Definition: The correlation coefficient r measures the strength and direction of a linear relationship between two quantitative variables, ranging from negative 1 to positive 1.
What does r tell you?
The value r is always between -1 and 1. The sign shows direction, and the size shows how tightly points hug a line. Values near 0 mean little linear association. It has no units and does not change if you swap x and y.
How do you interpret it?
Squaring r gives r2, the fraction of variation in y explained by the linear relationship with x.
Worked example: for hours of study and quiz score, r = 0.8. Then r2 = 0.64, so about 64% of the variation in scores is accounted for by the line. For r = -0.8, the strength is the same but the trend goes downward.
Formula in words: r is the average of the products of the z-scores of x and y, using n – 1 in the denominator.
What mistakes are common?
Claiming that correlation proves causation, and using r for curved patterns. A perfect curve can still give r near 0. Mathbot looks at the scatterplot before quoting r.