Pearson product moment correlation

Pearson Product Moment
Correlation
Welcome to the Pearson Product
Moment Correlation Learning
Module

• The Pearson Product Moment Correlation is the most
widely used statistic when determining the
relationship between two variables that are
continuous.

• The Pearson Product Moment Correlation is the most
widely used statistic when determining the
relationship between two variables that are
continuous.
Variable A Variable B

• By continuous we mean a variable that can take any
valuable between two points.

• Here is an example:

• 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.

• The Pearson Product Moment Correlation will either
indicate a strong relationship

• Or a weak even nonexistent relationship

• Strong relationships can either be positive

• Or negative

• The Pearson Product Moment Correlation or simply
Pearson Correlation values range from -1.0 to 1.0

• The Pearson Product Moment Correlation or simply
Pearson Correlation values range from -1.0 to 1.0
-1 +10

• A Pearson Correlation of 1.0 has a perfect postive
relationship. Note two qualities here:

(1) direction

(1) direction
(2) strength

(1) direction
(2) strength
• A +1.0 Pearson Correlation’s direction is positive and it’s
strength is very or perfectly strong.

(1) direction
(2) strength
• A -1.0 Pearson Correlation’s direction is negative and it’s

(1) direction
(2) strength
• A 0.0 Pearson Correlation has no direction and has no
strength.

(1) direction
(2) strength
strength.
strength is moderately weak.

(1) direction
(2) strength
strength.
strength is moderately weak.
strength is very weak.

• There is another quality as well. With a Pearson
correlation you are considering the relationship
between only two variables.

• Three’s a crowd:

• Three’s a crowd:
• Bottom line: The Pearson Correlation is used only
when exploring the relationship between two
variables.

• Let’s look at a fictitious problem to illustrate how the
Pearson Correlation is calculated.

• Imagine you are conducting a study to determine the
relationship between the average daily temperature
and the average daily ice cream sales in a particular
city.

• Imagine the data set looks like this:

• 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

• Notice how as one variable goes up (temperature)
the other variable increases (ice cream sales)

• Notice how as one variable goes up (temperature)
the other variable increases (ice cream sales)
Ave Daily Temp
900
800
700
600
500
560
480
350
320
230

• One way to look at this relationship is to rank order
both variable values like so:

Ave Daily Temp
900
800
700
600
500
560
480
350
320
230

Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
1st

Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
1st 1st

Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
1st 1st
2nd 2nd

Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
1st 1st
2nd
3rd 3rd
2nd

Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
1st 1st
2nd
3rd 3rd
2nd
4th 4th

Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
1st 1st
2nd
5th 5th
4th 4th
3rd 3rd
2nd

• Notice how their rank orders are identical. And
because their standard deviations are similar as well,
these variables have a +1.0 Pearson Correlations.
Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
1st 1st
2nd
5th 5th
4th 4th
3rd 3rd
2nd

• What would a perfectly negative correlation (-1.0)
look like?

• What would a perfectly negative correlation (-1.0)
look like?
Ave Daily Temp
900
800
700
600
500
230
320
350
480
560
1st
1st
2nd
5th
5th
4th
4th
3rd 3rd
2nd

• What would a zero correlation (0.0) look like?

Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
1st
1st
2nd
5th 5th
4th
4th
3rd
3rd
2nd

• Note – Pearson Correlation is not just a comparison of rank ordered data
(that is what a Phi coefficient does) but the rank order is one factor that is
considered with a Pearson Correlation. Another factor is the degree to
which the standard deviations are similar.
Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
1st
1st
2nd
5th 5th
4th
4th
3rd
3rd
2nd

• 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

• 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
𝑟 = ∑ (𝑍 𝑋 ∙ 𝑍 𝑌)
𝑛

• Let’s calculate the Pearson Correlation, for the
following data set:

following data set:
Ave Daily Temp
900
800
700
600
500
560
480
350
320
230

following data set:
• It is very 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.
Ave Daily Temp
900
800
700
600
500
560
480
350
320
230

• When computing a Pearson Correlation you will
normally have two variables that DO NOT USE THE
SAME METRIC:

SAME METRIC:
Ave Daily Temp
900
800
700
600
500
560
480
350
320
230

SAME METRIC:
Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
The metric here is degrees

SAME METRIC:
Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
The metric here is degrees The metric here is number
of ice cream sales

• 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.

• 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
560
480
350
320
230

Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
+1.5
+0.8
-0.3
-0.6
-1.3

Ave Daily Temp
900
800
700
600
500
560
480
350
320
230
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
+1.5
+0.8
-0.3
-0.6
-1.3
Different Metric
(raw scores)

• 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
560
480
350
320
230
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
+1.5
+0.8
-0.3
-0.6
-1.3
Different Metric
(raw scores)

• Once the values are standardized we multiply them

𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝑛

Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
+1.5
+0.8
-0.3
-0.6
-1.3
𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝑛

Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
+1.5
+0.8
-0.3
-0.6
-1.3
X
X
X
X
X
𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝑛

Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
+1.5
+0.8
-0.3
-0.6
-1.3
X
X
X
X
X
1.9
0.4
0.0
0.6
2.1
=
=
=
=
=
𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝑛

Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
+1.5
+0.8
-0.3
-0.6
-1.3
X
X
X
X
X
1.9
0.4
0.0
0.6
2.1
=
=
=
=
=
𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝑛
These are called cross products
because we are multiplying
across two values

Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
+1.5
+0.8
-0.3
-0.6
-1.3
X
X
X
X
X
1.9
0.4
0.0
0.6
2.1
=
=
=
=
=
𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝑛
1.9 + 0.4 + 0.0 + 0.6 + 2.1 = 5.0

• Finally, divide that number (5.0) by the number of
observations

observations
𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝑛

observations
𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝑛
The number of observations
(in this case 5)
Ave Daily Temp
+1.4
+0.7
0.0
-0.7
-1.4
+1.5
+0.8
-0.3
-0.6
-1.3
1
2
3
4
5

𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝟓

𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝟓
(in this case 5)
𝑟 =
𝟓
𝟓

𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝟓
(in this case 5)
𝑟 =
𝟓
𝟓
Sum of the cross products
1.9 + 0.4 + 0.0 + 0.6 + 2.1 =
5.0

𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝟓
(in this case 5)
𝑟 =
𝟓
𝟓
1.9 + 0.4 + 0.0 + 0.6 + 2.1 =
5.0
𝑟 = +𝟏. 𝟎

𝑟 =
∑ (𝒁 𝑿 ∙ 𝒁 𝒀)
𝟓
(in this case 5)
𝑟 =
𝟓
𝟓
1.9 + 0.4 + 0.0 + 0.6 + 2.1 =
5.0
𝑟 = +𝟏. 𝟎
This is the Pearson Correlation
which in this case is a perfect
positive relationship

• In summary:
• The Pearson Product Moment Correlation can range
from -1 to 0 to +1.

• In summary:
• The Pearson Product Moment Correlation can range
from -1 to 0 to +1.
-1 +10

• A correlation of 0 indicates no association between
the variables of interest.

• A correlation of 0 indicates no association between
the variables of interest.
• The direction (positive or negative) simply indicates a
positive or negative (inverse) relationship between
the variables.

• If POSITIVE, when values increase on one variable,
they tend to increase on another variable.

Variable 1
10
9
8
7
Variable 2
5
4
3
2
-1 +10

Variable 1
10
9
8
7
Variable 2
5
4
3
2
Pearson
Correlation = +1.0
-1 +10

• If NEGATIVE, when values increase on one variable,
they tend to decrease on another variable.

Variable 1
10
9
8
7
Variable 2
5
4
3
2
-1 +10

Variable 1
10
9
8
7
Variable 2
5
4
3
2
Pearson
Correlation = -1.0
-1 +10

• The strength of the relationship depends on the
decimal value.

decimal value.
-1 +10

decimal value.
-1 +10 0.2
weak

decimal value.
-1 +10 0.8
strong

decimal value.
-1 +10
0.2
weak

decimal value.
-1 +10
0.8
strong

• There is a tendency to interpret the Pearson Product
Moment Correlation with causal language as though
changes in one variable causes changes in the other.

• Whether to interpret the Pearson Product Moment
Correlation as prediction or causation depends on
the nature of the research design rather than the
nature of the statistic.

• First, analyze the nature of the research design
before interpreting the Pearson Product Moment
Correlation with causal or prediction language.

• First, analyze the nature of the research design
before interpreting the Pearson Product Moment
Correlation with causal or prediction language.
• So, if your research question is focused on the
relationship between two continuous variables the
Pearson Product Moment Correlation would be the
appropriate statistical method to use.

Pearson product moment correlation

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