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# In a set of 2n distinct observations, each of the observations below the medianof all the observations is increased by 5 and each of the remaining observationsis decreased by 3. Then the mean of the new set of observations will be:$(A)$ increases by 1$(B)$ decreases by 1$(C)$ decreases by 2$(D)$ increases by 2  Verified
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Hint:- Find median of 2n numbers and then split 2n numbers in two parts.

As, there are 2n numbers. So, let these numbers be,
$\Rightarrow {x_1},{\text{ }}{x_2},{\text{ }}{x_3},{\text{ }}......{\text{ }}{x_{n - 1}},{\text{ }}{x_n},{\text{ }}{x_{n + 1}},{\text{ }}.......{\text{ }}{x_{2n - 1}},{\text{ }}{x_{2n}}$
And as we know that mean of numbers is,
$\Rightarrow$Mean $= {\text{ }}\dfrac{{{\text{Sum of the numbers}}}}{{{\text{Total number of numbers}}}}$
So, mean of 2n numbers given above will be,
Let mean of 2n numbers $= {\text{ }}\bar X$
$\Rightarrow$Then, $\bar X = \dfrac{{{x_1} + {x_2} + ..... + {x_n} + {x_{n + 1}} + ...... + {x_{2n}}}}{{2n}}{\text{ }}$ (1)
As, given in the question that total terms are 2n which is even.
$\Rightarrow$So, median of 2n terms will be $\dfrac{{{x_n} + {x_{n + 1}}}}{2}.$
So, median will lie between terms ${{\text{x}}_n}$ and ${{\text{x}}_{n + 1}}$
As, we are given that each observation below median is increased by 5
$\Rightarrow$So, for each i from ${\text{i}} = 1$to ${\text{i}} = n$.
$\Rightarrow $${{\text{x}}_i} becomes {{\text{x}}_i} + 5 As, we are given that each observation above median is decreased by 3. \Rightarrow So, for each i from {\text{i}} = n + 1to {\text{i}} = 2n. \Rightarrow$${{\text{x}}_i}$ becomes ${{\text{x}}_i} - 3$
So, now mean of 2n observations can be written as,
$\Rightarrow$New mean$= \dfrac{{({x_1} + 5) + ({x_2} + 5) + ......... + ({x_n} + 5) + ({x_{n + 1}} - 3) + .....({x_{2n}} - 3)}}{{2n}}$
Solving above equation. It becomes,
$\Rightarrow$So, new mean$= \dfrac{{{x_1} + {x_2} + ..... + {x_n} + 5n + {x_{n + 1}} + ...... + {x_{2n}} - 3n}}{{2n}}$
Solving the above equation using equation 1. It becomes,
$\Rightarrow$New mean$= \dfrac{{{x_1} + {x_2} + ..... + {x_n} + {x_{n + 1}} + ...... + {x_{2n}}}}{{2n}} + \dfrac{{2n}}{{2n}}{\text{ = }}\bar X + 1$
So, the new mean of 2n numbers is 1 more than the previous mean.
$\Rightarrow$Hence, the correct option will be A.

Note:- Whenever we came up with this type of problem then first find that what will be the new numbers after the given conditions and then you should find their mean by the formula Mean $= {\text{ }}\dfrac{{{\text{Sum of the numbers}}}}{{{\text{Total number of numbers}}}}$
and then we can easily find the change in mean.