Question

Median of the data, arranged in ascending order 7, 10, 15, x, y, 27, 30 is 17 and when one more observation 50 is added to the data, the median has become 18. Find x and y.

Answer 1

Here, before adding 50, the total number of observations = 7
7, 10, 15, x, y, 27, 30
The above is the set of observations arranged in ascending order.
Here, n = 7
So, since the number of observations is odd, the median $=(\frac{n+1}{2}{)}^{th}term=17\Rightarrow (\frac{7+1}{2}{)}^{thterm=4thterm=17}$
The fourth term is x here.
S0, x = 17
Now, when 50 is added to the above set of observations, the number of observations,n becomes 8.
Now, this contains an even number of observations.
7, 10, 15, 17, y, 27, 30, 50
So, median$=\frac{\frac{n^{th}}{2}term+(\frac{n}{2}+1{)}^{th}term}{2}=18\Rightarrow \frac{\frac{8^{th}}{2}term+(\frac{8}{2}+1{)}^{th}term}{2}=18\Rightarrow \frac{4^{th}term+5^{th}term}{2}=18\Rightarrow \frac{17+y}{2}=18\Rightarrow 17+y=36\Rightarrow y=19$