Question

Two datasets arranged in descending order are; {8, , 4,1} and {9, , 5,2}. If the medians of the two given datasets are equal, what is the value of ( − )2?

Answer 1

We have been given that two datasets arranged in descending order are: $\{8, x, 4,1\}$ and $\{9, y, 5,2\}$. We are asked to find the value of $(y-x)^2$, if the medians of the two given datasets are equal.

First of all, we will arrange our given data sets in ascending order.

$\{1,4,x,8\}$ and $\{2,5,y,9\}$

Since our data sets have 4 data points, so median will be average of middle two terms.

$\frac{4+x}{2}$ and $\frac{5+y}{2}$

Now we will equate both median as we are told that median of both data sets are equal.

$\frac{5+y}{2}=\frac{4+x}{2}$

Cross multiply:

$2(5+y)=2(4+x)$

Divide both sides by 2:

$\frac{2(5+y)}{2}=\frac{2(4+x)}{2}$

$5+y=4+x$

$5-5+y-x=4-5+x-x$

$y-x=-1$

Let us square both sides of our equation.

$(y-x)^2=(-1)^2$

$(y-x)^2=1$

Therefore, the value of $(y-x)^2$ would be 1.