Question

Find the median of 33,31,48,45,41,92,78,51,61.

Answer 1

Hint: Here we arrange all the observations in ascending order or descending order so we either move higher up the scale or we move down the scale and then using the formula for median we choose the middlemost term from the observations.
* Median is the term that is exactly at the middle of the observations, when they are arranged in the ascending order.
* If number of observations is odd then median is calculated by \[{\left( {\dfrac{{n + 1}}{2}} \right)^{th}}term\]
If number of observations is even then, median is calculated by \[\dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}}term + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}term}}{2}\]
* A number is odd if it can be expressed in the form \[2n \pm 1\], example: \[5 = 2 \times 2 + 1\] is an odd number.
* A number is even if it can be expressed in the form \[2n\], example: \[18 = 2 \times 9\] is an even number.

Complete step by step solution:
Given observations are \[33,31,48,45,41,92,78,51,61\]
Therefore, total number of observations is \[9\]
Since, these observations are not in any ascending or descending order, therefore first we arrange them in ascending order (increasing order).
\[31,33,41,45,48,51,61,78,92\]
Now, the number of observations is \[9\] which is odd .
Therefore, using the formula for median when number of observations is odd
Median \[ = {\left( {\dfrac{{n + 1}}{2}} \right)^{th}}\]term
Substituting the value of \[n = 9\] in the formula
Median \[ = {\left( {\dfrac{{9 + 1}}{2}} \right)^{th}}\]term
               \[ = {\left( {\dfrac{{10}}{2}} \right)^{th}}\]term
               \[ = {5^{th}}\] term
Now taking the \[{5^{th}}\] term of the observations from Left side or from the right side.

Therefore, \[{5^{th}}\] term of the observations is \[48\].
Therefore, the term \[48\] is the middle most number from the set of these observations.

Note:
Students are likely to make mistakes in calculating the median when they have not arranged the observations in ascending or descending order. Usually students rush straight to the formula after counting the number of observations but that doesn’t work in both the cases.
Also, In case of probability distribution, median is the separating value between the higher half of the data and the lower half of the data.