# Drastic drop in Somers' D ? Why?

I came across to find the correlation between the ratings assigned by two coaches to a same group of 40 players.

I have tabulated the results as below:

The Somers' D is 50%.

However, for the case below,

The Somers' D is 94.7%.

My question is, why both scenarios are having 2 deviations but the first scenario has so much lower Somers' D compared to the second scenario?

Say you have a set of pairs $$(x_i,y_i)$$, Somers D is linked to Kendall's tau, which is defined by :
$$\tau=\frac{N_C-N_D}{n(n-1)/2}$$
Where $$N_C$$ is the number of concordant pairs (ranks of both elements agree, $$x_i>x_j$$ and $$y_i>y_j$$ or $$x_i and $$y_i) and $$N_D$$ the umber of pairs that are not. First it is possible that there might be a problem with the special case where $$x_i=x_j$$ or $$y_i=y_j$$, of which you have a lot of examples. This case might be handled differently depending on your implementation. Overall it quantifies the number of difference between the number of concordant and discordant pairs. By construction Kendall's tau is symetric in X and Y. (And this is probably what you wanted to use.)
Somers D, treat the variables differently. Somers D of $$Y$$ with respect to $$X$$ is defined as : $$D(Y,X)=\tau(X,Y)/\tau(X,X)$$
That is it quantifies the difference between the number of concordant and discordant pairs, divided by the number of pairs with $$X$$ values in the pair being unequal. You can see that it is not symmetric. It explain that there is a difference between your cases.