Question

The mean of $15$ students in an examination are $25,19,17,24,23,29,31,40,19,20,22,$ $26,17,35,21$ . Find the median score.

Answer 1

Hint: From the given question, we have to find the median from the given mean of some students in the examination by using a data handling concept. First, we have to discuss some properties of median based on the data.
Data handling is an art. Sometimes the raw data (data as they are) will not be useful to get the required information. In order to get proper useful information, we have to process very important useful information from the given data.

Formula used: When the given data are arranged in ascending or descending order, we can find a value which is centrally located in the arranged order. This central value or the middle most value is called the median of the data.
First, we arrange the entire data in the ascending order. Let ${\text{N}}$ be the number of observations. If ${\text{N}}$ is an odd integer, then there is only one middle term and it is the${\left( {\dfrac{{{\text{N}} + 1}}{2}} \right)^{{\text{th}}}}$ term of the given set of observations. This is the median. If ${\text{N}}$ is an even integer, there are two middle terms. They are ${\dfrac{{\text{N}}}{2}^{{\text{th}}}}$ term and ${\left( {\dfrac{{\text{N}}}{2} + 1} \right)^{{\text{th}}}}$ term. Hence the median is the average of two terms.
The mean of several items is the value equally shared out among the items.
\[{\text{Mean = }}\dfrac{{{\text{Total of all items}}}}{{{\text{Number of items}}}}\].
\[ \Rightarrow {\text{Total of all items = Mean}} \times {\text{Number of items}}\]

Complete step-by-step solution:
Given that the marks of $15$ students in an examination are $25,19,17,24,23,29,31,40,19,20,22,$
$26,17,35,21$ .
First, we have to arrange the given observations in the ascending order. Then the given observation be
$17,17,19,19,20,21,22,23,24,25,26,29,31,35,40$ .
Here, the number of observations is ${\text{N}} = 15$ .
Thus, the given number of observations is odd.
According to the condition of the median, If${\text{N}}$ is an odd integer, then there is only one middle term and it is the ${\left( {\dfrac{{{\text{N}} + 1}}{2}} \right)^{{\text{th}}}}$ term of the given set of observations and that is the required median.
Median \[ = {\left( {\dfrac{{{\text{N}} + 1}}{2}} \right)^{{\text{th}}}}{\text{term}}\]
\[ \Rightarrow {\left( {\dfrac{{15 + 1}}{2}} \right)^{{\text{th}}}}{\text{term}}\]
Hence,
\[ \Rightarrow {\left( {\dfrac{{16}}{2}} \right)^{{\text{th}}}}{\text{term}}\]
\[ \Rightarrow {8^{{\text{th}}}}{\text{term}} = 23\]
Hence, we get the median of the given observation is $23$.

$\therefore $ The median score of the given mean of $15$ students in an examination is $23$.

Note: We have to remember that, the mean is equivalent to average. The process of sharing out equally is the basis of average. We call the averaged or quantity as the arithmetic average (or arithmetic mean or simply average or mean). The median is sometimes used as opposed to the mean when there are outliers in the sequence that might skew the average of the values.