Question

Following is arranged in ascending order. If the median of the data is $63$ , find the volume of $x$ in a series $29,32,48,50,x,x + 2,72,78,84,95$
$\left( a \right){\text{ 60}}$
$\left( b \right){\text{ 62}}$
$\left( c \right){\text{ 63}}$
$\left( d \right){\text{ 64}}$

Answer 1

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.