2. The Pearson Product Moment Correlation is the most
widely used statistic when determining the relationship
between or independence of two variables that are
continuous.
3. The Pearson Product Moment Correlation is the most
widely used statistic when determining the relationship
between or independence of two variables that are
continuous.
Variable A Variable B
4. By continuous we mean a variable that can take any
valuable between two points.
5. By continuous we mean a variable that can take any
valuable between two points.
Here is an example:
6. By continuous we mean a variable that can take any
valuable between two points.
Here is an example:
Suppose the fire department mandates that all fire fighters must
weigh between 150 and 250 pounds. The weight of a fire fighter
would be an example of a continuous variable; since a fire
fighter's weight could take on any value between 150 and 250
pounds.
7. By continuous we mean a variable that can take any
valuable between two points.
Here is an example:
Suppose the fire department mandates that all fire fighters must
weigh between 150 and 250 pounds. The weight of a fire fighter
would be an example of a continuous variable; since a fire
fighter's weight could take on any value between 150 and 250
pounds.
8. The Pearson Product Moment Correlation will either
indicate a strong relationship
9. The Pearson Product Moment Correlation will either
indicate a strong relationship
Variable A Variable B
11. Or a weak even nonexistent relationship
Variable A Variable B
12. Or a weak even nonexistent relationship
Variable A Variable B
This is what is meant by
independence!
13. The Pearson Product Moment Correlation or simply
Pearson Correlation values range from -1.0 to +1.0
14. The Pearson Product Moment Correlation or simply
Pearson Correlation values range from -1.0 to +1.0
-1 0 +1
15. A Pearson Correlation of 1.0 has a perfect positive
relationship. Note two qualities here:
16. A Pearson Correlation of 1.0 has a perfect positive
relationship. Note two qualities here:
(1) direction
17. A Pearson Correlation of 1.0 has a perfect positive
relationship. Note two qualities here:
(1) direction
(2) strength
18. A Pearson Correlation of 1.0 has a perfect postive
relationship. Note two qualities here:
(1) direction
(2) strength
A +1.0 Pearson Correlation’s direction is positive and it’s
strength is very or perfectly strong.
19. A Pearson Correlation of 1.0 has a perfect postive
relationship. Note two qualities here:
(1) direction
(2) strength
A +1.0 Pearson Correlation’s direction is positive and it’s
strength is very or perfectly strong.
A -1.0 Pearson Correlation’s direction is negative and it’s
strength is very or perfectly strong.
20. A Pearson Correlation of 1.0 has a perfect postive
relationship. Note two qualities here:
(1) direction
(2) strength
A +1.0 Pearson Correlation’s direction is positive and it’s
strength is very or perfectly strong.
A -1.0 Pearson Correlation’s direction is negative and it’s
strength is very or perfectly strong.
A 0.0 Pearson Correlation has no direction and has no strength.
21. A Pearson Correlation of 1.0 has a perfect postive
relationship. Note two qualities here:
(1) direction
(2) strength
A +1.0 Pearson Correlation’s direction is positive and it’s
strength is very or perfectly strong.
A -1.0 Pearson Correlation’s direction is negative and it’s
strength is very or perfectly strong.
A 0.0 Pearson Correlation has no direction and has no strength.
This is would be evidence of
independence between two
variables
22. A Pearson Correlation of 1.0 has a perfect postive
relationship. Note two qualities here:
(1) direction
(2) strength
A +1.0 Pearson Correlation’s direction is positive and it’s
strength is very or perfectly strong.
A -1.0 Pearson Correlation’s direction is negative and it’s
strength is very or perfectly strong.
A 0.0 Pearson Correlation has no direction and has no strength.
A +0.3 Pearson Correlation’s direction is positive and it’s
strength is moderately weak.
23. A Pearson Correlation of 1.0 has a perfect postive
relationship. Note two qualities here:
(1) direction
(2) strength
A +1.0 Pearson Correlation’s direction is positive and it’s
strength is very or perfectly strong.
A -1.0 Pearson Correlation’s direction is negative and it’s
strength is very or perfectly strong.
A 0.0 Pearson Correlation has no direction and has no strength.
A +0.3 Pearson Correlation’s direction is positive and it’s
strength is moderately weak.
A -0.1 Pearson Correlation’s direction is negative and it’s
strength is very weak.
24. There is another quality as well. With a Pearson
correlation you are considering the relationship
between or independence of only two variables.
25. There is another quality as well. With a Pearson
correlation you are considering the relationship
between or independence of only two variables.
26. There is another quality as well. With a Pearson
correlation you are considering the relationship
between or independence of only two variables.
Three’s a crowd:
27. There is another quality as well. With a Pearson
correlation you are considering the relationship
between or independence of only two variables.
Three’s a crowd:
28. There is another quality as well. With a Pearson
correlation you are considering the relationship
between or independence of only two variables.
Three’s a crowd:
Bottom line: The Pearson Correlation is used only when
exploring the relationship between or independence of
two variables.
29. Let’s look at a fictitious problem to illustrate how the
Pearson Correlation is calculated.
30. Imagine you are working for a company that is trying
convince patrons that ice-cream is a dessert for all
seasons. They ask you to conduct a study to determine
if the average daily temperature and the average daily
ice cream sales are independent of one another.
31. Imagine you are working for a company that is trying
convince patrons that ice-cream is a dessert for all
seasons. They ask you to conduct a study to determine
if the average daily temperature and the average daily
ice cream sales are independent of one another.
33. Imagine the data set looks like this:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
34. Notice how as one variable goes up (temperature) the
other variable increases (ice cream sales)
35. Notice how as one variable goes up (temperature) the
other variable increases (ice cream sales)
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
36. Notice how as one variable goes up (temperature) the
other variable increases (ice cream sales)
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
37. One way to look at this relationship is to rank order
both variable values like so:
38. One way to look at this relationship is to rank order
both variable values like so:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
39. One way to look at this relationship is to rank order
both variable values like so:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st
40. One way to look at this relationship is to rank order
both variable values like so:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
41. One way to look at this relationship is to rank order
both variable values like so:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
42. One way to look at this relationship is to rank order
both variable values like so:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
2nd 2nd
43. One way to look at this relationship is to rank order
both variable values like so:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
2nd
3rd 3rd
2nd
44. One way to look at this relationship is to rank order
both variable values like so:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
2nd
3rd 3rd
2nd
4th 4th
45. One way to look at this relationship is to rank order
both variable values like so:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
2nd
2nd
3rd 3rd
4th 4th
5th 5th
46. Notice how their rank orders are identical. And
because their standard deviations are similar as well,
these variables have a +1.0 Pearson Correlation.
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
2nd
2nd
3rd 3rd
4th 4th
5th 5th
47. Notice how their rank orders are identical. And
because their standard deviations are similar as well,
these variables have a +1.0 Pearson Correlation.
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
2nd
2nd
3rd 3rd
4th 4th
5th 5th
Meaning that higher values for one
variable are associated with higher
values for another variable
48. Notice how their rank orders are identical. And
because their standard deviations are similar as well,
these variables have a +1.0 Pearson Correlation.
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
2nd
2nd
3rd 3rd
4th 4th
5th 5th
Meaning that higher values for one
variable are associated with higher
values for another variable
49. Notice how their rank orders are identical. And
because their standard deviations are similar as well,
these variables have a +1.0 Pearson Correlation.
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
2nd
2nd
3rd 3rd
4th 4th
5th 5th
Meaning that higher values for one
variable are associated with higher
values for another variable
50. Notice how their rank orders are identical. And
because their standard deviations are similar as well,
these variables have a +1.0 Pearson Correlation.
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
2nd
2nd
3rd 3rd
4th 4th
5th 5th
Or
51. Notice how their rank orders are identical. And
because their standard deviations are similar as well,
these variables have a +1.0 Pearson Correlation.
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
2nd
2nd
3rd 3rd
4th 4th
5th 5th
Meaning that lower values for one
variable are associated with lower
values for another variable
52. Notice how their rank orders are identical. And
because their standard deviations are similar as well,
these variables have a +1.0 Pearson Correlation.
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
1st 1st
2nd
2nd
3rd 3rd
4th 4th
5th 5th
Meaning that lower values for one
variable are associated with lower
values for another variable
53. What would a perfectly negative correlation (-1.0) look
like?
59. What would a perfectly negative correlation (-1.0) look
like?
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
230
320
350
480
560
1st
1st
2nd
5th
5th
4th
4th
3rd 3rd
2nd
Meaning that higher values for one
variable are associated with lower
values for another variable
60. What would a perfectly negative correlation (-1.0) look
like?
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
230
320
350
480
560
1st
1st
2nd
5th
5th
4th
4th
3rd 3rd
2nd
Meaning that higher values for one
variable are associated with lower
values for another variable
62. What would a zero correlation (0.0) look like?
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
5th 5th
1st
1st
2nd
4th
4th
3rd
3rd
2nd
63. What would a zero correlation (0.0) look like?
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
5th 5th
1st
1st
2nd
4th
4th
3rd
3rd
2nd
64. What would a zero correlation (0.0) look like?
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
5th 5th
1st
1st
2nd
4th
4th
3rd
3rd
2nd
65. What would a zero correlation (0.0) look like?
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
5th 5th
1st
1st
2nd
4th
4th
3rd
3rd
2nd
66. What would a zero correlation (0.0) look like?
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
5th 5th
1st
1st
2nd
4th
4th
3rd
3rd
2nd
67. What would a zero correlation (0.0) look like?
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
5th 5th
1st
1st
2nd
4th
4th
3rd
3rd
2nd
68. What would a zero correlation (0.0) look like?
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
5th 5th
1st
1st
2nd
4th
4th
3rd
3rd
2nd
If this is the result than we can
conclude that temperature and ice
cream are independent of one
another
69. The Pearson Product Moment Correlation (PPMC) is
calculated as the average cross product of the z-scores
of two variables for a single group of people. Here is
the equation for the PPMC
70. The Pearson Product Moment Correlation (PPMC) is
calculated as the average cross product of the z-scores
of two variables for a single group of people. Here is
the equation for the PPMC
푟 = Σ (푍푋 ∙ 푍푌)
푛
72. Let’s calculate the Pearson Correlation, for the
following data set:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
73. Let’s calculate the Pearson Correlation, for the
following data set:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
It is important to note that the Pearson Correlation can be
computed in a matter of seconds using statistical
software. The next set of slides is designed to help you
see what is happening conceptually as well as
computationally with the Pearson Correlation.
74. When computing a Pearson Correlation you will
normally have two variables that DO NOT USE THE
SAME METRIC:
75. When computing a Pearson Correlation you will
normally have two variables that DO NOT USE THE
SAME METRIC:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
76. When computing a Pearson Correlation you will
normally have two variables that DO NOT USE THE
SAME METRIC:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
The metric
here is degrees
77. When computing a Pearson Correlation you will
normally have two variables that DO NOT USE THE
SAME METRIC:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
The metric here is
number of ice
cream sales
The metric
here is degrees
78. So we have to get these two variables on the same
metric. This is done by calculating the z scores or
standardized scores for the values from each variable.
79. So these raw score values in separate metrics are
transformed into standardized values which converts
them into the same metric:
80. So these raw score values in separate metrics are
transformed into standardized values which converts
them into the same metric:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
81. So these raw score values in separate metrics are
transformed into standardized values which converts
them into the same metric:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
82. So these raw score values in separate metrics are
transformed into standardized values which converts
them into the same metric:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
Ave Daily Ice Cream Sales
+1.5
+0.8
-0.3
-0.6
-1.3
83. So these raw score values in separate metrics are
transformed into standardized values which converts
them into the same metric:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
Ave Daily Ice Cream Sales
+1.5
+0.8
-0.3
-0.6
-1.3
Different Metric
(raw scores)
84. So these raw score values in separate metrics are
transformed into standardized values which converts
them into the same metric:
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
Ave Daily Ice Cream Sales
+1.5
+0.8
-0.3
-0.6
-1.3
Same Metric
(z or standard
scores)
85. • Note – this is done by subtracting each value from
it’s mean (e.g., 900 minus 700 = 200) and dividing it
by it’s standard deviation (e.g., 200 / 14.1 = 1.4)
Ave Daily Temp
900
800
700
600
500
Ave Daily Ice Cream Sales
560
480
350
320
230
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
Ave Daily Ice Cream Sales
+1.5
+0.8
-0.3
-0.6
-1.3
86. • Once the values are standardized we multiply them
87. • Once the values are standardized we multiply them
푟 =
Σ (풁푿 ∙ 풁풀)
푛
88. • Once the values are standardized we multiply them
푟 =
Σ (풁푿 ∙ 풁풀)
푛
89. • Once the values are standardized we multiply them
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
Ave Daily Ice Cream Sales
+1.5
+0.8
-0.3
-0.6
-1.3
푟 =
Σ (풁푿 ∙ 풁풀)
푛
90. • Once the values are standardized we multiply them
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
Ave Daily Ice Cream Sales
+1.5
+0.8
-0.3
-0.6
-1.3
X
X
X
X
X
푟 =
Σ (풁푿 ∙ 풁풀)
푛
91. • Once the values are standardized we multiply them
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
Ave Daily Ice Cream Sales
+1.5
+0.8
-0.3
-0.6
-1.3
X
X
X
X
X
Cross Products
1.9
0.4
0.0
0.6
2.1
=
=
=
=
=
푟 =
Σ (풁푿 ∙ 풁풀)
푛
92. • Once the values are standardized we multiply them
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
Ave Daily Ice Cream Sales
+1.5
+0.8
-0.3
-0.6
-1.3
X
X
X
X
X
Cross Products
1.9
0.4
0.0
0.6
2.1
=
=
=
=
=
푟 =
Σ (풁푿 ∙ 풁풀)
푛
These are called cross products
because we are multiplying
across two values
93. • Once the values are standardized we multiply them
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
Ave Daily Ice Cream Sales
+1.5
+0.8
-0.3
-0.6
-1.3
X
X
X
X
X
Cross Products
1.9
0.4
0.0
0.6
2.1
=
=
=
=
=
푟 =
Σ (풁푿 ∙ 풁풀)
푛
1.9 + 0.4 + 0.0 + 0.6 + 2.1 = 5.0
Then we sum the cross products
95. • Finally, divide that number (5.0) by the number of
observations
푟 =
Σ (풁푿 ∙ 풁풀)
푛
96. • Finally, divide that number (5.0) by the number of
observations
푟 =
Σ (풁푿 ∙ 풁풀)
푛
The number of observations
(in this case 5)
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
Ave Daily Ice Cream Sales
+1.5
+0.8
-0.3
-0.6
-1.3
1
2
3
4
5
98. 푟 =
Σ (풁푿 ∙ 풁풀)
ퟓ
The number of observations
(in this case 5)
푟 =
ퟓ
ퟓ
99. 푟 =
Σ (풁푿 ∙ 풁풀)
ퟓ
The number of observations
(in this case 5)
푟 =
ퟓ
ퟓ
Sum of the cross products
1.9 + 0.4 + 0.0 + 0.6 + 2.1 =
5.0
100. 푟 =
Σ (풁푿 ∙ 풁풀)
ퟓ
The number of observations
(in this case 5)
푟 =
ퟓ
ퟓ
Sum of the cross products
1.9 + 0.4 + 0.0 + 0.6 + 2.1 =
5.0
푟 = +ퟏ. ퟎ
101. 푟 =
Σ (풁푿 ∙ 풁풀)
ퟓ
The number of observations
(in this case 5)
푟 =
ퟓ
ퟓ
Sum of the cross products
1.9 + 0.4 + 0.0 + 0.6 + 2.1 =
5.0
푟 = +ퟏ. ퟎ
This is the Pearson Correlation
which in this case is a perfect
positive relationship