Correlation: meaning, types and Karl Pearson's coefficient

1. Meaning and utility

Correlation is the relationship between two variables such that a change in one is accompanied by a change in the other. Croxton and Cowden: “When the relationship is of a quantitative nature, the appropriate statistical tool for discovering and measuring the relationship and expressing it in a brief formula is known as correlation.” The coefficient of correlation (r) measures its degree and direction.

Utility: it measures the degree of relationship in one figure; it is the first step to regression and prediction (if price and demand are correlated, demand can be estimated from price); it helps business decisions (advertising and sales, rainfall and crop yield); it reduces uncertainty in forecasting; and in economics it helps to test theories (income and consumption).

Correlation is not causation. Two variables may move together because both depend on a third (ice-cream sales and drowning deaths both rise in summer), by pure chance (spurious or nonsense correlation), or because each affects the other.

2. Kinds of correlation

• Positive and negative. Positive: both move in the same direction (income and expenditure). Negative: they move in opposite directions (price and demand).
• Simple, multiple and partial. Simple: two variables. Multiple: one variable with two or more others together (yield with rainfall and fertiliser). Partial: two variables with the effect of others held constant.
• Linear and non-linear (curvilinear). Linear: the ratio of change is constant and the points lie near a straight line. Non-linear: the ratio varies and the points follow a curve.
• Degrees: perfect (r = ±1), high (±0.75 to ±1), moderate (±0.25 to ±0.75), low (0 to ±0.25), and no correlation (r = 0).

Methods of study: scatter diagram (graphic), Karl Pearson’s coefficient, Spearman’s rank correlation, and the concurrent deviation method.

Scatter diagram: plot each pair (x, y) as a dot. Dots rising from bottom left to top right show positive correlation; falling, negative; a straight-line band, high correlation; a shapeless cloud, none. It is simple and shows non-linear patterns, but gives no exact figure.

3. Karl Pearson’s coefficient

r = Σxy / √(Σx² × Σy²), where x and y are deviations from the actual means. With deviations dx, dy from assumed means (or raw values):

r = [nΣdxdy − (Σdx)(Σdy)] / [√(nΣdx² − (Σdx)²) × √(nΣdy² − (Σdy)²)].

Properties: −1 ≤ r ≤ +1; it is independent of change of origin and scale (so assumed means and step deviations give the same r); it is a pure number; r is the geometric mean of the two regression coefficients, r = ±√(b_yx × b_xy); r² (coefficient of determination) gives the share of variation in y explained by x.

Merits: gives both direction and degree in one figure; based on all observations; the most widely used measure. Demerits: assumes a linear relationship; affected by extreme values; laborious; easily misinterpreted as proving causation.

4. Worked: temperature and germination time (January 2026)

Temperature (X): 57, 42, 40, 38, 42, 45, 42, 44, 40. Germination time (Y): 10, 26, 30, 41, 29, 27, 27, 19, 18. Take 47 and 26 as assumed means.

• Numerator = 9(−268) − (−33)(−7) = −2,412 − 231 = −2,643.
• √(9 × 367 − 33²) = √(3,303 − 1,089) = √2,214 = 47.05.
• √(9 × 621 − 7²) = √(5,589 − 49) = √5,540 = 74.43.
• r = −2,643 / (47.05 × 74.43) = −2,643 / 3,502.2 = −0.755.

Interpretation: there is a high negative correlation: the higher the temperature, the shorter the germination time.

Probable error (to judge significance): PE = 0.6745 × (1 − r²) / √n = 0.6745 × (1 − 0.5695) / 3 = 0.6745 × 0.4305 / 3 = 0.097. Since |r| (0.755) exceeds 6 PE (0.58), the correlation is significant. (Rule: if r < PE it is not significant; if r > 6 PE it is significant.)

How this is examined

Karl Pearson’s r with assumed means: January 2026 (ten marks); from 10 pairs of raw data: August 2024. Theory: “Define correlation. Explain the kinds of correlation” (January 2025), “Explain correlation analysis” (June 2025, ten marks), “Write a short note on types of correlation” (January 2026). In the essay, cover meaning, utility, kinds, methods and the properties of r, and add the warning that correlation does not prove causation.