Correlation
Class 11 Economics · Statistics for Economics · Chapter 6
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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 \(+1\).
Correlation
Key terms
Correlation
The statistical relationship between two variables, measuring the direction and intensity of their co-movement (covariation, not causation).
Positive correlation
A relationship in which two variables move in the same direction, such as income and consumption.
Negative correlation
A relationship in which two variables move in opposite directions, such as price and demand.
Scatter diagram
A graph of paired values of two variables that shows the direction and closeness of their relationship without a numerical value.
Karl Pearson's coefficient
A numerical measure of the degree of linear correlation, lying between \(-1\) and \(+1\) and free of units.
Spearman's rank correlation
A measure of correlation based on the ranks of items, useful for qualitative data and data with extreme values.
Extra questions & answers
Try one from this chapter
Correlation measures the direction and intensity of: