Pearson Product-Moment Correlation

What does this test do?

The Pearson product-moment correlation coefficient (or Pearson correlation coefficient, for short) is a measure of the strength of a linear association between two variables and is denoted by r. Basically, a Pearson product-moment correlation attempts to draw a line of best fit through the data of two variables, and the Pearson correlation coefficient, r, indicates how far away all these data points are to this line of best fit (i.e., how well the data points fit this new model/line of best fit).

What values can the Pearson correlation coefficient take?

The Pearson correlation coefficient, r, can take a range of values from +1 to -1. A value of 0 indicates that there is no association between the two variables. A value greater than 0 indicates a positive association; that is, as the value of one variable increases, so does the value of the other variable. A value less than 0 indicates a negative association; that is, as the value of one variable increases, the value of the other variable decreases. This is shown in the diagram below:

Pearson Coefficient - Different Values

How can we determine the strength of association based on the Pearson correlation coefficient?

The stronger the association of the two variables, the closer the Pearson correlation coefficient, r, will be to either +1 or -1 depending on whether the relationship is positive or negative, respectively. Achieving a value of +1 or -1 means that all your data points are included on the line of best fit – there are no data points that show any variation away from this line. Values for r between +1 and -1 (for example, r = 0.8 or -0.4) indicate that there is variation around the line of best fit. The closer the value of r to 0 the greater the variation around the line of best fit. Different relationships and their correlation coefficients are shown in the diagram below:

Different values for the Pearson Correlation Coefficient

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Are there guidelines to interpreting Pearson's correlation coefficient?

Yes, the following guidelines have been proposed:

  Coefficient, r
Strength of Association Positive Negative
Small .1 to .3 -0.1 to -0.3
Medium .3 to .5 -0.3 to -0.5
Large .5 to 1.0 -0.5 to -1.0

Remember that these values are guidelines and whether an association is strong or not will also depend on what you are measuring.

Can you use any type of variable for Pearson's correlation coefficient?

No, the two variables have to be measured on either an interval or ratio scale. However, both variables do not need to be measured on the same scale (e.g., one variable can be ratio and one can be interval). Further information about types of variable can be found in our Types of Variable guide. If you have ordinal data, you will want to use Spearman's rank-order correlation or a Kendall's Tau Correlation instead of the Pearson product-moment correlation.

Do the two variables have to be measured in the same units?

No, the two variables can be measured in entirely different units. For example, you could correlate a person's age with their blood sugar levels. Here, the units are completely different; age is measured in years and blood sugar level measured in mmol/L (a measure of concentration). Indeed, the calculations for Pearson's correlation coefficient were designed such that the units of measurement do not affect the calculation. This allows the correlation coefficient to be comparable and not influenced by the units of the variables used.

What about dependent and independent variables?

The Pearson product-moment correlation does not take into consideration whether a variable has been classified as a dependent or independent variable. It treats all variables equally. For example, you might want to find out whether basketball performance is correlated to a person's height. You might, therefore, plot a graph of performance against height and calculate the Pearson correlation coefficient. Lets say, for example, that r = .67. That is, as height increases so does basketball performance. This makes sense. However, if we plotted the variables the other way around and wanted to determine whether a person's height was determined by their basketball performance (which makes no sense), we would still get r = .67. This is because the Pearson correlation coefficient makes no account of any theory behind why you chose the two variables to compare. This is illustrated below:

Not influenced by Dependent and Independent Variables

Does the Pearson correlation coefficient indicate the slope of the line?

It is important to realize that the Pearson correlation coefficient, r, does not represent the slope of the line of best fit. Therefore, if you get a Pearson correlation coefficient of +1 this does not mean that for every unit increase in one variable there is a unit increase in another. It simply means that there is no variation between the data points and the line of best fit. This is illustrated below:

The Pearson Coefficient does not indicate the slope of the line of best fit.


What assumptions does Pearson's correlation make?

The first and most important step before analysing your data using Pearson’s correlation is to check whether it is appropriate to use this statistical test. After all, Pearson’s correlation will only give you valid/accurate results if your study design and data "pass/meet" seven assumptions that underpin Pearson’s correlation.

In many cases, Pearson’s correlation will be the incorrect statistical test to use because your data "violates/does not meet" one or more of these assumptions. This is not uncommon when working with real-world data, which is often "messy", as opposed to textbook examples. However, there is often a solution, whether this involves using a different statistical test, or making adjustments to your data so that you can continue to use Pearson’s correlation.

We briefly set out the seven assumptions below, three of which relate to your study design and how you measured your variables (i.e., Assumptions #1, #2 and #3 below), and four which relate to the characteristics of your data (i.e., Assumptions #4, #5, #6 and #7 below):

Note: We list seven assumptions below, but there is disagreement in the statistics literature whether the term "assumptions" should be used to describe all of these (e.g., see Nunnally, 1978). We highlight this point for transparency. However, we use the word "assumptions" to stress their importance and to indicate that they should be examined closely when using a Pearson’s correlation if you want accurate/valid results. We also use the word "assumptions" to indicate that where some of these are not met, Pearson’s correlation will no longer be the correct statistical test to analyse your data.

Since assumptions #1, #2 and #3 relate to your study design and how you measured your variables, if any of these three assumptions are not met (i.e., if any of these assumptions do not fit with your research), Pearson’s correlation is the incorrect statistical test to analyse your data. It is likely that there will be other statistical tests you can use instead, but Pearson’s correlation is not the correct test.

After checking if your study design and variables meet assumptions #1, #2 and #3, you should now check if your data also meets assumptions #4, #5, #6 and #7 below. When checking if your data meets these four assumptions, do not be surprised if this process takes up the majority of the time you dedicate to carrying out your analysis. As we mentioned above, it is not uncommon for one or more of these assumptions to be violated (i.e., not met) when working with real-world data rather than textbook examples. However, with the right guidance this does not need to be a difficult process and there are often other statistical analysis techniques that you can carry out that will allow you to continue with your analysis.

Note: If your two continuous, paired variables (i.e., Assumptions #1 and 2) follow a bivariate normal distribution, there will be linearity, univariate normality and homoscedasticity (i.e., Assumptions #4, #5 and #6 below; e.g., Lindeman et al., 1980). Unfortunately, the assumption of bivariate normality is very difficult to test, which is why we focus on linearity and univariate normality instead. Homoscedasticity is also difficult to test, but we include this so that you know why it is important. We include outliers at the end (i.e., Assumption #7) because they cannot only lead to violations of the linearity and univariate normality assumptions, but they also have a large impact on the value of Pearson’s correlation coefficient, r (e.g., Wilcox, 2012).

You can check whether your data meets assumptions #4, #5 and #7 using a number of statistics packages (to learn more, see our guides for: SPSS Statistics, Stata and Minitab). If any of these seven assumptions are violated (i.e., not met), there are often other statistical analysis techniques that you can carry out that will allow you to continue with your analysis (e.g., see Shevlyakov and Oja, 2016).

On the next page we discuss other characteristics of Pearson's correlation that you should consider.

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