Question

The following observations have been arranged in ascending order, if the median of the data is 63.
Find the value of $x$:
$29,32,48,50,x,x + 2,72,78,84,95$

Answer 1

Hint: According to the question we have to determine the value of $x$ if the median of the data is 63 and the observation is $29,32,48,50,x,x + 2,72,78,84,95$. So, first of all to determine the median we have to determine the total number of observations. Which can be determine the with the help of the formula to find the median but before that as we know that the given observations are even in count so, we have to determine the median for the even number of observation and to determine the median for even number of data we have to use the formula as mentioned below:

Formula used: $ \Rightarrow $Median$ = {\dfrac{n}{2}^{th}}$and ${\left( {\dfrac{n}{2} + 1} \right)^{th}}$observations.
Where n are the even numbers of observations.
 So, if the number of given observations are odd then we have to apply the formula (A) to determine the median. Then after determining the observation number we have to check for the median.
After that we have to substitute the value of median to determine the value of x.

Complete step-by-step solution:
Step 1: First of all to determine the median we have to determine the total number of observations and as we know that the total number of observations are even in number.
$
   \Rightarrow 29,32,48,50,x,x + 2,72,78,84,95 \\
   \Rightarrow n = 10
 $
Step 2: Now, to determine the median for the even numbers we have to use the formula (A) as mentioned in the solution hint.
$ \Rightarrow $Median
$ = {\dfrac{{10}}{2}^{th}}$
$ = {5^{th}}$observation
And,
$
   = {\left( {\dfrac{{10}}{2} + 1} \right)^{th}} \\
   = {(5 + 1)^{th}}
 $
$ = $${6^{th}}$ observation
Step 3: Now, as mentioned in the question that the median for the given data is 63 so, we have to determine the median with the help of the observations as obtained in the solution step 2. Hence,
$ \Rightarrow $Median $ = \dfrac{{{5^{th}} + {6^{th}}}}{2}$
Now, on substituting all the values in the expression obtained just above,
$
   \Rightarrow 63 = \dfrac{{x + x + 2}}{2} \\
   \Rightarrow x + 1 = \dfrac{{63 \times 2}}{2} \\
   \Rightarrow x + 1 = 63 \\
   \Rightarrow x = 63 - 1 \\
   \Rightarrow x = 62
 $

Hence, with the help of the formula (A) as mentioned in the solution hint we have determined the value of $x = 62$.

Note: To determine the median for the given data it is necessary that we have to find the total number of given observation and of the total number of given observation is odd then to find the median we have to use the formula ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$
If the total number of observations are even then there will be two median which can be obtain by finding the observations which are ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$ and ${\left( {\dfrac{n}{2} + 1} \right)^{th}}$ observations.