Question

The median of the data set 16, 5, 11, 19, x, 4 is 9. Find the value of x.

Answer 1

Hint: We need to find the value of x if the median number for the numbers 16, 5, 11, 19, x, 4 is 9. We start to solve the question by arranging the given numbers in descending order. Then, we use the formula of median given by $\dfrac{\left( {{\left( \dfrac{n}{2} \right)}^{th}}term+\left( {{\left( \dfrac{n}{2} \right)}^{th}}+1 \right)term \right)}{2}$ to find the value of the variable x.

Complete step by step answer:
We are given a data set and the median of the data set is 9.We need to find the value of x in the data set. We will be solving the given question by arranging the data set in descending order and then finding the value of x using the median formula given by$\dfrac{\left( {{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \left( \dfrac{n}{2} \right)+1 \right)}^{th}}term \right)}{2}$
The median is the middlemost number of the data set. It is the center value in a sorted list of numbers.
We have to arrange the 16, 5, 11, 19, x, 4 is 9 numbers in descending order.
Descending order involves arranging the numbers from largest to smallest.
Arranging the numbers in descending order, we get,
$\Rightarrow 19,16,11,x,5,4$
In our case, the total number of observations given is 6.
As the total number of observations is even, the median for the data set is the given by $\dfrac{\left( {{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \left( \dfrac{n}{2} \right)+1 \right)}^{th}}term \right)}{2}$ where n is the total number of observations.
Here,
n=6
$\Rightarrow median=\dfrac{\left( {{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \left( \dfrac{n}{2} \right)+1 \right)}^{th}}term \right)}{2}$
Substituting the value of n in the above formula, we get,
$\Rightarrow median=\dfrac{\left( {{\left( \dfrac{6}{2} \right)}^{th}}term+{{\left( \left( \dfrac{6}{2} \right)+1 \right)}^{th}}term \right)}{2}$
Simplifying the above equation, we get,
$\Rightarrow median=\dfrac{\left( {{3}^{rd}}term+{{\left( 3+1 \right)}^{th}}term \right)}{2}$
Let us evaluate it further.
$\Rightarrow median=\dfrac{\left( {{3}^{rd}}term+{{4}^{th}}term \right)}{2}$
From the above data set arranged in descending order, we know that the ${{3}^{rd}}$ term is 11 and the ${{4}^{th}}$ term is x.
Substituting the same, we get,
$\Rightarrow median=\dfrac{\left( 11+x \right)}{2}$
From the given question, we know that the value of the median is 9.
Substituting the same in the above equation, we get,
$\Rightarrow 9=\dfrac{\left( 11+x \right)}{2}$
Moving the number 2 to the other side of the equation, we get,
$\Rightarrow 18=11+x$
Moving the number 2 to the other side of the equation, we get,
$\Rightarrow 18-11=x$
Simplifying the above equation, we get,
$\therefore x=7$
$\therefore$ The value of x in the data set is 7.

Note: The median is the mid-value in the sorted list of numbers. We must not forget to arrange the given numbers in descending order before computing the median for the data. The formula of median depends upon the total number of observations(n).
If n is odd, then $median={{\left( \dfrac{n+1}{2} \right)}^{th}}term$
If n is even, then $median=\dfrac{\left( {{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \left( \dfrac{n}{2} \right)+1 \right)}^{th}}term \right)}{2}$