-line graph: Same as a bar chart but uses lines instead of columns.
Line graph: Data is ‘plotted’ on a graph and the individual points are joined with a line or
lines.
Pictograms: Data represented by a picture or diagram. The simplest form has one picture to
represent one object.
Mapping diagrams: Show the relations between members of different sets. Arrows used to
show the relationship.
Scatter graphs: Used to show two different aspects of collected data. Usually used with
independent variables about the same thing, eg height and weight.
2
Class intervals: Grouping data into equal ‘classes’ to give a clearer idea of the distribution e.g.
Marks out of 40 may be grouped as 1- 10, 11 –20 etc.
Venn diagrams: A diagram used to show the relationship between sets.
Carroll diagram: A diagram used to classify data; definition on one side with not the definition
on the other.
Tree diagram: Used for decision making and shows possible outcomes of events.
Mean: The idea of ‘fair shares’. The mean average is found by totalling the data and then
dividing by the number of pieces of data that there are.
Mode: What there is most of.
Median: The middle one.
Range: The range of a set of data is the difference between the greatest and the smallest value
in the set.
Discrete data: The data stands alone and is not related to each other.
Continuous data: Without a break in values. Usually arises with measurement e.g. heights.
When represented in bar chart the columns should be adjacent (touching).
Axis (p1 axes): One of the reference lines on a graph.
3
Collecting Data
Encourage children to design their own data collection sheet and evaluate it
ie
could we collect the information in another way?
Is this the best way of writing it down?
-
-
(pictures: R Thompson)
Children should be taught how to tally when dealing with large numbers
cars
bicycles
lorries
4
Mapping Diagrams
A mapping is a relation in which:1)
Each member of the first set must be used once and only once.
2)
To each member of the first set there corresponds one and only one member of the
second set.
3)
It is not essential for each member of the second set to be used.
a)
One-to-one mapping
The arrow reads
‘is the sister of’.
There is only one arrow leaving each member of the first set and only one arriving at each
member of the second set. This is called a one-to one correspondence or mapping.
b)
Many-to-one mapping
Our Favourite Fruit
The arrow reads ‘prefers this fruit’. There is only one arrow leaving each member of the first
set but more than one arriving at some of the members of the second set. This is called
a many-to one mapping.
Other Relations
One-to-many relation
Many-to-many relation
Neither of the
above adhere to the definitions given for mapping diagrams, but come up under the broad
category of relation diagrams.
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Pictograms
a)
One picture representing one frequency of that data.
b)
One
a group of units.
picture (symbol) representing
How we travel to school
walk
by car
by bus
by bicycle
Here the half symbol is seen to represent a specific amount
c)
One picture (symbol) representing a group of units where the partial representation of
the symbols represents less than the number.
Raffle tickets sold by each class
20 tickets
=
less than 20 tickets.
Class A
Class B
Class C
Class D
Notice that in this type of pictogram specific amounts cannot always be counted exactly.
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Block graphs. Bar Charts and Bar Lines
Usually used to represent discrete data (not relating to each other) e.g. colour, pets, vehicles
this means that there should be gaps between the blocks.
-
Children’s Favourite
Blocks should always be the same size. Block graphs most appropriate when representing
small amounts of data. The frequency axis (either vertical or horizontal) often has the numeral
‘inside the square’ rather than ‘on the line’.
Bar Charts
A bar chart requires that the data is collected before being drawn, unlike a block graph which
can be built up ‘as you go along’.
Instead of drawing separate symbols, long oblongs (bars) of the same width can represent the
data. It is the length (height) of the column which ‘catches the eye’ and gives the required
information. Children need to be introduced to the notion of scaling on the frequency axis
when using larger amounts of data.
The columns can be separate when using discrete data which has no natural sequence between
the bars
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How our parents travel to work
The columns can be adjacent when
order.
Types of travel
using data which is represented in an
Number of Children
Our shoe sizes
NB
Shoe sizes are discrete data as there are gaps between the sizes. The columns in the
above bar chart are ‘touching’ to indicate the order of shoe size.
9
Bar Lines
This type of graph is made up of lines instead of oblongs and the lines may be arranged
vertically or horizontally.
Bar-line graph of favourite comics
Disneytime
Supergirl
Zappo
Beano
_____________________________________
2
4
6
Number of
Children
10
8
10
Venn, Carroll and Tree Diagrams
CARROLL
One attribute one question
-
two areas
Two
questions -four
attributes -two
areas
CARROLL
CARRO
LL
VENN
TREE
Three attributes -three questions eight areas
VENN
thick
red
TREE
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CARROL
Matrices
A matrix (p1 matrices) uses the idea of rows and columns. Games such as ‘What’s In The
Square?’ can be a useful introduction to coordinates as well as a valuable way of displaying
information.
which blocks are missing?
Place the blocks so that there is one
difference either side, two
differences up and down.
Using a 4 x 3 x 2 system of Logic Blocks (take a half set of
blocks of one thickness), place the blocks so that there are no
two of the same shape, colour and size next to each other
either sideways or up and down, Or, perhaps, so there are
three differences sideways and only two up and down, etc.
(
12
Pictures: R Thompson
Line Graphs
The meaning of intermediate values needs to be discussed with children.
Sometimes it is impossible to say what happens between known points. ie Temperature taken every
hour.
Here it is not certain that it changes constantly or even ir it goes up and down
between known temperatures.
it
Sometimes it is possible to know something of what is going on between known points ie A graph to
show the growth of a bean plant (measured every Friday)
Here the height of the plant as it grows will rarely go downwards, but may not rise steadily
between known points.
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Sometimes it is possible to be certain of all values between known
‘points i.e. A graph to show the relationship between squares and their perimeters.
Here it is possible to ‘read’ the fractional possibilities at any point as the relationship between
the two axes is constant.
Conversion graphs are another example where the intermediate values on the line graph have a
known meaning.
14
A Trip to the Park
se this distance line and the story to complete
the graph below.
Anil left home at 10 past 9.
He walked to Tim’s house. It took him 10 minutes.
Tim wasn’t ready, so Anil had to wait 20 minutes at Tim’s house.
They both walked to the park. They arrived there at 10 am and sat fishing by the lake.
At 12 noon they left the park and walked back to the cafe for lunch.
It took them 10 minutes.
They stayed 30 minutes at the cafe for lunch.
In 10 minutes they were back at Tim’s house.
They played there until 2 pm.
Then Anil left on his 10 minute journey home.
A graph to show Anil’s day out
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After the girls had left, Alan and David decided to go to the show as well.
One weekend on holiday, Carol and Barbara decided to visit Truro
Show.
They chose to cycle rather than walk.
They set off at 9.30 am to walk at a steady speed the 4 miles to Truro.
They set of at 10.15 am and arrived at Truro at 10.45 am.
At 11.00 am, when they were 1 mile from Truro, they had a 15 minute
rest.
They cycled at a steady speed.
The boys did not stop for a rest.
They finally arrived at Truro at 11.45 am.
Draw the boys’ journey on the same graph as the girls’ journey.
?
Draw a graph to show their journey.
1.
At what time did the boys pass the girls?
2.
How far had the girls travelled when the boys passed them
17
Scattergraphs
Use to show two different aspects of collected data.
Most often used with continuous data.
The arm length and height of individuals are points on the graph. An individual whose height is 139
cm and arm length 57 cm is shown as a cross. The completed graph can be used to show whether there
is a broad connection, or correlation, between the two quantities.
Scatter graphs could also be drawn about discrete data i.e. eye colour, hair colour. The data is about
unrelated data so would be called a discrete graph.
Statistics
Scatter Graphs
John collected some data about 15 children. He recorded their measurements to the nearest
centimetre or kilogram.
Height (H cm)
Arm length (L cm)
Stride length (S cm)
Mass (M kg)
120
50
40
36
126
57
45
37
145
64
52
42
154
62
60
43
139
56
46
42
160
61
60
48
148
60
45
46
130
59
48
38
Height (H cm)
Arm length (L cm)
Stride length (S cm)
Mass (M kg)
142
63
50
45
132
52
41
40
152
64
58
43
150
61
57
44
158
67
48
44
156
66
51
47
124
54
42
37
135
63
51
42
He wondered if there was any relationship between the arm lengths and the heights. To find out
he started to draw a graph of the ordered pairs (H,L).
Arm Length (L cm)
70
65
60
55
50
100
110
120
130
140
Height (H cm)
150
160
1.
Copy and complete the scatter graph for (H,L).
2.
Do you notice any general trends from your scatter graph? If so, write about them.
3.
Using the data from the charts, draw scatter graphs for each of these ordered pairs.
Write about them
(a) (L,S)
(b) (L,M)
(c) (M,S)
(d) (H,M)
(e) (S,M)
20
Pie Charts
Simple Pie Charts
Paper circles are very useful to introduce children to the representation of data in a pie chart, and has
obvious links with fractional values.
i.e. The favourite fruit of 24 children.
where ½ of the group chose apples
¼ of the group chose bananas
1
/8 of the group chose pears
1
/8 of the group chose plums
However, rarely in real-life does data conveniently fall into such neat categories!
Use of Angles
This pie chart was constructed using the following data
Our Favourite TV Programmes
Neighbours (N)
12
5
Blue Peter (B)
Thundercats (T)
9
The Chart Show (C) 4
30
Method (using fractions of 360°)
i.e.
360
/30 = 12°
Neighbours 12 x 12°
Blue Peter
5 x 12°
Thundercats 9 x 12°
The Chart Show4 x 12°
30
144°
60°
= 108°
= 48°
360°
=
=
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Misleading Representations
These diagrams are not clear and honest illustrations of data. Say what you think each diagram
sets out to show and why it is misleading.
b)
What the nation drinks
Tea
Coffee
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Chocolate