The Pearson Product-Moment Correlation Coefficient (also known as r) is a value that measures the correlation between two variables $\mathit{X}$ and $\mathit{Y}$.

The coefficient value ranges from $-1$to $1$, where a positive value represents a positive correlation, a value of $0$ represents no correlation and a negative value represents a negative correlation.

Values of ±$1$ are considered total correlations and only occur when all of the data points lie on a straight line.

Data sets with values of r near 0 are considered to have weak correlations, while data sets with values of r near $\+1$ or $-1$ are considered to have strong correlations.

The formula used to calculate the Pearson Product-Moment Correlation Coefficient (r) on a given sample is

$r \= \frac{n\cdot \left(\sum x_i \cdot y_i\right)-\left(\sum x_i\right)\left(\sum y_i\right)}{\sqrt{n\cdot \sum x_i^2-{\left(\sum x_i\right)}^2}\cdot \sqrt{n \cdot \sum y_i^2-{\left(\sum y_i\right)}^2}}$

where n represents the amount of data points and $x_i\,y_i$ are the samples indexed with $i$.

The coefficient of determination measures how much of the variation in the value of y can be explained by the relationship between x and y.

This value, as a percent, can be calculated with  $r^2\:$

Example: if $r \= 0.5$ then the coefficient of determination of the data set is  0.5²  which is  0.25.

This means that 25% of the variation in the value of y can be explained by the relationship between x and y. The other 75% remains unexplained.