Coefficient of Correlation (r)
(when deviations are taken from zero)
Rank Difference Correlation Coefficient ( ρ )
r =[ Σxy - N * MxMy ] /√[ (ΣX2 - NM2x)*(ΣY2 - NM2y)] -----(1)
σx = √[(ΣX2/N) - M2x)]
σy = √[(ΣY2/N) - M2y)]
* by subtraction of fixed amount from X and Y.
r* = (NΣX'Y' - ΣX'*ΣY' ) / (√([NΣX'2 - (ΣX' )2] *√[NΣY'2 - (ΣY' )2] ) ----(2)
where X'=X-a and Y'=Y-b;
Equn. 1 & Equn.2 are alternate forms.
Correlation coefficient from Ranking score (not tied ) : ρ =1 - [6 * ΣD2 / N( N2-1 )] where D = x - y;
When in stead of absolute x-score or y-score, Judge X and Judge Y give a ranking score for traits A, B,C etc such as rank 1,2,3,4,5,........ a coefficient of correlation is found out
provided rank for similar trends are not tied, such that say for trait A, both Judge X & Judge Y scores are 2. If one treats the ranked data as absolute data, one gets the r under *
When ranks are treated as scores and there are no ties, ρ = r . ρ provides a quick and convenient way of estimating the correlation when N is small or when we only have ranks.
In case of large N, the ρ is still useful for exploratory purpose.