Correlation

How two variables move together: scatter diagrams, positive and negative correlation, Karl Pearson’s coefficient r and Spearman’s rank correlation.

Key ideas

Correlation measures the direction and intensity of a linear relationship between two variables.
Positive: both move the same way (income and consumption); negative: one rises as the other falls (price and quantity demanded).
A scatter diagram shows the pattern at a glance: points rising, falling, on a line, or scattered with no pattern.
r lies between −1 and +1, has no unit, and does not change if a constant is added or every value is scaled.
r = 0 means no linear relationship; a curved relation may still exist.
Spearman’s method uses ranks, so it suits qualities such as beauty or honesty and data with extreme values.
Correlation does not prove cause and effect: it may be spurious or due to a third factor.

Pearson’s r (deviations from the means)	r = Σxy ÷ √(Σx² × Σy²)
From covariance	r = covariance ÷ (σx × σy)
Spearman’s rank correlation	rs = 1 − 6Σd² ÷ (n³ − n), d = difference in ranks
Range of r	−1 ≤ r ≤ +1
Reading r	+1 perfect positive; −1 perfect negative; near ±0.9 high; near ±0.1 low

Adding the rank differences instead of their squares for Σd² (or forgetting that the plain differences add to zero).
Using n² − 1 or n² in Spearman’s formula instead of n³ − n.
Accepting a value of r above 1 or below −1: it means a calculation slip.
Reading a high r as proof that one variable causes the other.

Find Karl Pearson’s coefficient of correlation for X: 17, 6, 17, 9, 6 and Y: 94, 61, 95, 67, 63 (to two decimal places).

Working: Σxy = 380, Σx² = 126, Σy² = 1160; r = Σxy ÷ √(Σx² × Σy²)

Answer: 0.99