CBSE Class 11 — Notes, Chapters & Practice Quizzes

Chapter 6: Correlation

Summary

Correlation analysis studies the relationship between two variables, measuring both the direction and the intensity of their co-movement. It measures covariation, not causation: a correlation may reflect a cause-and-effect link, a coincidence (spurious correlation, like shoe size and pocket money), or the influence of a hidden third factor (temperature behind both ice-cream sales and drowning deaths). Correlation is positive when the variables move in the same direction (income and consumption) and negative when they move in opposite directions (price of apples and their demand). Three techniques measure correlation. A scatter diagram plots the paired values and shows the direction and closeness of the relationship visually but gives no number; if all points lie on a line, correlation is perfect. Karl Pearson's coefficient of correlation gives a precise numerical value of the degree of a linear relationship: \(r=\dfrac{\sum xy}{N\,\sigma_x \sigma_y}\), where \(x=X-\bar{X}\) and \(y=Y-\bar{Y}\). It has no unit, lies between \(-1\) and \(+1\), and is unaffected by change of origin and scale; \(r=0\) means no linear relation and \(r=\pm1\) means a perfect linear one. Spearman's rank correlation, \(r_s=1-\dfrac{6\sum D^2}{n^3-n}\), is used when items can only be ranked (qualities like beauty or honesty), when the relationship is clearly non-linear, or when data have extreme values, since it is not distorted by them; a correction factor is added for repeated ranks. Both coefficients lie between \(-1\) and \(+1\).

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