Correlation

Where a measure of dispersion describes a single variable, correlation studies the relationship between two variables — whether, and how strongly, they move together. Does advertisement expenditure move with sales? Does rainfall move with agricultural output in Andhra Pradesh? Correlation summarises this relationship in a single figure, the correlation coefficient ($r$), which always lies between $-1$ and $+1$:

• $r = +1$ — perfect positive correlation: the two variables move in the same direction, in exact proportion.
• $r = -1$ — perfect negative correlation: the two variables move in exactly opposite directions.
• $r = 0$ — no correlation: no linear relationship exists between the two variables.
• Any value strictly between these extremes indicates the degree of positive or negative correlation — the closer $|r|$ is to 1, the stronger the relationship, and the closer it is to 0, the weaker.

Karl Pearson's Coefficient of Correlation, the most widely used method, uses the actual numerical values of both series. By the deviation-from-actual-mean method:

where $x = X - \bar{X}$ and $y = Y - \bar{Y}$ are deviations of each series from its own mean. An equivalent direct-value formula, useful when the mean is not a whole number, is:

Spearman's Rank Correlation Coefficient is used instead when data is available only as ranks rather than actual measured values (for example, two judges' rankings of contestants, or a ranking of preferences), or when the underlying values cannot be meaningfully measured:

$r_s = 1 - \frac{6 \Sigma d^2}{n(n^2 - 1)}$ …