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Pearson's Product-Moment Correlation using SPSS

Introduction

The Pearson product-moment correlation coefficient (Pearson’s correlation, for short) is a measure of the strength and direction of association that exists between two variables measured on at least an interval scale. For example, you could use a Pearson’s correlation to understand whether there is an association between exam performance and time spent revising; whether there is an association between depression and length of unemployment; and so forth. A Pearson’s 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). You can learn more here, which we recommend if you are not familiar with this test.

This "quick start" guide shows you how to carry out a Pearson's correlation using SPSS, as well as interpret and report the results from this test. However, before we introduce you to this procedure, you need to understand the different assumptions that your data must meet in order for a Pearson's correlation to give you a valid result. We discuss these assumptions next.

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Assumptions

When you choose to analyse your data using Pearson’s correlation, part of the process involves checking to make sure that the data you want to analyse can actually be analysed using Pearson’s correlation. You need to do this because it is only appropriate to use Pearson’s correlation if your data "passes" four assumptions that are required for Pearson’s correlation to give you a valid result. In practice, checking for these four assumptions just adds a little bit more time to your analysis, requiring you to click of few more buttons in SPSS when performing your analysis, as well as think a little bit more about your data, but it is not a difficult task.

Before we introduce you to these four assumptions, do not be surprised if, when analysing your own data using SPSS, one or more of these assumptions is violated (i.e., is not met). This is not uncommon when working with real-world data rather than textbook examples, which often only show you how to carry out Pearson’s correlation when everything goes well! However, don’t worry. Even when your data fails certain assumptions, there is often a solution to overcome this. First, let’s take a look at these four assumptions:

You can check assumptions #2, #3 and #4 using SPSS. We suggest testing these assumptions in this order because it represents an order where, if a violation to the assumption is not correctable, you will no longer be able to use Pearson’s correlation (although you may be able to run another statistical test on your data instead). Just remember that if you do not run the statistical tests on these assumptions correctly, the results you get when running a Pearson's correlation might not be valid. This is why we dedicate a number of sections of our enhanced Pearson's correlation guide to help you get this right. You can find out about our enhanced content as a whole here, or more specifically, learn how we help with testing assumptions here.

In the section, Procedure, we illustrate the SPSS procedure to perform a Pearson’s correlation assuming that no assumptions have been violated. First, we set out the example we use to explain the Pearson’s correlation procedure in SPSS.

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Example

A researcher wants to know whether a person's height is related to how well they perform in a long jump. The researcher recruited untrained individuals from the general population, measured their height and had them perform a long jump. The researcher then investigated whether there is an association between height and long jump performance.

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Setup in SPSS

In SPSS, we created two variables so that we could enter our data: Height (i.e., the person's height) and Jump_Dist (i.e., long jump distance). In our enhanced Pearson's correlation guide, we show you how to correctly enter data in SPSS to run a Pearson's correlation. You can learn about our enhanced data setup content here. Alternately, we have a generic, "quick start" guide to show you how to enter data into SPSS, available here.

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Test Procedure in SPSS

The six steps below show you how to analyse your data using Pearson’s correlation in SPSS when none of the four assumptions in the previous section, Assumptions, have been violated. At the end of these six steps, we show you how to interpret the results from this test. If you are looking for help to make sure your data meets assumptions #2, #3 and #4, which are required when using Pearson’s correlations, and can be tested using SPSS, you can learn more about our enhanced guides here.

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Output

SPSS generates quite a few tables for a Pearson’s correlation, but only one for the main Pearson’s correlation procedure that you ran in the previous section. If your data passed assumptions #2 (linear relationship), #3 (no outliers) and #4 (normality), which we explained earlier in the Assumptions section, you will only need to interpret this one table. However, since you should have tested your data for these assumptions, you will also need to interpret the SPSS output that was produced when you tested for them (i.e., you will have to interpret (a) the scatterplot you used to check for a linear relationship between your two variables, (b) the "casewise diagnostics" table that highlights if you had any significant outliers, and (c) the output SPSS produces for your Shapiro-Wilk test of normality). If you do not know how to do this, we show you in our enhanced Pearson’s correlation guide. Remember that if your data failed any of these assumptions, the output that you get from the Pearson’s correlation procedure (i.e., the table we discuss below), will no longer be correct.

However, in this "quick start" guide, we focus on the results from the Pearson’s correlation procedure only, assuming that your data met all the relevant assumptions. Therefore, when running the Pearson’s correlation procedure, you will be presented with the Correlations table in the output viewer as shown below:

The Pearson Product-Moment Correlation

Published with written permission from SPSS Inc., an IBM Company.

The results are presented in a matrix such that, as can be seen above, the correlations are replicated. Nevertheless, the table presents the Pearson correlation coefficient, the significance value and the sample size that the calculation is based on.

In this example, we can see that the Pearson correlation coefficient, r, is 0.777, and that this is statistically significant (p < 0.0005). For interpreting multiple correlations, see our enhanced Pearson’s guide.

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Understanding the Output

In our example above, you might present the results are follows:

A Pearson product-moment correlation was run to determine the relationship between an individual's height and their performance in a long jump (distance jumped). The data showed no violation of normality, linearity or homoscedasticity (you will need to have checked for these). There was a strong, positive correlation between height and distance jumped, which was statistically significant (r = .777, n = 27, p < .0005).

In our enhanced Pearson’s correlation guide, we also show you how to write up the results from your assumptions tests and Pearson’s correlation output if you need to report this in a dissertation/thesis, assignment or research report. We do this using the Harvard and APA styles. We also show you how to write up your results if you have performed multiple Pearson’s correlations. You can learn more about our enhanced content here.

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