Correlation

In economics, two quantities often move together — for example, as the price of a good rises, its demand falls; as income rises, saving rises. Correlation measures the relationship between two variables: whether they move together, and how closely. It does not prove that one causes the other; it only shows that they are related.

Correlation can be of these types:

• Positive correlation — the two variables move in the same direction (both rise together or both fall together), e.g. income and consumption.
• Negative correlation — the two move in opposite directions (one rises as the other falls), e.g. price and demand.
• Zero (no) correlation — no clear relationship.

The simplest way to see correlation is a scatter diagram — we plot each pair of values as a dot on a graph (one variable on each axis). The pattern of dots reveals the relationship: dots rising from left to right show positive correlation; dots falling from left to right show negative correlation; scattered dots with no pattern show no correlation. The more closely the dots cluster around a straight line, the stronger the correlation.

A scatter diagram shows the direction of correlation but not its exact strength. For an exact numerical measure we use Karl Pearson's coefficient of correlation (r). Using deviations from the mean (dx = X − X̄, dy = Y − Ȳ), the formula is:

r = Σ(dx·dy) ÷ √(Σdx² × Σdy²)

The value of r always lies between −1 and +1:

• r = +1 → perfect positive correlation.
• r = −1 → perfect negative correlation.
• r = 0 → no correlation.
• Values near ±1 mean strong correlation; values near 0 mean weak correlation.

Worked example. Find r for X: 10, 20, 30, 40, 50 and Y: 20, 40, 60, 80, 100.

• Means: X̄ = 150÷5 = 30; Ȳ = 300÷5 = 60.
• dx: −20, −10, 0, 10, 20. dy: −40, −20, 0, 20, 40.
• Σ(dx·dy) = 800 + 200 + 0 + 200 + 800 = 2000.
• Σdx² = 400 + 100 + 0 + 100 + 400 = 1000; Σdy² = 1600 + 400 + 0 + 400 + 1600 = 4000.
• r = 2000 ÷ √(1000 × 4000) = 2000 ÷ √4000000 = 2000 ÷ 2000 = +1.

So X and Y have perfect positive correlation — here Y is always exactly twice X. Pearson's r is the most widely used measure, but it assumes a linear relationship and (unlike the median) is affected by extreme values.

Sometimes data cannot be measured in exact numbers but can be ranked — for example, the ranking of contestants by two judges, or items rated by beauty, honesty or efficiency. For such qualitative or ranked data we use Spearman's rank correlation coefficient (R):

R = 1 − [ 6ΣD² ÷ N(N² − 1) ]

where D is the difference between the two ranks of each item and N is the number of items. Like Pearson's r, R also lies between −1 and +1.

Worked example. Two judges rank 5 paintings as follows — Judge A: 1, 2, 3, 4, 5 and Judge B: 2, 1, 4, 3, 5. Find the rank correlation.

• Differences D = A − B: −1, 1, −1, 1, 0.
• D²: 1, 1, 1, 1, 0; ΣD² = 4. N = 5, so N(N² − 1) = 5 × (25 − 1) = 5 × 24 = 120.
• R = 1 − (6 × 4) ÷ 120 = 1 − 24÷120 = 1 − 0.2 = +0.8.

So the two judges' rankings have a high positive agreement (R = +0.8). Interpreting a correlation coefficient: a value close to +1 means a strong direct relationship, close to −1 a strong inverse relationship, and close to 0 a weak or no relationship. But always remember the golden caution: correlation is not causation — a high correlation between two things (like ice-cream sales and drowning) may be due to a third common factor (summer heat), not because one causes the other.