Correlation
From Plus magazine 04/06/2008 (see http://plus.maths.org/latestnews/may-aug08/oilcricket/index.html )
Questions to think about:
•
What do the graphs show?
•
What else would you like to know?
•
If you were a journalist, what headline might you write?
Australia cricket % test win
120
% tests won
100
80
60
40
20
0
1980
1985
1990
1995
2000
2005
Year
Monthly oil price US dollars
$120.00
$100.00
Price
$80.00
$60.00
$40.00
$20.00
$0.00
0
10
20
30
40
50
60
70
80
Months after Jan 2002
Year
2001
2002
2003
2004
2005
2006
2007
Inflation adjusted
average oil price ($)
27.29
26.61
31.62
41.84
53.77
60.73
64.92
Australia cricket
% test win
57.1
90.9
75
71.4
60
100
100
Correlation and causation
Correlation does not imply causation. Can you think of any examples of variables
where you might get a positive (or negative) correlation between them but it is clear
that they are not directly affecting each other?
Calculating correlation
Using quadrants to get an idea of correlation
•
•
•
•
Find the mean of each set of
data.
Plot the point representing both
means.
Draw vertical and horizontal
lines through the “mean point”
to divide the page into 4
quadrants.
The quadrants where most
points lie give you an idea
about correlation.
Calculating Pearson’s product moment correlation coefficient
• It is possible to use ICT to calculate correlation coefficients. What are the
advantages and disadvantages of doing so?
•
Here are three alternative formulae for calculating Pearson’s product moment
correlation coefficient:
1
⎛ x − x ⎞⎛ y − y ⎞
∑ xy − nxy
∑( x − x )( y − y ) r =
r=
∑
⎜ s ⎟⎜ s ⎟ r =
2
2
n −1 ⎝ x ⎠⎝ y ⎠
(x − x) ×( y − y)
( x − nx 2 ) × ( y 2 − ny 2 )
∑
∑
∑
Which will be easiest to use?
Which will be easiest for students to understand?
What do all the letters and symbols stand for?
If you show students more than one formula, how can you convince them that they
give the same answer?
Calculate the correlation coefficient for oil price and Australian cricket % win.
x
57.1
90.9
75
71.4
60
100
100
y
27.29
26.61
31.62
41.84
53.77
60.73
64.92
An example of positive correlation
There are 50 states in the USA and a Federal District; the District of Columbia (DC).
The 2003 data for percentage of adults with a college education and number of
murders per 100 000 people shows a positive correlation.
What might the graph look like?
An example of negative correlation
Electronegativity is a measure of an atom’s ability to “attract electrons”. It is
measured on the Pauling scale, which goes up to 4. For elements in the periodic
table, the correlation between atomic number and electronegativity is negative.
What might the graph look like?
The problem solving approach
You are given some data for a random sample of 180 year 11 students from Census
at School. There are a number of questions you could investigate. You will be
looking at whether foot length is a good predictor of height.
Use the stages in the cycle above to plan your approach to this problem. If you could
use a computer to handle the data, you would be able to use all of it but you will not
be able to do this in the limited time available.
………….Plan……….
In solving problems it is crucial to be able to identify the target population from the
statement of the problem in context. What could be the target population? Think of
more than one possible answer.
What questions do you need to find answers to in order to solve the problem of
finding the relationship between foot lengths and heights?
………Collect…….…
What data will you use in order to solve the problem of finding the relationship
between foot lengths and heights? Does this depend on the target population?
……….Process…….
How will you display your data? What calculations will you make?
………Discuss…….
How will you discuss the solution to the problem? What inferences can you draw?
number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
gender
M
M
M
F
M
F
M
M
F
M
F
M
F
F
M
F
M
M
F
F
F
F
M
F
F
F
M
F
F
M
F
M
M
F
F
F
M
M
F
M
M
F
M
F
M
height
176
201
169
111
181
163
200
198
166
111
169
209
200
179
197
120
188
173
159
155
162
162
178
169
162
169
193
159
159
184
172
164
165
159
163
170
195
179
161
176
176
163
167
168
176
footLength
29
35
35
12
27
26
30
31.5
22.5
12
21
35
35
29
29
19
32
35
24.5
20
23
23.8
26
25
24
24
29
24.5
26
28.5
28
25
25.8
23
23
24.5
29.5
26
20
23
27
25
26
25
30
bellyButton
110
135
66
50
104
100
130
110
98
111
80
135
50
120
92
60
124
71
100
90
95
100
107
104
106
103
111
98
100
113
100
101
100
100
101
105
120
110
96
105
112
104
101
105
61
number
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
gender
F
F
F
M
M
F
F
M
M
M
F
M
M
F
M
F
F
M
M
M
F
M
M
M
M
F
M
F
F
F
F
F
M
F
F
F
M
F
F
F
M
F
M
F
M
height
159
160
158
175
174
160
169
172
186
181
162
200
181
174
182
162
160
172
180
180
200
179
180
162
175
149
182
178
170
162
154
177
183
156
115
173
179
161
161
158
175
166
170
172
177
footLength
22
27
19
15
24
25
25
25
23
29
24.5
30
27
25.5
26
25
22
26
27
27
12
28
28
24
30
35
28.7
24.4
23
28
23
20.5
15
22.5
22
22
27.5
23
24
35
31
26
20
24.5
29
bellyButton
96
97
81
100
107
95
107
86
91
117
101
100
108
108
112
97
97
107
112
112
100
107
112
101
78
66
114
108
105
99
94
106
118
103
92
98
114
92
99
108
108
101
109
106
109
number
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
gender
M
F
M
F
M
F
F
F
F
M
M
F
M
F
F
M
M
F
F
M
F
M
M
M
F
F
M
F
M
M
F
M
M
F
M
M
M
F
M
F
F
M
F
F
M
M
height
179
155
170
167
184
165
175
170
163
178
173
159
183
173
160
176
173
169
155
186
178
170
173
181
161
156
186
153
174
175
167
200
176
164
178
168
160
159
178
158
165
172
168
170
166
169
footLength
27
23
27
23.5
28.5
22.5
25
26
24.5
27
27
25
29
25
20
29
27
24.5
22
28
27.5
25
26
28.5
24.5
20
20
23.5
29
26
24.8
26
26.5
15
26
25.5
27
30
29
23
23
27.5
25
25
27.5
24
bellyButton
106
97
105
81
111
101
114
106
102
111
105
90
90
106
101
112
105
110
96
130
114
106
105
113
101
95
120
95
113
109
103
102
108
100
106
106
60
88
109
99
99.5
110
103
105
98
102
number
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
gender
F
F
F
F
F
M
M
M
M
M
F
F
F
F
F
F
M
M
M
M
F
M
F
F
F
M
F
F
F
F
F
F
M
M
M
M
M
F
M
M
M
M
M
F
height
156
163
172
173
166
173
170
167
174
185
161
153
158
160
172
168
184
185
170
166
166
166
166
168
152
156
168
171
171
169
161
173
180
173
180
200
178
160
178
175
177
200
187
162
footLength
23
25.5
25
22
24
25
26
25
25
29
25
22
25
20
23
25.5
29
26.5
26.5
29
24
27
24
24
24.2
26.5
28
25
25
25.5
23
25
26.5
29
28
30
27
26
30
25.2
26
31
26
23
bellyButton
97
98
102
88
103
107
104
105
88
115
101
94
96
85
112
98
102
117
103
102
70
105
103
70
95
112
99
108
116
108
97
110
110
106
104
100
110
100
130
125
114
135
132
105