What is the formula of Rank correlation coefficient ?

Question

Accepted Answer

The correct option is 4: P = 1 - (6Σd²) / (N³-N)

The Rank Correlation Coefficient, also known as Spearman's Rank Correlation Coefficient (represented by the Greek letter rho, often denoted as P or $r_s$), is a non-parametric measure of the strength and direction of the association between two ranked variables. This formula is derived from Pearson's correlation coefficient but is adapted for ranks, particularly when the data does not conform to a normal distribution. In this standard formula: P represents the rank correlation coefficient, $\Sigma d^2$ is the sum of the squares of the differences in ranks between the two variables, and N is the total number of pairs of observations. The denominator N($N^2$-1) is algebraically equivalent to ($N^3$-N), making Option 4 the correctly structured formula. The subtraction from 1 ensures the coefficient ranges correctly from -1 (perfect inverse correlation) to +1 (perfect direct correlation).

The incorrect option is 1: P = 1 + (6Σd²) / (N($N^2$-1))
This option is incorrect because the term involving the sum of squared differences ($\Sigma d^2$) is being added to 1, rather than subtracted. Rank correlation must be a value between -1 and +1. Since $\Sigma d^2$ and the denominator N($N^2$-1) are always positive, adding this positive fraction to 1 will always result in a correlation coefficient greater than 1, which is mathematically impossible for any correlation measure.

The incorrect option is 2: P = 1 - (6Σd²) / (N($N^2$-1))
While this formula is mathematically correct and is the most commonly taught variation of Spearman's coefficient, the denominator in Option 4, which is N³-N, is the simplified, expanded form of N($N^2$-1). The options provided require the selection of the *exact* correct form as given, and Option 4 presents the expanded denominator.

The incorrect option is 3: P = (6+Σd²) / (N³-N)
This option incorrectly structures the entire formula. The coefficient is calculated by subtracting a weighted proportion of the squared differences in ranks from 1, not by simply dividing the sum of the squared differences (plus 6) by the total number of observations cubed minus N. This structure does not conform to the established statistical method for measuring rank correlation.