An investigator has missed a value while collecting data in an experiment. Denoting the missing value by $x$, the observations are: $10,4,11,6,17,15,9,8,x$. What should be the value of $x$, if he wants $\text{mean}=\text{median}=\text{mode}$ for this data set?

Given observations are:

$10,4,11,6,17,15,9,8,x$

Sum of known observations:

$10+4+11+6+17+15+9+8=80$

There are total $9$ observations.

Check option $2$, that is $x=10$.

Then data becomes:

$10,4,11,6,17,15,9,8,10$

Arranging in ascending order:

$4,6,8,9,10,10,11,15,17$

Median is the middle value.

Since there are $9$ observations, median is the $5^{\text{th}}$ value.

So,

$\text{Median}=10$

Mean is:

$\text{Mean}=\frac{80+10}{9}$

$\text{Mean}=\frac{90}{9}=10$

Also, $10$ occurs twice, so

$\text{Mode}=10$

Hence,

$\text{mean}=\text{median}=\text{mode}=10$