Pearson's Correlation Coefficient

Why x_i−x̄ exists: this asks one question per person: is their practice-hours above or below average? Positive means "practiced more than typical," negative means "less." It's the same gap-from-mean idea as variance — but now tracked for two variables at once, so we can compare their patterns.

Why y_i−ȳ exists: the matching question for bowling speed — above or below that person's group average? On its own it tells you nothing about practice hours. The relationship only appears once you look at both deviations for the same person together.

Why multiply the two deviations? Multiplying two same-sign numbers gives positive; two opposite-sign numbers give negative. So each person contributes a positive amount if they're above-average on both variables or below-average on both — and a negative amount if they're above on one but below on the other. Sum that across everyone and the sign tells you the overall direction of the relationship. Try the live demo below: drag points between quadrants and watch r flip sign.

Why divide by √Σdx²·√Σdy²? The numerator alone is in mixed units (hours × mph) and its size depends on how spread out the data happens to be — more spread means a bigger raw number even with the same underlying relationship. Dividing by each variable's own spread cancels the units and rescales everything to a universal −1-to-1 scale, so r means the same thing regardless of whether you measured in hours or minutes, mph or km/h.