The can be used to measure the linear association between two variables.

This correlation coefficient always takes on a value between -1 and 1 where:

• -1: Perfectly negative linear correlation between two variables.
• 0: No linear correlation between two variables.
• 1: 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-r²

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

To calculate the p-value for a Pearson correlation coefficient in R, you can use the cor.test() function.

cor.test(x, y)

The following example shows how to use this function in practice.

Example: Calculate P-Value for Correlation Coefficient in R

The following code shows how to use the cor.test() function to calculate the p-value for the correlation coefficient between two variables in R:

#create two variables
x <- c(70, 78, 90, 87, 84, 86, 91, 74, 83, 85)
y <- c(90, 94, 79, 86, 84, 83, 88, 92, 76, 75)

#calculate correlation coefficient and corresponding p-value
cor.test(x, y)

	Pearson's product-moment correlation

data:  x and y
t = -1.7885, df = 8, p-value = 0.1115
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.8709830  0.1434593
sample estimates:
       cor 
-0.5344408

From the output we can see:

• The Pearson correlation coefficient is -0.5344408.
• The corresponding p-value is 0.1115.

Since the correlation coefficient is negative, it indicates that there is a negative linear relationship between the two variables.

However, since the p-value of the correlation coefficient is not less than 0.05, the correlation is not statistically significant.

#create two variables
x <- c(70, 78, 90, 87, 84, 86, 91, 74, 83, 85)
y <- c(90, 94, 79, 86, 84, 83, 88, 92, 76, 75)

#calculate p-value for correlation between x and y
cor.test(x, y)$p.value

[1] 0.1114995

The p-value for the correlation coefficient is 0.1114995.

This matches the p-value from the previous output.