Coefficient of Correlation (r)

from

raw scores

(when deviations are taken from zero)

AND

Rank Difference Correlation Coefficient ( ρ )

 no. of subjects subtract from x - subtract from y - subject/ frequency score-x or Judge X score-y or Judge Y A B C D E F G H I J K L M N O Mean X (Mx) * Mean Y (My) * co-efficient of correlation-r * SD-    σx SD-    σy Rank Difference Correlation Coefficient(non tied)-  ρ

Formulae:

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.