4.2 Pearson’s
product–moment correlation
coefficient
Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012
Pearson’s product–moment correlation coefficient
When the relationship between two variables needs to be studied, a
good way of representing the data graphically is to use a scatter
diagram. With a scatter diagram, a series of points is plotted. The x- and
y-coordinates of each point are taken from the values of the variables.
The data and scatter diagram below show the height of young children
(cm) plotted against their mass (kg).
24
Mass (kg)
22
20
Height
Mass
Height
Mass
34
3.8
86
11.1
40
7.0
87
16.4
45
9.0
95
20.9
46
4.2
96
16.2
52
10.1
96
14.0
59
6.2
101
19.5
63
9.9
108
15.9
64
16.0
109
12.0
71
15.8
117
19.4
73
9.9
121
14.3
18
16
14
12
10
8
6
4
2
Height (cm)
10
20
30
40
50
60
70
80
90
100
110
120
130
Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012
Pearson’s product–moment correlation coefficient
24
From the graph it is
clear that there is a
relationship between the
two variables.
Generally, as height
increases so does
mass.
Mass (kg)
22
20
18
16
14
12
10
8
6
4
2
Height (cm)
10
20
30
40
50
60
70
80
90
100
110
120
130
This can be
emphasized by
plotting a line of best
fit. This should pass
through the point x, y 
which represents the
mean values of the
height and mass.
To see how strong a correlation there is between the two variables,
Pearson’s product–moment correlation coefficient can be calculated.
Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012
Pearson’s product–moment correlation coefficient
The value of Pearson’s product–moment coefficient (r ) gives an
indication of the level of correlation between two variables. It has a
value in the range –1 ≤ r ≤ 1.
s xy
r
The value of r is calculated using the formula
sx s y
where s x is the standard deviation of x, s y is the
standard deviation of y and s xy is the covariance
of x and y
A value of r near –1 implies a strong negative correlation between
the two variables.
A value of r near +1 implies a strong positive correlation between the
two variables.
A value of r near 0 implies there is little or no correlation between the
two variables.
Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012
Pearson’s product–moment correlation coefficient
For the data given earlier, the results can be entered in a
GDC calculator and the value of r calculated.
r = 0.767
This value of r shows that there is a moderate positive
correlation between a child’s height and mass.
Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012