Pearson's correlation coefficient, normally denoted as r, is a statistical value that measures the linear relationship between two variables. It ranges in value from +1 to -1, indicating a perfect positive and negative linear relationship respectively between two variables. The calculation of the correlation coefficient is normally performed by statistical programs, such SPSS and SAS, to provide the most accurate possible values for reporting in scientific studies. The interpretation and use of Pearson's correlation coefficient varies based on the context and purpose of the respective study in which it is calculated.

### Items you will need

- Scientific calculator or statistical program
- Critical Values of the Correlation Coefficient Table

Identify the dependent variable to be tested between two independently derived observations. One of the requirements of Pearson's correlation coefficient is that the two variables being compared must be observed or measured independently to eliminate any biased results.

Calculate Pearson's correlation coefficient. For large amounts of data, the calculation can become very tedious. In addition to various statistical programs, many scientific calculators have the ability to calculate the value. The actual equation is provided in the Reference section.

Report a correlation value close to 0 as indication that there is no linear relationship between the two variables. As the correlation coefficient approaches 0, the values become less correlated which identifies variables that may not be related to one another.

Report a correlation value close to 1 as indication that there is a positive, linear relationship between the two variables. A value greater than zero that approaches 1 results in greater positive correlation between the data. As one variable increases a certain amount, the other variable increases in a corresponding amount. The interpretation must be determined based on the context of the study.

Report a correlation value close to -1 as indication that there is a negative, linear relationship between the two variables. As the coefficient approaches -1, the variables become more negatively correlated indicating that as one variable increases, the other variable decreases by a corresponding amount. The interpretation again must be determined based of the context of the study.

Interpret the correlation coefficient based on the context of the particular data set. The correlation value is essentially an arbitrary value that must be applied based on the variables being compared. For example, a resulting r value of 0.912 indicates a very strong and positive linear relationship between two variables. In a study comparing two variables that are not normally identified as related, these results provide evidence that one variable may positively affect the other variable, resulting in cause for further research between the two. However, the exact same r value in a study comparing two variables that are proven to have a perfectly positive linear relationship may identify an error in the data or other potential problems in the experimental design. Thus, it is important to understand the context of the data when reporting and interpreting Pearson's correlation coefficient.

Determine the significance of the results. This is accomplished using the correlation coefficient, degrees of freedom and a Critical Values of the Correlation Coefficient table. The degrees of freedom is calculated as the number of paired observations minus 2. Using this value, identify the corresponding critical value in the correlation table for either a 0.05 and 0.01 test identifying 95 and 99 percent confidence level respectively. Compare the critical value to the previously calculated correlation coefficient. If the correlation coefficient is greater, the results are said to be of significance.

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- Confidence intervals for the correlation coefficient may also be of use in population studies.