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

Answer 1

The correct option is C the median decreases

Let's assume $x_1<x_2<x_3<.....<x_{11}$.

Here, the median $=x_6$

Now, we are replacing $x_{11}\left(\text{the largest term}\right)$ by median of $x_1,x_2,...,x_{10}.$
Median of $(x_1,x_2,...,x_{10})=\frac{x_5+x_6}{2}$

The new dataset is $x_1,x_2,...,x_5,\frac{x_5+x_6}{2},x_6,...,x_{10}$

The new median $=\frac{x_5+x_6}{2}<x_6$