## Can you establish cause-and-effect?

No, the Pearson correlation cannot determine a cause-and-effect relationship. It can only establish the strength of linear association between two variables. As stated earlier, it does not even distinguish between independent and dependent variables.

## How do I report the output of a Pearson product-moment correlation?

You need to state that you used the Pearson product-moment correlation and report the value of the correlation coefficient, *r*, as well as the degrees of freedom (df). You should express the result as follows:

where the degrees of freedom (df) is the number of data points minus 2 (*N* – 2). If you have not tested the significance of the correlation then leave out the degrees of freedom and *p*-value such that you would simply report: *r* = -0.52.

## Can I determine whether the association is statistically significant?

Yes, the easy way to do this is through a statistical programme, such as SPSS Statistics. To learn how to run a Pearson correlation in SPSS Statistics, go to our guide: Pearson's correlation in SPSS Statistics. You need to be careful how you interpret the statistical significance of a correlation. If your correlation coefficient has been determined to be statistically significant this does not mean that you have a strong association. It simply tests the null hypothesis that there is no relationship. By rejecting the null hypothesis, you accept the alternative hypothesis that states that there is a relationship, but with no information about the strength of the relationship or its importance.

## What is the Coefficient of Determination?

The coefficient of determination, *r*^{2}, is the square of the Pearson correlation coefficient *r* (i.e., *r*^{2}). So, for example, a Pearson correlation coefficient of 0.6 would result in a coefficient of determination of 0.36, (i.e., *r*^{2} = 0.6 x 0.6 = 0.36). The coefficient of determination, with respect to correlation, is the proportion of the variance that is shared by both variables. It gives a measure of the amount of variation that can be explained by the model (the correlation is the model). It is sometimes expressed as a percentage (e.g., 36% instead of 0.36) when we discuss the proportion of variance explained by the correlation. However, you should not write *r*^{2} = 36%, or any other percentage. You should write it as a proportion (e.g., *r*^{2} = 0.36).

To learn how to run a Pearson correlation in SPSS Statistics, go to our guide: Pearson's correlation in SPSS Statistics.

**Bibliography** and **Referencing**

Please see the list below:

Book | Cohen, J. (1988). *Statistical power analysis for the behavioral sciences* (2nd ed.). New York: Psychology Press. |

Book | Cohen, B. H. (2013). *Explaining psychological statistics* (4th ed.). Hoboken, NJ: John Wiley & Sons. |

Journal Article | Edgell, S. E., & Noon. S. M. (1984). Effect of violation of normality on the t test of the correlation coefficient. *Psychological Bulletin*, *95*(3), 576-583. |

Book | Hogg, R. V., McKean, J., & Craig, A. T. (2014). *Introduction to mathematical statistics* (7th. ed.). Harlow, Essex: Pearson Education. |

Book | Lindeman, R. H., Merenda, P. F., & Gold, R. Z. (1980). *Introduction to bivariate and multivariate analysis*. Glenview, IL: Scott, Foresman and Company. |

Book |
Nunnally, J. C. (1978). *Psychometric theory* (2nd ed.). New York: McGraw-Hill Book Company. |

Journal Article | Pearson, K. (1895). Contributions to the mathematical theory of evolution. *Psychological transactions of the Royal Society of London A*, *186*, 343-414. |

Journal Article | Pearson, K. (1920). Notes on the history of correlations. *Biometrika*, *13*, 25-45. |

Book | Shevlyakov, G. L., & Oja, H. (2016). *Robust correlation: Theory and applications*. Chichester, West Sussex: John Wiley & Sons. |

Book | Wilcox, R. (2012). *Introduction to robust estimation and hypothesis testing* (3rd ed.). Oxford: Academic Press. |

## Reference **this article**

Laerd Statistics (2020). Pearson's product moment correlation. *Statistical tutorials and software guides*. Retrieved **Month**, **Day**, **Year**, from https://statistics.laerd.com/statistical-guides/pearson-correlation-coefficient-statistical-guide.php

For example, if you viewed this guide on **29**^{th} April 2020, you would use the following reference:

Laerd Statistics (2020). Pearson's product moment correlation. *Statistical tutorials and software guides*. Retrieved April, 29, 2020, from https://statistics.laerd.com/statistical-guides/pearson-correlation-coefficient-statistical-guide.php