Question

The mean of 2, 12, 4, 9, 5 and 16 is $x$. The median of 4, 3, $x,x-1$, 12 and 16 is y. Find the values of $x$ and $y$.

Answer 1

Hint: To obtain the value of two variables we will use the mean and median formula. Firstly by using the mean formula we will get the value of $x$. Then we will put the value of $x$ in the second data given and find the median of the second data to get the value of $y$ and hence get our desired answer.

Complete step by step answer:
The data set is given to us as:
$2,12,4,9,5,16$……$\left( 1 \right)$
Mean $=x$….$\left( 2 \right)$
We know the formula for mean is given as below:
Mean= Sum of all the element/Number of element
Putting value from equation (1) and (2) in above formula we get,
$\begin{align}
  & x=\dfrac{2+12+4+9+5+16}{6} \\
 & \Rightarrow x=\dfrac{48}{6} \\
\end{align}$
$\therefore x=8$…$\left( 3 \right)$
Next data set is given to us is as follows:
$4,3,x,x-1,12,16$
Substituting the value from the equation (30 above we get,
$\begin{align}
  & 4,3,8,8-1,12,16 \\
 & \Rightarrow 4,3,8,7,12,16 \\
\end{align}$
We to find the median we first have to arrange the data set in ascending order so we get,
$3,4,7,8,12,16$….$\left( 4 \right)$
Median of above data set is $=y$….$\left( 5 \right)$
Now as we have even terms so the median will be sum of middle two terms divided by 2 as follows:
Median $=\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}$
Putting value from equation (4) and (5) above we get,
$\begin{align}
  & y=\dfrac{7+8}{2} \\
 & \Rightarrow y=\dfrac{15}{2} \\
\end{align}$
$\therefore y=7.5$….$\left( 6 \right)$
So from equation (3) and (6) we have $x=8$ and $y=7.5$
Hence value of $x$ and $y$ is 8 and $7.5$

Note: Mean is also known as average in other topics of mathematics it is found by adding all the data points and dividing them by the numbers of data points. Median is the middle number of a data point which is found by ordering all data points and selecting the middle term of them and if there are more than one middle term we find the average of them and get our median.