How to Calculate a Pearson Correlation Coefficient by Hand

A Pearson Correlation Coefficient measures the linear association between two variables.

It always takes on a value between -1 and 1 where:

  • -1 indicates a perfectly negative linear correlation between two variables
  • 0 indicates no linear correlation between two variables
  • 1 indicates a perfectly positive linear correlation between two variables

The formula to calculate a Pearson Correlation Coefficient, denoted r, is:

Source: Wikipedia

This tutorial provides a step-by-step example of how to calculate a Pearson Correlation Coefficient by hand for the following dataset:

Step 1: Calculate the Mean of X and Y

First, we’ll calculate the mean of both the X and Y values:

Step 2: Calculate the Difference Between Means

Next, we’ll calculate the difference between each of the individual X and Y values and their respective means:

Step 3: Calculate the Remaining Values

Next, we’ll calculate the remaining values needed to complete the Pearson Correlation Coefficient formula:

Step 4: Calculate the Sums

Next, we’ll calculate the sums of the the last three columns:

Step 5: Calculate the Pearson Correlation Coefficient

Now we’ll simply plug in the sums from the previous step into the formula for the Pearson Correlation Coefficient:

Pearson correlation coefficient by hand

The Pearson Correlation Coefficient turns out to be 0.947.

Since this value is close to 1, this is an indication that X and Y are strongly positively correlated.

In other words, as the value for X increases the value for Y also increases in a highly predictable fashion.

Additional Resources

An Introduction to the Pearson Correlation Coefficient
How to Find a Confidence Interval for a Correlation Coefficient

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