Question

The following data has been arranged in ascending order:
$24,27,28,31,34, x ,37,40,42,45.$
If the median of the data is 35 find \[x\]. In the above data, if 45 is changed to 33, find the new median.

Answer 1

Hint: In the given question, we have some numbers. We have been given the median of the data. We have to calculate the unknown value of the data. It can be done using the formula of median, putting the values, and solving for the answer. Then, we have to calculate the new median when one entry is changed.

Complete step by step solution:
The given numbers are $24,27,28,31,34, x,37,40,42,45.$
Here, the number of observations, \[n = 10\], i.e., even.
So, we are going to use the following formula to calculate the value,
\[median = \dfrac{{{x_{n/2}} + {x_{\left( {n/2 + 1} \right)}}}}{2}\]
So, we have,
\[35 = \dfrac{{34 + x}}{2} \Rightarrow 70 = 34 + x\]
Hence, \[x = 36\].
Now, when \[45\] is changed to \[33\], we have,
$24,27,28,31,33,34,36,37,40,42$
So, \[Median = \dfrac{{33 + 34}}{2} = 33.5\]

Note: In this question, we were given some numbers. We were also given the median of the data. We had to calculate the unknown value of the data. We did that by using the formula of median, putting the values, and solving for the answer. Then, we calculated the new median when one entry was changed by using the same formula. So, it is very important that we know where, which, how and what formula to use in what context.