Chapter 6 — Correlation

Correlation analysis is a means for examining relationships between two variables systematically. It studies and measures the direction and intensity of relationship among variables. Importantly, correlation measures covariation, not causation — the presence of correlation between two variables simply means that when one variable changes, the other changes in a definite direction, either the same (positive) or opposite (negative).

Correlation is commonly classified into positive and negative correlation. Positive correlation exists when variables move together in the same direction — for example, when income rises, consumption also rises. Negative correlation exists when variables move in opposite directions — for example, when the price of apples falls, its demand increases.

The three important tools used to study correlation are: (1) Scatter diagrams, which visually present the nature of association without giving any specific numerical value; (2) Karl Pearson's coefficient of correlation, which gives a precise numerical value of the degree of linear relationship between two variables; and (3) Spearman's rank correlation, which measures the linear association between ranks assigned to individual items according to their attributes.

A scatter diagram is a useful technique for visually examining the form of relationship without calculating any numerical value. The values of two variables are plotted as points on a graph paper. The degree of closeness of the scatter points and their overall direction enable examination of the relationship. If all points lie on a line, the correlation is perfect; if the scatter points are widely dispersed, the correlation is low.

Key properties of r include: it has no unit (it is a pure number); its value lies between –1 and +1 (values outside this range indicate a calculation error); a negative value indicates an inverse relation; if r = 0 the two variables are uncorrelated (no linear relation); if r = 1 or r = –1 the correlation is perfect; and the magnitude of r is unaffected by change of origin and change of scale.

If r = 0, the two variables are uncorrelated — there is no linear relation between them, though other types of relation may exist. If r = 1 or r = –1, the correlation is perfect and there is an exact linear relation between the variables. A high value of r (close to +1 or –1) indicates a strong linear relationship, while a low value (close to zero) indicates a weak linear relation.

No. Correlation measures covariation, not causation. Correlation should never be interpreted as implying a cause-and-effect relation. For example, the relation between the arrival of migratory birds in a sanctuary and birth rates in a locality cannot be given any cause-and-effect interpretation — the relationship is simple coincidence. A third variable may also be the cause behind a high correlation between two other variables.

Spearman's rank correlation should be used in the following situations: (1) when variables like height and weight cannot be precisely measured but individuals can be ranked; (2) when dealing with attributes such as fairness, honesty, or beauty that cannot be measured numerically; (3) when there is a non-linear relationship whose direction is clear; and (4) when data contains extreme values, since Spearman's coefficient is not affected by them.

The step deviation method reduces the burden of calculation when variable values are large. It uses the property that r is independent of change in origin and scale. Variables X and Y are transformed as U = (X – A)/B and V = (Y – C)/D, where A and C are assumed means and B and D are common factors of the same sign. Since rUV = rXY, the correlation can be calculated on the simpler transformed values.

When ranks are repeated, a common rank is given to the repeated items — the common rank is the mean of the ranks those items would have assumed if slightly different from each other. A correction factor is then added to the formula for each set of repeated ranks. The correction term for m repeated ranks is (m³ – m)/12, and all such correction factors are summed and added to ΣD² in the formula before computing rs.

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Karl Pearson's coefficient of correlation r is defined as the ratio of the covariance of X and Y to the product of their standard deviations. In terms of deviations from the mean, r = Σxy / (N · σx · σy), where x = X – X̄ and y = Y – Ȳ are deviations from their respective means. An equivalent computational formula is r = (NΣXy – (ΣX)(ΣY)) / √[(NΣX² – (ΣX)²)(NΣY² – (ΣY)²)].

In Example 1, the correlation between years of schooling of farmers and annual yield per acre was calculated using seven data points. The resulting Karl Pearson's coefficient r was 0.644, indicating that years of education of farmers and annual yield per acre are positively correlated. The chapter notes this implies that more years farmers invest in education, the higher the yield per acre, underlining the importance of farmers' education.