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: We can check whether the number of observations is odd or even by counting. Then we can write the equation for median in terms of x. Then we can equate it to the given value of median. Then we can solve for x to get the required solution.

Complete step-by-step answer:
We have the raw data,
29, 32, 48, 50, x, \[x + 2\] , 72, 78, 84, 95
It is given that the data is in ascending order.
We know that for a data in ascending order, the median is the middle most observation.
We can count the number of observations.
There are 10 observations. Then the median is given by the average of the middle most terms.
If n is even, the median is given by the mean of ${\left( {\dfrac{n}{2}} \right)^{th}}$ and ${\left( {\dfrac{n}{2} + 1} \right)^{th}}$ observation.
We have $n = 10$ , so $\dfrac{n}{2} = 5$ and $\dfrac{n}{2} + 1 = 6$
Therefore, median is the mean of the 5th and 6th term.
From the given observations, 5th term is x and 6th term is \[x + 2\]
 $ \Rightarrow Median = \dfrac{{x + x + 2}}{2}$
On simplification we get,
 $ \Rightarrow Median = \dfrac{{2x + 2}}{2}$
Hence we have,
 $ \Rightarrow Median = x + 1$
It is given that the median is 63
 $ \Rightarrow x + 1 = 63$
On rearranging, we get,
 $ \Rightarrow x = 63 - 1$
On simplification we get,
 $ \Rightarrow x = 62$
So, the required value of x is 62.

Note: Mean mode and median are known as the measures of central tendency. It gives an idea about the data. It is not affected by extreme values and does not depend on all the observations. For finding the median, we must arrange the observation in ascending or descending order to find the middle most term. For n observations arranged in ascending or descending order,
If n is even, the median is given by the mean of ${\left( {\dfrac{n}{2}} \right)^{th}}$ and ${\left( {\dfrac{n}{2} + 1} \right)^{th}}$
If n is odd, the median is ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$ term