\"Skewing the average\" Suppose the numbers 1, 2,3,4,.. ,2018 are all written on
ID: 3069637 • Letter: #
Question
"Skewing the average" Suppose the numbers 1, 2,3,4,.. ,2018 are all written on a piece of paper. You are allowed to perform the following operation: choose two numbers, erase both of them from the piece of paper, and then write the average (mean) of the two numbers on the paper. For example, you could choose the numbers 2 and 4 and then after this operation the paper would have 1, 3, 3,5, , 2018 written on it. You may then apply this operation to the new list of numbers (erasing two of them and then writing the mean of the two erased numbers), and keep going until there is only one number written on the paper. Is it possible that the final number on the paper is 2? Justify your answer.Explanation / Answer
No, it is not possible that the final number on the paper is 2.
Because, if we continue the process untill we get the single number then the final list consist all the numbers equal to an average of all the given number in the original dataset.
Let's take small example as 1, 2, 3, 4.
Do the above process randomly.
Suppose we choose 1 and 2
Then we get the new list as 1.5, 1.5, 3, 4
Next if we choose 1.5 and 3 then we get (1.5+3)/2 = 2.25
So the new list is 1.5, 2.25, 2.25, 4
Suppose we take 2.25 and 4 so we get the new list as 1.5, 2.25, 3.125, 3.125
Then, if we take 1.5 and 4 then we get 2.3125, 2.25, 3.125, 2.3125
Then we take 2.25 and 3.125, so we get 2.3125, 2.6875, 2.6875, 2.3125
If we take 2.3125 and 2.6875 then the new list is 2.5, 2.5, 2.6875,2.3125
If we take last two numbers then the new list may be 2.5, 2.5, 2.5, 2.5
Note that 2.5 is the is the average of 1, 2, 3, 4.
So if we continue the process untill we get the single number then this sigle numer must be the average of all the given number in the original dataset.
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