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**Hint:**So for solving this very question, we will use the concept of the median. The formula we will use in this question is of median and it is given by $\dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}}observation + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}observation}}{2}$ . and by equating it with the value of median we will get the value of $x$ .

**Formula used:**

Median is given by,

$Median = \dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}}observation + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}observation}}{2}$

Here, $n$ will be the number of terms in the observations.

**Complete step-by-step answer:**So we have an observation which is in ascending order is given as $29,32,48,50,x,x + 2,72,78,84,95$ . Also, we have the value of the median given as $63$ .

Since the number of observations in this question is $n = 10$ and is an even number.

Therefore, by using the median formula given by

$Median = \dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}}observation + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}observation}}{2}$

On substituting the values, we get

$ \Rightarrow 63 = \dfrac{{{{\left( {\dfrac{{10}}{2}} \right)}^{th}}observation + {{\left( {\dfrac{{10}}{2} + 1} \right)}^{th}}observation}}{2}$

And on solving the numerator braces, we get the equation as

$ \Rightarrow 63 = \dfrac{{{{\left( 5 \right)}^{th}}observation + {{\left( 6 \right)}^{th}}observation}}{2}$

As we can see the values from the observation at the above place, so on substituting the values, we get the equation as

$ \Rightarrow 63 = \dfrac{{x + \left( {x + 2} \right)}}{2}$

By doing the cross-multiplication, we get

$ \Rightarrow 63 \times 2 = 2x + 2$

Taking the constant term one side and solving for it, we get the equation as

$ \Rightarrow 126 - 2 = 2x$

And on solving the subtraction, we get

$ \Rightarrow 124 = 2x$

And solving for the value of $x$ , we get

$ \Rightarrow x = 62$

**Hence, the option $\left( b \right)$ is correct.**

**Note:**So just like the median we can also calculate the mean of the observation. And is defined as Mean (or arithmetic mean) is a type of average. It is processed by adding the qualities and partitioning by the number of qualities. The word 'Average' is an equivalent for the number-crunching mean - which is the value gotten by partitioning the sum of a bunch of amounts by the quantity having the number in the set.

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