what is a 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 positive relationship. Note two qualities here:

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

(1) direction


(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 +0.3 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.

• 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

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Ave Daily Ice Cream Sales

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• 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)

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


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• Notice how their rank orders are identical. And because their standard deviations are similar as well, these variables have a +1.0 Pearson Correlation.

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Meaning that higher values for one variable are associated with higher

values for another variable


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Or


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Meaning that lower values for one variable are associated with lower


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


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Meaning that higher values for one variable are associated with lower


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


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• 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.

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• 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 ∙𝑍𝑌 )𝑛

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


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• 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.

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


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The metric here is degrees


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The metric here is number of ice

cream sales

The metric here is degrees

• 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:


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Ave Daily Temp

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Ave Daily Temp

+1.4

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0.0

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+1.5

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-0.6

-1.3


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Ave Daily Temp

+1.4

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+1.5

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-1.3

Different Metric (raw scores)


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+1.5

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Same Metric (z or standard

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

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Ave Daily Temp

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0.0

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+1.5

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-0.3

-0.6

-1.3

• Once the values are standardized we multiply them


𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝑛


Ave Daily Temp

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+0.7

0.0

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-1.4


+1.5

+0.8

-0.3

-0.6

-1.3

𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝑛


Ave Daily Temp

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0.0

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+1.5

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-1.3

XXXXX

𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝑛


Ave Daily Temp

+1.4

+0.7

0.0

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+1.5

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-1.3

XXXXX

Cross Products

1.9

0.4

0.0

0.6

2.1

=====

𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝑛


Ave Daily Temp

+1.4

+0.7

0.0

-0.7

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+1.5

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-0.3

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XXXXX

Cross Products

1.9

0.4

0.0

0.6

2.1

=====

𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝑛

These are called cross products because we are multiplying

across two values


Ave Daily Temp

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+1.5

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XXXXX

Cross Products

1.9

0.4

0.0

0.6

2.1

=====

𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝑛

1.9 + 0.4 + 0.0 + 0.6 + 2.1 = 5.0Then we sum the cross products

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


𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝑛


𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝑛

The number of observations (in this case 5)

Ave Daily Temp

+1.4

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+1.5

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12345

𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝟓

𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝟓


𝑟=𝟓𝟓

𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝟓


𝑟=𝟓𝟓

Sum of the cross products1.9 + 0.4 + 0.0 + 0.6 + 2.1 =

5.0

𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝟓


𝑟=𝟓𝟓


5.0

𝑟=+𝟏 .𝟎

𝑟=∑(𝒁 𝑿 ∙𝒁𝒀 )

𝟓


𝑟=𝟓𝟓


5.0

𝑟=+𝟏 .𝟎This is the Pearson Correlation which in this case is a perfect

positive relationship

• In summary:

• 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

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2

-1 +10


Variable 1

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Variable 2

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PearsonCorrelation = +1.0

-1 +10

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


Variable 1

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Variable 2

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-1 +10


Variable 1

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Variable 2

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PearsonCorrelation = -1.0

-1 +10

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


-1 +10


-1 +10 0.2weak


-1 +10


-1 +10 0.8strong


-1 +10


-1 +100.2

weak


-1 +10


-1 +100.8

strong


-1 +10

• 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.

what is a pearson product moment correlation?

Education

pearson correlations

thepearson correlation

continuous variable

variable b

perfect postiverelationship

strong relationshipvariable

strong relationships

direction2 strength