Question

Consider the following data $41,43,127,99,61,92,71,58,57$ . If $58$ is replaced by $85$, what will be the new median?

Answer 1

Remember to always arrange the sequence in ascending or descending order before finding out the median. According to the definition of the median, it is the term in the middle position of the sequence. For an odd number of terms, the median will be $\dfrac{{{{\left( {N + 1} \right)}^{th}}}}{2}term$.

Complete step by step solution:
Here in this problem, we are given nine numbers, i.e. $41,43,127,99,61,92,71,58,57$ . Then the number $58$ is replaced by the number $85$ . With this information, we need to find the new median of the quantities.
Before starting with the solution, we In statistics and probability theory, a median is a value separating the higher half from the lower half of a data sample, a population or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic advantage of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed so much by a small proportion of extremely large or small values, and so it may give a better idea of a "typical" value.
In practice, the median for a set of values can be found by finding the number at the middle position (in case of an odd number of values) and the average of two middle values (in case of even number of values)
{\text{Median = }}\left\\{ {\begin{array}{*{20}{l}} {\dfrac{{{{\left( {N + 1} \right)}^{th}}}}{2}term{\text{ }};{\text{ when N is odd}}} \\\ {\dfrac{{{{\left( {\dfrac{N}{2}} \right)}^{th}}term + {{\left( {\dfrac{N}{2} + 1} \right)}^{th}}term}}{2};{\text{ when N is even}}} \end{array}} \right.
But before applying this formula, we must have the data in ascending or descending order
$\Rightarrow 41,43,127,99,61,92,71,85\left( {{\text{replaced by 58}}} \right),57{\text{ }} \Rightarrow {\text{ }}41,43,57,61,71,85,92,99,127$
Since we have a number of quantities as nine which is an odd number, we get:
$\Rightarrow {\text{Median}} = \dfrac{{{{\left( {N + 1} \right)}^{th}}}}{2}term{\text{ ; where N is the number of observations}}$
Therefore, for the above set of observations:
$\Rightarrow {\text{Median}} = \dfrac{{{{\left( {N + 1} \right)}^{th}}}}{2}term = \dfrac{{{{\left( {9 + 1} \right)}^{th}}}}{2}term = {5^{th}}term$

Hence, we got that the median will be the fifth term in the sequence, which is $71$.

Note:
In questions involving the concepts of statistics like mean, median and mode, be careful with the order of the sequence and utilise the fundamental definitions of these terms. An alternative approach can be taken by arranging the sequence in descending order and just find the middle term. Since we have an odd number of terms, the middle term will be the median according to the definition.