Karl Pearson's Coefficient of Correlation

Karl Pearson's Coefficient of Correlation (denoted $r$) is the most widely used numerical measure of the degree of linear correlation between two quantitative variables $X$ and $Y$.

Actual mean (deviation) method — deviations are taken from the actual arithmetic means $\overline{X}$ and $\overline{Y}$: with $x=X-\overline{X}$ and $y=Y-\overline{Y}$,

Direct (raw-score) method — avoids computing the means first, working directly with the raw totals:

Step-deviation (short-cut) method — convenient when the actual means are not whole numbers or the values are large: choose any convenient assumed means $A$ (for $X$) and $B$ (for $Y$), and (optionally) common divisors $h_x, h_y$; compute $u=\dfrac{X-A}{h_x}$ and $v=\dfrac{Y-B}{h_y}$, then

A key property that makes the short-cut method valid: $r$ is independent of the change of origin and scale — subtracting any assumed mean and dividing by any constant divisor never changes the value of $r$. This is exactly why the three methods above always give the same numerical answer for the same data, and it is the standard way to cross-verify a computed $r$ (compute it once by the direct method and once by the step-deviation method — the two must agree).

Merits: gives one precise number capturing both the direction (sign) and the strength (magnitude) of a linear relationship; the most rigorous and widely used measure for quantitative data. …