Which correlation coefficient is resistant to outlier?

Spearman Rank correlation coefficient is resistant to outlier.

Formula for it is as below:

For example X=[x1,x2,x3,x4,x5,x6] and Y=[y1,y2,y3,y4,y5,y6]

Now to find r_x we need to order data of X in ascending order with marking its position in another arrray.

Example X=[1,3,0,5,3,6].

Now lets sort X and mark sorted array position.

Array is like X[1]=1,X[2]=3,X[3]=0,X[4]=5,X[5]=3,X[6]=6.

Now lets sort it 0(X[3]),1(X[1]),3(X[3]),3(X[4]),5(X[5]),6(X[6]) .

so it will be like 0,1,3,3,5,6. Now convert values to indexes of sorted order means X = [0(p[1]),1(p[2]),3(p[3]),3(p[4]),5(p[5]),6(p[6])]

Now check original sequence of X = 1,3,0,5,3,6. Now replace X’s value with sorted index like value 1 is at position 2 in sorted array p[2]. There are two 3 values and its sorted array position is p[3] and p[4] so, for two same value we will take mean of position means 3+4/2=3.5 so for both 3 values positions is 3.5 . Like wise for other values we will get sorted array postion. So, final values according to sorted values indexs for 1,3,0,5,3,6 will be 2,3.5,1,5,3.5,6. These values area called rank of values r₁=2,r₂=3.5,r₃=1,r₄=5,r₅=3.5,r₆=6. so r_x=[2,3.5,1,5,3.5,6]

Now come to Spearman Rank correlation coefficient. Its denote by p_rank. Its equation is as below

p_rank = 1 – (6Σ^N(rx_i -ry_i)²/N(N²-1))

Here N is size of array and rx_i or ry_i are rx₁ or ry₁ to rx_n or ry_n

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