One way to quantify the relationship between two variables is to use the Pearson correlation coefficient, which is a measure of the linear association between two variables*.*

It always takes on a value between -1 and 1 where:

- -1 indicates a perfectly negative linear correlation between two variables
- 0 indicates no linear correlation between two variables
- 1 indicates a perfectly positive linear correlation between two variables

To determine if a correlation coefficient is statistically significant, you can calculate the corresponding t-score and p-value.

The formula to calculate the t-score of a correlation coefficient (r) is:

**t** = r * √n-2 / √1-r^{2}

The p-value is calculated as the corresponding two-sided p-value for the t-distribution with n-2 degrees of freedom.

**Example: Correlation Test in R**

To determine if the correlation coefficient between two variables is statistically significant, you can perform a correlation test in R using the following syntax:

**cor.test(x, y, method=c(“pearson”, “kendall”, “spearman”))**

where:

**x, y:**Numeric vectors of data.**method:**Method used to calculate correlation between two vectors. Default is “pearson.”

For example, suppose we have the following two vectors in R:

x <- c(2, 3, 3, 5, 6, 9, 14, 15, 19, 21, 22, 23) y <- c(23, 24, 24, 23, 17, 28, 38, 34, 35, 39, 41, 43)

Before we perform a correlation test between the two variables, we can create a quick scatterplot to view their relationship:

#create scatterplot plot(x, y, pch=16)

There appears to be a positive correlation between the two variables. That is, as one increases the other tends to increase as well.

To see if this correlation is statistically significant, we can perform a correlation test:

#perform correlation test between the two vectors cor.test(x, y) Pearson's product-moment correlation data: x and y t = 7.8756, df = 10, p-value = 1.35e-05 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.7575203 0.9799783 sample estimates: cor 0.9279869

The correlation coefficient between the two vectors turns out to be **0.9279869**.

The test statistic turns out to be **7.8756 **and the corresponding p-value is **1.35e-05**.

Since this value is less than .05, we have sufficient evidence to say that the correlation between the two variables is statistically significant.

**Additional Resources**

The following tutorials provide additional information about correlation coefficients:

An Introduction to the Pearson Correlation Coefficient

What is Considered to Be a “Strong” Correlation?

The Five Assumptions for Pearson Correlation