Correlation Coefficient Calculator

What is the Correlation Coefficient?

The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It is widely used in data analysis, scientific research, and financial modeling to determine how changes in one variable correspond to changes in another.

Pearson Correlation Coefficient Formula

The Pearson product-moment correlation coefficient (denoted as $r$) is calculated using the following mathematical formula:

$$r = \frac{n \sum XY - (\sum X)(\sum Y)}{\sqrt{\left[n \sum X^2 - (\sum X)^2\right] \left[n \sum Y^2 - (\sum Y)^2\right]}}$$

Where:

• $n$ is the total number of paired data values.
• $\sum X$ and $\sum Y$ are the sums of the variables $X$ and $Y$.
• $\sum X^2$ and $\sum Y^2$ are the sums of the squared variables.
• $\sum XY$ is the sum of the product of paired variables.

Interpreting the correlation Coefficient ($r$)

The value of $r$ always ranges from $-1$ to $+1$:

• $r = 1$: Perfect positive linear relationship.
• $r = -1$: Perfect negative linear relationship.
• $r = 0$: No linear relationship between the variables.
• Values closer to $+1$ or $-1$ indicate stronger linear relationships, whereas values closer to $0$ indicate weaker linear relationships.