The mean of n observations is X. If the first item is increased by 1, second by 2 and so on, then the new mean is
一
A. X +n
一
B. X+n/2
一
C. X+n+1/2
D. None of these
Answers
Given, mean of n observation is x
now, mean = sum of all
observation/total no. of observation
=> x= sum of all observation/n
=> sum of all observation = nx
=> X1+X2+X3.......+Xn = nx ____(1)
If terms are changed
i.e. X1+1, X2+2, .............Xn+n
sum of all observation=[ (X1+1)+(X2+2)+..............(Xn+n) ]
= [(X1+X2+X3.......+Xn)+(1+2+3...+n) ]
=[nx+ sum of first n terms] (from 1)
=[nx + n/2(2(1)+(n-1)(1))]
=[nx+ n/2(2+n-1)]
=[nx +n/2(n+1)]
=n[x+1/2(n+1)]
=n[x+n/2+1/2] _______(2)
Now, new mean = sum of all new
observation/total no. of observation
=>new mean =n[x+n/2+1/2]/n
=>new mean =x+n/2+1/2
So, correct answer is none of these.
New Mean= X + (n+1)/2 if The mean of n observations is X and first item is increased by 1, second by 2 and so on,
Step-by-step explanation:
The mean of n observations is X
=> Sum of n observations = nX
let say n terms are
a₁ , a₂ , a₃ ....................................aₙ
=> a₁ + a₂ + a₃ +................................+ aₙ = nX
the first item is increased by 1, second by 2 and so on
=> new n terms would be
a₁+1 , a₂+3 , a₃+3 ....................................aₙ+n
Sum of n terms would be
= a₁+1 , a₂+3 , a₃+3 ....................................aₙ+n
= (a₁ + a₂ + a₃ +................................+ aₙ ) + (1 + 2 + 3 +...........................+ n)
= nX + n(n+1)/2
= n ( X + (n+1)/2)
New Mean = n ( X + (n+1)/2)/n
=> New Mean= X + (n+1)/2
New Mean= X + (n+1)/2
option C is correct
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