The document provides an introduction to measuring relationships between variables through covariance and Pearson's r. It defines correlation as a statistical method to describe and measure the relationship between two variables. Covariance is introduced as a way to measure how two variables vary together or oppose each other by taking the average of their cross-product deviations from the mean. However, covariance depends on the scale of measurement. Pearson's r standardizes the covariance by converting the variables to z-scores based on their standard deviations, allowing comparison of relationships between variables measured on different scales. It represents the covariability of the two variables divided by their separate variability.

1. Introduction to measures of relationship:
covariance, and Pearson r
Ivan Jacob Agaloos Pesigan

2. Outline
•
•
•
•
•
•
Defining correlation
Describing relationships: Scatter plots
Measuring relationships: Covariance
Measuring relationships: Pearson r
Measuring relationships: Issues
Uses of Correlation
2

3. Defining correlation
Introduction to measures of relationship:
covariance, and Pearson r
3

4. Defining correlation
• A correlation is a statistical method used to
describe and measure the relationship
between two variables.
• A relationship exists when changes in one
variable tend to be accompanied by
consistent and predictable changes in
the other variable.
4

5. Defining correlation
Although you should not make too much of the
distinction between relationships and
differences (if treatments have different
means, then means are related to treatments),
the distinction is useful in terms of the interests
of the experimenter and the structure of the
experiment.
5

6. Describing Relationships:
Scatter Plots
Introduction to measures of relationship:
covariance, and Pearson r
6

7. Child
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Vocabulary
5
8
7
9
10
8
6
6
10
9
7
7
9
6
8
Digit Symbol
12
15
14
18
19
18
14
17
20
17
15
16
16
13
16
7

8. 16
14
12
Digit Symbol
18
20
A strong, positive linear relationship
5
6
7
8
9
10
Vocabulary
8

9. Child
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Age
7.5
6.5
7.5
6
7
6.5
5.5
7
5.5
6.5
7
5
5
5
5.5
6
Time to complete
21
43
26
37
25
34
35
33
44
35
41
41
50
56
51
40
9

10. 40
35
30
25
20
Time to complete
45
50
55
A negative linear relationship
5.0
5.5
6.0
6.5
7.0
7.5
Age
10

11. Describing relationships:
Scatter plots
•
•
•
•
•
Linear relationship
Positive linear relationship
Negative linear relationship
Curvilinear relationship
No linear relationship
11

12. Linear relationship
A negative linear relationship
40
30
35
Time to complete
16
25
14
20
12
Digit Symbol
45
18
50
55
20
A strong, positive linear relationship
5
6
7
8
Vocabulary
9
10
5.0
5.5
6.0
6.5
7.0
7.5
Age
12

13. Curvilinear Relationship
A curvilinear relationship
14
8
10
12
Number of Errors
15
10
5
Time to Complete the Test
16
18
20
A negative, curved line relationship
1
2
3
Motivation
4
5
4.0
4.5
5.0
5.5
6.0
6.5
Age
13

14. No linear relationship
70
60
50
Depression Score
80
No linear relationship
60
80
100
IQ
120
140
14

15. Measuring relationships:
Covariance
Introduction to measures of relationship:
covariance, and Pearson r
15

16. Measuring relationships:
Covariance
85
80
75
70
Exam
90
95
100
IQ and final examination grade
100
105
110
115
IQ
120
125
16

17. Measuring relationships:
Covariance
• The simplest way to look at whether two
variables are associated is to look at whether
they covary.
• To understand what covariance is, we first
need to think back to the concept of
variance.
17

18. Measuring relationships:
Covariance
• Remember that the variance of a single
variable represents the average amount that
the data vary from the mean.
variance = s
2
X
2
X
∑( X − M )
=
variance = s =
i
2
X
nX −1
∑( X − X )
2
i
nX −1
18

19. Measuring relationships:
Covariance
• If we are interested in whether two variables
are related, then we are interested in whether
changes in one variable are met with
similar changes in the other variable.
• Therefore, when one variable deviates
from its mean we would expect the
other variable to deviate from its
mean in a similar way or the directly
opposite way.
19

20. Measuring relationships:
Covariance
Final Exam Scores per Student
80
85
Final Exam Scores
115
110
75
105
70
100
IQ Scores
90
120
95
125
100
IQ Scores per Student
2
4
6
Student
8
10
2
4
6
8
10
Student
20

21. Measuring relationships:
Covariance
15
10
5
0
Exam
20
25
30
IQ and Errors in the Final Examination
100
105
110
115
IQ
120
125
21

22. Measuring relationships:
Covariance
Final Exam Errors per Student
10
15
Final Exam Errors
115
110
5
105
0
100
IQ Scores
20
120
25
125
30
IQ Scores per Student
2
4
6
Student
8
10
2
4
6
8
10
Student
22

23. Student
IQ
Final Exam Score
1
97
71
2
98
71
3
108
74
4
114
84
5
115
86
6
119
86
7
120
87
8
121
90
9
122
96
10
127
99
23

24. Student
1
2
3
4
5
6
7
8
9
10
SUM
MEAN
IQ
97
98
108
114
115
119
120
121
122
127
1141
114.1
Final Exam Score
71
71
74
84
86
86
87
90
96
99
844
84.4
24

25. Student
IQ
X - Mx
1
2
3
4
5
6
7
8
9
10
SUM
MEAN
97
98
108
114
115
119
120
121
122
127
1141
114.1
-17.1
-16.1
-6.1
-0.1
0.9
4.9
5.9
6.9
7.9
12.9
Final Exam
Score
71
71
74
84
86
86
87
90
96
99
844
84.4
Y - MY
-13.4
-13.4
-10.4
-0.4
1.6
1.6
2.6
5.6
11.6
14.6
25

26. Measuring relationships:
Covariance
Final Exam Scores per Student
100
IQ Scores per Student
14.6
120
125
12.9
5.9
6.9
95
11.6
1.6
1.6
2.6
-0.4
75
105
80
-6.1
5.6
85
Final Exam Scores
115
0.9
110
-0.1
100
-17.1
-16.1
70
IQ Scores
90
4.9
7.9
2
4
6
Student
8
10
-10.4
-13.4
-13.4
2
4
6
8
10
Student
26

27. Measuring relationships:
Covariance
• When we multiply the deviations of one
variable by the corresponding deviations of a
second variable, we get what is known as the
cross-product deviations.
• As with the variance, if we want an average
value of the combined deviations for the two
variables, we must divide by the number of
observations (n− 1)
27

28. Student
IQ
1
97
2
98
3
108
4
114
5
115
6
119
7
120
8
121
9
122
10
127
SUM
1141
MEAN
114.1
X - Mx
-17.1
-16.1
-6.1
-0.1
0.9
4.9
5.9
6.9
7.9
12.9
Final
Exam
Score
71
71
74
84
86
86
87
90
96
99
844
84.4
Y - MY
-13.4
-13.4
-10.4
-0.4
1.6
1.6
2.6
5.6
11.6
14.6
(X - Mx)
(Y – MY)
229.14
215.74
63.44
0.04
1.44
7.84
15.34
38.64
91.64
188.34
851.6
cov = 94.62
28

29. Measuring relationships:
Covariance
• This averaged sum of combined deviations is
known as the covariance.
covariance = cov X ,Y
∑ ( X − M ) (Y − M )
=
covariance = cov X ,Y =
i
X
i
Y
n −1
∑ ( X − X ) (Y − Y )
i
i
n −1
29

30. Measuring relationships:
Covariance
covariance = cov X ,Y
cov X ,Y
∑ ( X - M ) (Y - M )
=
i
X
i
Y
n -1
851.6 851.6
=
=
≈ 94.622
10 -1
9
30

31. Measuring relationships:
Covariance
• Calculating the covariance is a good way to
assess whether two variables are related to each
other.
• A positive covariance indicates that as one
variable deviates from the mean, the other
variable deviates in the same direction.
• On the other hand, a negative covariance
indicates that as one variable deviates from the
mean (e.g. increases), the other deviates from the
mean in the opposite direction (e.g. decreases).
31

32. Measuring relationships:
Covariance
• There is, however, one problem with
covariance as a measure of the relationship
between variables and that is that it depends
upon the scales of measurement used.
• So, covariance is not a standardized
measure.
32

33. Measuring relationships:
Covariance
• For example, if we use the data above and
assume that they represented two variables
measured in miles then the covariance is 94.62
(as calculated above). If we then convert these
data into kilometers (by multiplying all values
by 1.609) and calculate the covariance again
then we should find that it increases to
244.97.
33

34. Measuring relationships:
Covariance
• This dependence on the scale of measurement
is a problem because it means that we cannot
compare covariances in an objective way – so,
we cannot say whether a covariance is
particularly large or small relative to another
data set unless both data sets were measured
in the same units.
34

35. Measuring Relationships:
Pearson r
Introduction to measures of relationship:
covariance, and Pearson r
35

36. Measuring relationships: Pearson r
• To overcome the problem of dependence on
the measurement scale, we need to convert
the covariance into a standard set of units.
• This process is known as standardization.
36

37. Measuring relationships: Pearson r
• A very basic form of standardization would be
to insist that all experiments use the same
units of measurement, say meters – that way,
all results could be easily compared.
• However, what happens if you want to
measure attitudes – you’d be hard pushed to
measure them in meters!
37

38. Measuring relationships: Pearson r
• Therefore, we need a unit of measurement
into which any scale of measurement can be
converted. The unit of measurement we use is
the standard deviation.
• If we divide any distance from the mean by the
standard deviation, it gives us that distance in
standard deviation units.
38

39. Measuring relationships: Pearson r
X i − MX X i − X
zX =
=
sX
sX
Yi − MY Yi − Y
zY =
=
sY
sY
39

40. Measuring relationships: Pearson r
• Since the properties of z scores form the foundation
necessary for understanding the Pearson product
moment correlation coefficient (r) they will be briefly
reviewed:
1. The sum of a set of z score (Σ z) and therefore
also the mean equal 0.
2. The variance (s2) of the set of z scores equals 1, as
does the standard deviation (s).
3. Neither the shape of the distribution of X, nor of
its relationship to any other variable, is affected by
transforming it to zX
40

41. Measuring relationships: Pearson r
rX ,Y
∑z
=
where
z
X Y
n −1
X i − MX X i − X
zX =
=
sX
sX
cov X ,Y
=
Yi − MY Yi − Y
zY =
=
sY
sY
rX ,Y
=
∑ ( X − M ) (Y − M )
=
X
i
( )( )
sX sY n −1
∑ ( X − X ) (Y − Y )
i
i
(s s ) (n −1)
i
X
i
Y
(n −1)
∑ ( X − X ) (Y − Y )
i
i
(n −1)
then
then
i
∑ ( X − M ) (Y − M )
=
rX ,Y =
Y
cov X ,Y
sX sY
covariability of X and Y
variability of X and Y separately
degree to which X and Y vary together
=
degree to which X and Y vary separately
rX ,Y =
rX ,Y
X Y
41

42. Student
IQ
zx = (X –
Mx)/sx
1
97
-1.69
2
98
-1.59
3
108
-0.60
4
114
-0.01
5
115
0.09
6
119
0.48
7
120
0.58
8
121
0.68
9
122
0.78
10
127
1.27
SUM
1141
MEAN
114.1
s = 10.14
Final
Exam
Score
71
71
74
84
86
86
87
90
96
99
844
84.4
zy = (Y –
MY)/sy
-1.37
-1.37
-1.06
-0.04
0.16
0.16
0.27
0.57
1.19
1.49
s = 9.77
zxzy
2.31
2.18
0.64
0.00
0.01
0.08
0.15
0.39
0.93
1.90
8.60
r = 0.96
42

43. Student
IQ
X - Mx
1
97
-17.1
2
98
-16.1
3
108
-6.1
4
114
-0.1
5
115
0.9
6
119
4.9
7
120
5.9
8
121
6.9
9
122
7.9
10
127
12.9
SUM
1141
MEAN
114.1
s = 10.14
Final
Exam
Score
71
71
74
84
86
86
87
90
96
99
844
84.4
Y - MY
-13.4
-13.4
-10.4
-0.4
1.6
1.6
2.6
5.6
11.6
14.6
(X - Mx)
(Y – MY)
229.14
215.74
63.44
0.04
1.44
7.84
15.34
38.64
91.64
188.34
851.6
s = 9.77
cov = 94.62
43

44. Measuring relationships: Pearson r
rX ,Y =
rX ,Y
cov X ,Y
sX sY
94.622
=
10.14 9.77
(
)(
)
94.622
=
= 0.96
99.00331457
44

45. Measuring relationships: Pearson r
radj
(
=1−
1− r
2
(n −1)
)
n−2
45

46. Measuring relationships: Pearson r
There are three general strategies for determining the size of
the population effect which a research is trying to detect:
1. To the extent that studies which have been carried out by
the current investigator or others are closely related to the
present investigation, the ESs found in these studies reflect
the magnitude which can be expected.
2. In some research areas an investigator may posit some
minimum population effect that would have either practical
of theoretical significance.
3. A third strategy in deciding what ES values to use in
determining the power of a study is to use certain suggested
conventional definitions of “small”, “medium”, and “large”
effects.
46

47. Measuring relationships: Pearson r
Size of Effect /
Magnitude
r
r2 (% of variance)
small
0.1
0.01 (1%)
medium
0.3
0.09 (9%)
large
0.5
0.25 (25%)
47

48. Measuring relationships: Pearson r
r = +1.00
r = -1.00
Perfect negative linear relationship
0
Y
-3
-3
-2
-2
-1
-1
0
Y
1
1
2
2
3
Perfect positive linear relationship
-3
-2
-1
0
1
X
2
3
-2
-1
0
1
2
3
X
48

49. Measuring relationships: Pearson r
r = .80
r = .30
Moderate positive linear relationship
Y
-3
-2
-2
-1
-1
0
Y
0
1
1
2
2
3
Strong positive linear relationship
-3
-2
-1
0
X
1
2
-2
-1
0
1
2
3
X
49

50. Measuring relationships: Pearson r
r = .10
r = 0
No linear relationship
-3
-3
-2
-2
-1
-1
Y
Y
0
0
1
1
2
2
Weak positive linear relationship
-3
-2
-1
0
X
1
2
-2
-1
0
1
2
3
X
50

51. Measuring relationships: Pearson r
• Pearson Product Moment Correlation Coefficient:
– It is a pure number and independent of the units of measurement.
– Its absolute value varies between zero, when the variables have no
linear relationship, and 1, when each variable is perfectly predicted by
the other. The absolute value thus gives the degree of relationship.
– Its sign indicates the direction of the relationship. A positive sign
indicates a tendency for high values of one variable to occur with high
values of the other, and low values to occur with low. A negative sign
indicates a tendency for high values of one variable to be associated
with low values of the other. Reversing the direction of measurement
of one of the variables will produce a coefficient of the same value bur
of opposite sign. Coefficients with equal value but opposite sign (for
example, +.50 and -.50) thus indicate equally strong linear relationship,
but in opposite directions.
51

52. Measuring Relationships: Issues
Introduction to measures of relationship:
covariance, and Pearson r
52

53. Restriction of Range
-3
-2
-1
0
Y
1
2
3
Restriction of range
-2
-1
0
1
X
2
53

54. Heterogeneous Subsamples
male
female
160
140
120
100
Weight
180
200
Relationship of Height and Weight
Blue Line = Males; Red Line = Females; Black Line = Overall
62
64
66
68
70
72
74
Height
54

55. Outliers
55

56. Correlation and Causation
56

57. Uses of Correlation
Introduction to measures of relationship:
covariance, and Pearson r
57

58. Uses of Correlation
• Prediction. If two variables are known to be
related in a systematic way, then it is possible
to use one of the variables to make accurate
predictions about the other.
58

59. Uses of Correlation
• Validity. Suppose that a psychologist
develops a new test for measuring intelligence.
How could you show that this test truly
measures what it claims; that is, how could
you demonstrate the validity of the test? One
common technique for demonstrating validity
is to use a correlation.
59

60. Uses of Correlation
• Reliability. In addition to evaluating the
validity of a measurement procedure,
correlations are used to determine reliability.
A measurement procedure is considered
reliable to the extent that it produces stable,
consistent measurements.
60

61. Uses of Correlation
• Theory Verification. Many psychological
theories make specific predictions about the
relationship between two variables.
61

62. People behind the concepts
Scatter plot
Bivariate plot
Pearson Product Moment
Correlation Coefficient
62

63. Slides Prepared by:
Ivan Jacob Agaloos Pesigan
Lecturer, Psychology Department
Ateneo de Manila University
Assistant Professor Lecturer, Psychology Department
De La Salle University
Lecturer, College of Education, Psychology
Department, Mathematics Department
Miriam College
63