Question

The median of the observations $22,24,33,37,x + 1,x + 3,46,47,57,58$ in ascending order is $42$ .
What are the values of $5$ th and $6$ th observations respectively?
a) $42,45$
b) $41,43$
c) $43,46$
d) $40,40$

Answer 1

Hint: The median of the observation is found by first arranging the given number in ascending order and the median is the average of the middle two numbers if the total number of observations is even. If the total number of observations are odd then the middle number will be the median.

Complete step by step answer:
Given,
The observation given to us is $22,24,33,37,x + 1,x + 3,46,47,57,58$ .
And it is already arranged in ascending order as it is given in the question. So we do not want to arrange them in ascending order.
The median of the observations is $42$ .
We need to find the values of $5$ th and $6$ th observations respectively.
Let Median of the observation be represented as M.
The total number of observations is $10$ . As $10$ is an even number, we need to take the middle two numbers and find the average.
The middle two numbers are $x + 1,x + 3$ .
The average of the middle two numbers is median.
The average of the two middle numbers is half the sum of two numbers.
$M = \dfrac{{x + 3 + x + 1}}{2}$
Substitute $M = 42$ in the above equation.
$42 = \dfrac{{x + 3 + x + 1}}{2}$
Add like terms in the equation,
$42 = \dfrac{{2x + 4}}{2}$
Take $2$out from the numerator on the right side of the equation.
$42 = \dfrac{{2(x + 2)}}{2}$
Cancel $2$ on numerator and denominator from the right side of the equation,
$42 = (x + 2)$
Bring $2$ two from the right side to left side of the equation,
$42 - 2 = x$
Subtract in order to get the value of $x$
$x = 40$
To find the values of $5$ th and $6$ th observations respectively, we need to substitute $x = 40$
value of $5$ th observation, $x + 1 = 40 + 1$
value of $6$ th observation, $x + 3 = 40 + 3$
The values of $5$ th and $6$ th observations respectively are $41,43$ .

So, the correct answer is “Option b”.

Note:
The value of the median can also be found by eliminating both first and last number simultaneously. In the end either we will remain with one or two terms. If we remain with one then that will be median or if we remain with two then we will take the average of them.