Question

Let $x_1,x_2,....x_{11}$ be $11$ distinct positive integers, If we replace the largest of these integers by the median of the other 10 integers, then

• A. the median remains the same
• B. the mean increases
• C. the median decreases
• D. the mean remains the same

Answer 1

Answer: (C)

the median decreases

Let $x_1,x_2,x_3,x_4,...........x_{11}$ are distinct positive integers

Suppose the increasing order of the numbers are

$\Rightarrow$ $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11}$

$x_{11}$ is largest number and the median is $x_6.$

Now, the median of first $10$ numbers $=\frac{x_5+x_6}{2}$

$\Rightarrow$ Let $a=\frac{x_5+x_6}{2}$

Now, we have to replace largest number $x_{11}$ by the median of first $10$ numbers i.e. $a$.

So, new increasing order will be

$\Rightarrow$ $x_1,x_2,x_3,x_4,x_5,a,x_6,x_7,x_8,x_9,x_{10}$

And new median will be $a$ i.e. $\frac{x_5+x_6}{2}$

Which is lesser than the median of first eleven number $x_6$

$\Rightarrow$ So, median decreases.