Example 1: There are 45 numbers
- 45 plus 1 is 46, then divide by 2 and you get 23
- So the median is the 23rd number in the sorted list.
Example2: There are 66 numbers
- 66 plus 1 is 67, then divide by 2 and you get 33.5 33 and a half? That means that the 33rd and 34th numbers in the sorted list are the two middle numbers.
- So to find the median: add the 33rd and 34th numbers together and divide by 2
Why do we have to add the number 1 regardless to find the median ?
Suppose you have $n$ objects, labeled $1,\ldots,n$.
If $n$ is odd, then the middle one is $\frac{n+1}{2}$. For example,
$$\fbox{1}\fbox{2}\fbox{3}\fbox{4}\fbox{5}\\ \overset{\uparrow}{\tfrac{5+1}{2}}$$
If $n$ is even, then there is no "middle" one, but the ones closest to the middle are $\frac{n}{2}$ and $\frac{n}{2}+1$. $$\fbox{1}\fbox{2}\fbox{3}\fbox{4}\fbox{5}\fbox{6}\\ \;\;\;\overset{\uparrow}{\tfrac{6}{2}}\overset{\uparrow}{\small(\tfrac{6}{2}+1)}$$ Those are the integers that are closest to $\frac{n+1}{2}$.
In general, if we consider the interval of real numbers $[1,x]$, the point that's equidistant from the ends (otherwise known as the middle) is the point $m$ such that $$\begin{align*} \underbrace{m-1}_{\text{distance from $m$ to $1$}}&=\underbrace{x-m}_{\text{distance from $x$ to $m$}}\\\\ 2m&=x+1\\\\ m&=\frac{x+1}{2} \end{align*}$$