Question

In: Statistics and Probability

*Regarding Pearsons Correlation Coefficent: explain why the average rank of a vector with n observations is...

*Regarding Pearsons Correlation Coefficent:

explain why the average rank of a vector with n observations is always n/2 + 1/2
assume that ties amongst observations are broken at random. so no two observations ever have same rank

Solutions

Expert Solution

So here we have to assume no two observation have same ranks. And if two observation is same say Xi and Xj is same then any one of them will be ranked k and other (k+1) and that will be selected randomly as given in question.

For example let us say we have following observations

Srudent names A B C D E F G H
marks 43 41 49 42 33 41 42 41

Marks can be arranged as - 49>43>42>41>33

Now to rank: C gets rank 1 and A ranked 2. Now D and G both got 42 and should be ranked 3. But to break the tie we will choose anyone of them randomly (say G ) and rank him 3 then D gets rank 4. Again we have tie at 41. Here also instead of giving B,F,H same rank 5 we will choose randomly and rank 5,6,7.

Say our final rank becomes

Srudent names A B C D E F G H
ranks 2 5 1 4 8 6 3 7

Here note that we are getting a set of natural number {1,2,3,4,5,6,7,8} as rank of students.


Clearly we will have set of natural number {1,2,3,4,5,....,n} as rank for n number of observations

Summing up we get sum of the rank     

Since we have n observation we will devide sum of rank by n and will get the required result that is


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