Laerd Statistics LoginCookies & Privacy

Pearson's correlation using Minitab

Introduction

The Pearson product-moment correlation, often shortened to Pearson correlation or Pearson's correlation, is used to assess the strength and direction of association between two continuous variables that are linearly related. Its coefficient, r, indicates the strength and direction of this relationship and can range from -1 for a perfect negative linear relationship to +1 for a perfect positive linear relationship. A value of 0 (zero) indicates that there is no relationship between the two variables.

For example, you could use a Pearson's correlation to understand whether there is an association between test performance and revision time (i.e., your two variables would be "test performance", measured as the exam mark achieved, and "revision time", measured in hours per week). If there was a strong, positive association, we could say that more time spent revising was associated with higher test performance. Alternately, you could use a Pearson's correlation to understand whether there is an association between blood pressure and time spent exercising (i.e., your two variables would be "blood pressure", measured in mm/Hg, and "time spent exercising", measured in hours per week). If there was a moderate, negative association, we could say that exercising more per week is associated with lower blood pressure.

In this guide, we show you how to carry out a Pearson's correlation using Minitab, 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.

Minitab

Assumptions

A Pearson's correlation has four assumptions. You cannot test the first of these assumptions with Minitab because it relates to your study design and choice of variables. However, you should check whether your study meets this assumption before moving on. If this assumption is not met, there is likely to be a different statistical test that you can use instead. Assumption #1 is explained below:

Assumptions #2, #3 and #4 relate to the nature of your data and can be checked using Minitab. You have to check that your data meets these assumptions because if it does not, the results you get when running a Pearson's correlation might not be valid. In fact, do not be surprised if your data violates one or more of these assumptions. This is not uncommon. However, there are possible solutions to correct such violations (e.g., transforming your data) such that you can still use a Pearson's correlation. Assumptions #2, #3 and #4 are explained below:

In practice, checking for assumptions #2, #3 and #4 will probably take up most of your time when carrying out a Pearson's correlation. However, it is not a difficult task and Minitab provides all the tools you need to do this.

In the section, Test Procedure in Minitab, we illustrate the Minitab procedure required 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 Minitab.

Minitab

Example

An educator wants to determine whether students' exam scores were related to revision time. For example, as students spent more time revising, did their exam score also increase (i.e., a positive relationship) or did the opposite happen (i.e., a negative relationship)? As such, scores in a maths exam were measured on a scale from 0 to 100 and the amount of time spent revising was measured in hours.

To carry out the analysis, the researcher recruited 40 students. The amount of time revising (i.e., the variable, Revision time) and the exam scores (i.e., the variable, Exam score) were recorded for all 40 participants. Expressed in variable terms, the researcher wanted to correlate Exam score and Revision time. A Pearson's correlation was used to determine whether there was a statistically significant relationship between exam score and revision time.

Note: The example and data used for this guide are fictitious. We have just created them for the purposes of this guide.

Minitab

Setup in Minitab

In Minitab, we entered our two variables into the first two columns ( and ). Under column we entered the name of one of the two variables, Exam score, as follows: . Then, under column we entered the name of the second of our two variables, Revision time, as follows: . Finally, we entered the scores for Exam score into the column and the scores for Revision time into the column. This is illustrated below:

Data setup for Pearson correlation in Minitab

Published with written permission from Minitab Inc.

Note: It does not matter which of your two variables you enter under C1 or C2. We have just entered the data into Minitab this way in our example.

Minitab

Test Procedure in Minitab

In this section, we show you how to analyse your data using a Pearson's correlation in Minitab when the four assumptions in the previous section, Assumptions, have not been violated. Therefore, the three steps required to run a Pearson's correlation in Minitab are shown below:

Minitab

Output of the Pearson's correlation in Minitab

The Minitab output for a Pearson's correlation is shown below:

Output for the dependent t-test in Minitab

The output contains two important pieces of information:

In this example, Pearson's correlation coefficient is 0.853 (i.e., the "Pearson correlation of Exam score and Revision Time = 0.853" line), which indicates a strong positive correlation between the two variables, Exam score and Revision time, according to Cohen (1998):

Coefficient ValueStrength of Association
0.1 < | r | < .3small correlation
0.3 < | r | < .5medium/moderate correlation
| r | > .5large/strong correlation

where | r | means the absolute value or r (e.g., | r | > .5 means r > .5 and r < -.5). Therefore, the Pearson correlation coefficient in this example (r = .853) suggests a strong correlation. If instead, r = -.853, you would also have had a strong correlation, albeit a negative one. This result shows that more time spent revising is associated with better (i.e., higher) exam scores. You can also see that Pearson's correlation coefficient is statistically significant as p < .0005 (i.e., the "P-Value = 0.000" line, which actually means p < .0005), which satisfies the common cut-off for statistical significance of p < .05.

Note: We present the output from the Pearson's correlation above. However, since you should have tested your data for the assumptions we explained earlier in the Assumptions section, you will also need to interpret the Minitab output that was produced when you tested for these assumptions. This includes: (a) the scatterplots you used to check if there was a linear relationship between your two variables (i.e., Assumption #2); (b) the same scatterplots that you will have used to check there were no significant outliers (i.e., Assumption #3); and (c) the Shapiro-Wilk test for normality to check whether your two variables were approximately normally distributed (i.e., Assumption #4). Also, remember that if your data failed any of these assumptions, the output that you get from the Pearson's correlation procedure (i.e., the output we discuss above) might no longer be valid, and you may have to carry out a different statistical test to analyse your data.

Minitab

Reporting the output of a Pearson's correlation

When you report the output of your Pearson's correlation, it is good practice to include:

Based on the results above, we could report the results of this study as follows:

A Pearson's product-moment correlation was run to assess the relationship between exam score and time spent revising in 40 students. There was a strong positive correlation between exam score and time spent revising, r(38) = .853, p < .0005.

In addition to reporting the results as above, a diagram (e.g., a scatterplot) can be used to visually present your results. This can make it easier for others to understand your results and is easily produced in Minitab.

Portions of information contained in this publication/book are printed with permission of Minitab Inc. All such material remains the exclusive property and copyright of Minitab Inc. All rights reserved.

1