Interpretation and Properties of the Correlation Coefficient

A computed correlation coefficient is only useful once it is correctly interpreted. Both Karl Pearson's $r$ and Spearman's $\rho$ share the same range and the same rules of interpretation.

Range: both $r$ and $\rho$ always lie between $-1$ and $+1$, inclusive: $-1 \le r \le +1$.

• $r = +1$: perfect positive correlation (every point lies exactly on an upward-sloping line).
• $r = -1$: perfect negative correlation (every point lies exactly on a downward-sloping line).
• $r = 0$: no linear correlation (this does not necessarily mean 'no relationship at all' — a strong non-linear/curvilinear relationship can still give $r=0$).
• Any value strictly between $0$ and $\pm1$ indicates a correlation of the corresponding sign, with a degree given by the size of $|r|$.

A commonly used scale for the degree of correlation (by the size of $|r|$, either sign):

Two properties worth remembering:

1. $r$ (and $\rho$) is a pure number — it carries no unit, regardless of the units in which $X$ and $Y$ are measured (rupees, kilograms, marks, etc.), which is what makes it possible to compare the strength of correlation across completely different pairs of variables.
2. $r$ is independent of change of origin and scale — adding/subtracting a constant to every value of $X$ or $Y$, or multiplying every value by a positive constant, never changes $r$ (this is the same property that makes the step-deviation method valid, in the section above). …