Pearson Correlation Coefficient

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What is the Pearson Correlation Coefficient?

In statistics, Pearson’s correlation coefficient, also known as Pearson’s r, is defined as the measure of the strength of correlation between two variables or data sets. There are many types of correlation coefficients, but Pearson’s correlation coefficient is the most popular. As it is based on the method of covariance, it is known as the best method to measure the association between two variables of interest. It gives information on the strength, or magnitude, of the relation as well as direction.

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Strength and Direction

The following explains the statistical significance of strength and direction in regard to Pearson’s correlation coefficient:

Strength of correlation

Strength, or magnitude, of correlation refers to the relationship correlation between two data sets. It reflects the consistency at which one variable changes in reaction to a change in the other. When data points fall in a line or very close to a line, they have a really strong correlation. In this case values may be +1 or -1, or very close to them. If data points fall farther away from each other and aren’t close to being points on a line, they have a weaker linear relationship.

Direction of correlation

The direction of the line reflects whether the two variables have a positive linear relationship or a negative linear relationship. The line has a positive linear relationship when it has an upward slope. A positive linear relationship means an increase in the value of one variable leads to an increase in the value of the other. On the other hand, the line has a negative linear relationship when it has a downward slope. In this case, an increase in the value of one variable will lead to a decrease in the value of the other, or vice versa.

Formula of Pearson’s Correlation Coefficient

The following is the formula of Pearson’s Correlation Coefficient:

Where:

n = the number of pairs of scores

Σ(xy) = the sum of the products of paired scores

(Σx) = the sum of x scores

(Σy) = the sum of y scores

Σx2 = the sum of squared x scores

Σy2 = the sum of squared y scores

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Interpreting Pearson’s Correlation Coefficient

The following depicts the guidelines for the interpretation of Pearson’s Correlation Coefficient:

Strength of Association	Positive	Negative
Small	0.1 to 0.3	-0.1 to -0.3
Medium	0.3 to 0.5	-0.3 to -0.5
Large	0.5 to 1	-0.5 to -1

The stronger the association between two variables, the closer the answer will be to +1 or -1 depending on whether it is a positive or negative linear correlation. The closer the answer is to 0, the more the variation there is between variables.

Examples of Pearson’s Correlation Coefficient

The following visual examples show a few types of correlations in order to understand Pearson’s correlation coefficient better.

Large Positive Correlation

Figure depicts a correlation coefficient of almost +1.
Scatterplots are plotted on a straight line.
When one variable increases, the other does too. This indicates a positive linear relationship.

Large Negative Correlation

Figure depicts a correlation coefficient of almost -1.
Scatterplots are plotted on a straight line.
When one variable increases, the other decreases. This indicates a negative linear relationship.

Medium Positive Correlation

Figure depicts a correlation coefficient between +0.8 and +1.
Scatter plots resemble a strong linear uphill pattern, however do not fall on a straight line.
Uphill pattern indicates that as one variable increases, the other does too. Hence, it is a positive correlation.

Medium Negative Correlation

Figure depicts a correlation coefficient between -0.8 and -1.
Scatterplots resemble a strong linear downward pattern, however do not fall on a straight line.
Downward pattern indicates that as one variable increases, the other decreases. Hence, it is a negative correlation.

Small Positive Correlation

Figure depicts a positive linear correlation of approximately +0.5.
Scatter plots are farther away from the straight line compared to earlier examples, but still indicate an uphill pattern. This signifies a positive correlation as when one variable increases, the other does too.

Small Negative Correlation

Figure depicts a negative linear correlation of approximately -0.5.
Scatter plots are farther away from the straight line compared to earlier examples, but still indicate a downward pattern. This signifies a negative correlation as when one variable increases, the other decreases.

Weak correlation/ No Correlation

Scatterplots don’t resemble a line.
Hard to judge whether it has a positive or negative correlation.