Calculating the median when the sample size is even

Question

In statistics the median is a point that divides an ordered data into two equal parts. By definition if $n$ is odd the median is $\frac{n}{2}{th}$ data point that is simply the middle point of the ordered data set. When $n$ is even the median is calculated by taking the mean of the two middle observations that is median=$\frac{(\frac{n}{2}){th} + (\frac{n}{2}+1){th}}{2}$. Now taking the two middle points results in a value that does not exist actually in data set. Then how can we declare this as a median point?