Question

Find the median of the first fifty whole numbers.

Answer 1

Hint: The median is the value of the middle-most observation(s). It is the measure of central tendency.
The median is that value of the given number of observations, which divides it into exactly two parts.
There are two possible cases:
When the number of observations $\left( n \right)$ is odd, the median is the value of the \[\mathop {\left( {\dfrac{{n + 1}}{2}} \right)}\nolimits^{th} \] observation.
For example, if n = 13, the value of the \[\mathop {\left( {\dfrac{{n + 1}}{2}} \right)}\nolimits^{th} \]observation will be the median.
When the number of observations $\left( n \right)$ is even, the median is the mean of the \[\mathop {\left( {\dfrac{n}{2}} \right)}\nolimits^{th} \] and the \[\mathop {\left( {\dfrac{n}{2} + 1} \right)}\nolimits^{th} \] observation.
For example, if n = 16, the mean of the values of the \[\mathop {\left( {\dfrac{{16}}{2}} \right)}\nolimits^{th} \]and the \[\mathop {\left( {\dfrac{{16}}{2} + 1} \right)}\nolimits^{th} \]observation, i.e., the mean of the values 8th and 9th observations will be the median.

Complete step-by-step answer:
Step 1: State the given data:
Whole numbers are the natural numbers including 0.
Therefore, the first fifty whole numbers are: 0, 1, 2, 3, ….., 49.
The total number of observations, $n = 50$ is even.
We know,
The median of the given data is the mean of the \[\mathop {\left( {\dfrac{n}{2}} \right)}\nolimits^{th} \] and the \[\mathop {\left( {\dfrac{n}{2} + 1} \right)}\nolimits^{th} \] observation.
i.e. the mean of the \[\mathop {\left( {\dfrac{{50}}{2}} \right)}\nolimits^{th} \] and the \[\mathop {\left( {\dfrac{{50}}{2} + 1} \right)}\nolimits^{th} \] observation
$ \Rightarrow $ The mean of the \[\mathop {\left( {25} \right)}\nolimits^{th} \] and the \[\mathop {\left( {26} \right)}\nolimits^{th} \] observation is the median.
Step 2: Find the \[\mathop {\left( {25} \right)}\nolimits^{th} \] and the \[\mathop {\left( {26} \right)}\nolimits^{th} \] values of observation.
The given data: 0, 1, 2, ….., 49 forms an AP, where the first term, $\mathop a\nolimits_1 = a = 0$ and common difference d = 1
Thus using the general form of AP:
$ \Rightarrow \mathop a\nolimits_n = a + \left( {n - 1} \right)d$
\[\because {\text{ }}\mathop {\left( {25} \right)}\nolimits^{th} \] observation is:
\[
   \Rightarrow \mathop a\nolimits_{25} = a + \left( {25 - 1} \right)d \\
   \Rightarrow \mathop a\nolimits_{25} = 0 + \left( {24} \right)1 \\
   \Rightarrow \mathop a\nolimits_{25} = 24 \\
 \]
\[\because {\text{ }}\mathop {\left( {26} \right)}\nolimits^{th} \] observation is:
\[
   \Rightarrow \mathop a\nolimits_{26} = a + \left( {26 - 1} \right)d \\
   \Rightarrow \mathop a\nolimits_{26} = 0 + \left( {25} \right)1 \\
   \Rightarrow \mathop a\nolimits_{26} = 25 \\
 \]
Step 3: Calculation of median
It is known The median of the given data is the mean of the \[\mathop {\left( {25} \right)}\nolimits^{th} \] and the \[\mathop {\left( {26} \right)}\nolimits^{th} \] observation.
i.e. mean of 24 and 25
$
   \Rightarrow {\text{ }}\dfrac{{25 + 26}}{2} \\
   \Rightarrow {\text{ }}\dfrac{{51}}{2} \\
   \Rightarrow {\text{ }}25.5 \\
 $
Final answer: The median of the first fifty whole numbers is 25.5.

Note: The data must be arranged in ascending or descending order for the calculation of the median. If it is not so then arrange it before calculating the median.
The above state formula for calculating the median is only for ungrouped data.
The other two measures of central tendency for ungrouped data are:
Mean: It is found by adding all the values of the observations and dividing it by the total number of observations.
So, mean $ = \dfrac{{{\text{sum of observations}}}}{{{\text{total number of observations}}}}$
Mode: The mode is the most frequently occurring observation.