This document introduces the rank correlation coefficient and how it is derived. Rank correlation measures the relationship between the ranks of individuals based on two characteristics. Spearman introduced rank correlation. The derivation shows how to calculate Spearman's rank correlation coefficient (ρ) using the deviations between ranks (di) and sample size (n). The coefficient ranges from -1 to 1, where 1 is total positive correlation, 0 is no correlation, and -1 is total negative correlation.

1. RANK CORRELATION COEFFICIENT
Syed . M Parveen
Dept.of Statistics
Maris Stella College
Vijayawada,AP.

2. Rank Correlation
Rank correlation was introduced by Spearman.
Let x1,x2,x3,…xn and y1,y2,y3,….yn be the ranks of n individuals of
the characteristics A and B respectively. The correlation obtained
for the ranks of individuals of two characteristics A and B is called
Rank Correlation.
Ranks of the individuals may or may not be the same.

3. Derivation of Spearman Rank Correlation Coefficient
Possibility 1. Ranks of the individuals are different ( not same).
Let the ranks of the individuals due to the characteristics A and B be 1,2,3,….n
(which are need not be in the same order) and in general xi ≠ y i .

4. x̅ = 1/n Ʃ xi ;
= 1/n *(1+2+3+….+n)
=n(n+1)/2n
=(n+1)/2 ………………………………..(1)
Similarly y̅ = (n+1)/2
x̅ = y̅
σx2 = var (x) = 1/n Ʃ xi
2 –( x̅)2
= 1/n * ( 12+22+32+……n2)-((n+1)/2)2
=(n(n-1)(2n-1)/6n)-((n+1)/2)2
=(n2-1)/12
Similarly σy2 = var (y) = (n2-1)/12

5. Let ρ be the rank correlation coefficient
ρ = Cov (X,Y)/ σ x *σ y
Cov (X,Y) = ρ * σ x * σ y
Let us consider deviation of the ranks as di
d i = xi – y i
d i = (xi- x̅)-( y i- y̅)
Squaring , summing over i from 1 to n and dividing with n, we
get
1/n Ʃ di
2 = 1/n Ʃ [ ( xi- x̅) – (y i- y̅)]2

6. =1/n Ʃ(xi- x̅)2 + 1/n Ʃ (yi- y̅)2 -2 *1/n Ʃ(xi- x̅)(yi- y̅)
=σx2+σy 2 -2*cov(X,Y) ……………………….[from equations (2), (4)]
=2* σx2 – 2 ρ * σx *σy (since σx2= σy 2 , σx = σy)
1/n* Ʃ di
2= 2* σx2( 1-ρ)
1 – ρ = (Ʃdi2) /2n σx2
= (Ʃdi2) / 2n((n2-1)/12)
= 6 (Ʃdi2) / n (n2-1)
ρ = 1 - 6 (Ʃdi2) / n (n2-1)