smc

We can see that $44^2=1936$ which is less than 2020. Therefore, there are $2020-44=1976$ of the $4040$ numbers greater than $2020$. Also, there are $2020+44=2064$ numbers that are less than or equal to $2020$. Since there are $44$ duplicates/extras, it will shift up our median's placement down $44$. Had the list of numbers been $1,2,3,\dots ,4040$, the median of the whole set would be $\frac{1+4040}{2}=2020.5$. Thus, our answer is $2020.5-44=\boxed{1976.5}$.