In: Statistics and Probability
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