Pearson correlation coefficient

In statistics, the Pearson correlation coefficient (also normalised covariance) $r$ measures how well two sets of data are linearly correlated. $r$ is computed as the average of the products of the z-scores of each data point:

$$r=\frac{1}{n-1}\sum _{i=1}^n\left(\frac{x_i-\bar{x}}{s_x}\right)\left(\frac{y_i-\bar{y}}{s_y}\right)=\frac{{\sum }_{i=1}^n{x}_i{y}_i-n\bar{x}\bar{y}}{\sqrt{{\sum }_{i=1}^n{x}_i^2-n{\bar{x}}^2}\sqrt{{\sum }_{i=1}^n{y}_i^2-n{\bar{y}}^2}}$$

where $\bar{x},\bar{y}$ are the arithmetic means and $s_x,s_y$ are the standard deviations. The right formula is easiest to hand compute. This is more succinctly defined in terms of the covariance:

$${\rho }_{X,Y}=\frac{\text{cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$$

The correlation coefficient is defined in the interval: $r\in [-1,1]$. Strongly correlated sets of data will produce values with a greater absolute value. For $r>0$, we have a positive association. For $r<0$, we have a negative association. If $r=0$, we have no correlation.

In most cases, $r$ is computed on a sample of the population, i.e., it is a sample correlation and estimates the actual population correlation.

See also

• Least squares