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).

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:

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:

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.

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.

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.

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:

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:

There are five assumptions that are made with respect to Pearson's correlation:

- The variables must be either interval or ratio measurements (see our Types of Variable guide for further details).
- The variables must be approximately normally distributed (see our Testing for Normality guide for further details).
- There is a linear relationship between the two variables (but see note at bottom of page). We discuss this later in this guide (jump to this section here).
- Outliers are either kept to a minimum or are removed entirely. We also discuss this later in this guide (jump to this section here).
- There is homoscedasticity of the data. This is discussed later in this guide (jump to this section here).

To test to see whether your two variables form a linear relationship you simply need to plot them on a graph (a scatterplot, for example) and visually inspect the graph's shape. In the diagram below, you will find a few different examples of a linear relationship and some non-linear relationships. It is not appropriate to analyse a non-linear relationship using a Pearson product-moment correlation.

Note: Pearson's correlation determines the degree to which a relationship is linear. Put another way, it determines whether there is a linear component of association between two continuous variables. As such, linearity is not actually an assumption of Pearson's correlation. However, you would not normally want to pursue a Pearson's correlation to determine the strength and direction of a linear relationship when you already know the relationship between your two variables is not linear. Instead, the relationship between your two variables might be better described by another statistical measure. For this reason, it is not uncommon to view the relationship between your two variables in a scatterplot to see if running a Pearson's correlation is the best choice as a measure of association or whether another measure would be better.

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