# Correlation Coefficient Calculator

## Calculate correlation coefficients step by step

For the given two sets of values, the calculator will find the Pearson correlation coefficient between them (either sample or population), with steps shown.

Related calculator: Sample/Population Covariance Calculator

Comma-separated.
Comma-separated.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Find the Pearson correlation coefficient between $\left\{1, 2, 3, 4, 5\right\}$ and $\left\{1, 3, 6, 5, 8\right\}$.

### Solution

The Pearson correlation coefficient is the ratio of the covariance and the product of the standard deviations: $r = \frac{cov(x,y)}{s_{x} s_{y}}$.

The standard deviation of $\left\{1, 2, 3, 4, 5\right\}$ is $s_{x} = \frac{\sqrt{10}}{2}$ (for steps, see standard deviation calculator).

The standard deviation of $\left\{1, 3, 6, 5, 8\right\}$ is $s_{y} = \frac{\sqrt{730}}{10}$ (for steps, see standard deviation calculator).

The covariance between $\left\{1, 2, 3, 4, 5\right\}$ and $\left\{1, 3, 6, 5, 8\right\}$ is $cov(x,y) = 4$ (for steps, see covariance calculator).

Thus, $r = \frac{cov(x,y)}{s_{x} s_{y}} = \frac{4}{\frac{\sqrt{10}}{2} \frac{\sqrt{730}}{10}} = \frac{8 \sqrt{73}}{73}$.

The Pearson correlation coefficient is $\frac{8 \sqrt{73}}{73}\approx 0.936329177569045$A.