We want to know the $2021^{th}$ term and the $2022^{th}$ term to get the median. We know that $44^{2}=1936\textless2021$, and $45^{2}=2025\textgreater2021$.  So, the number $1^{2}$, $2^{2}$, $3^{2}$, $\cdots $, $44^{2}$ are between $1$ to $1936$. $1936+44=1980$, which mean that $1936$ is the $1980^{th}$ number.  Thus, the $2021^{th}$ term will be $1936+41=1977$, and similarly the $2021^{th}$ term will be $1978$.  So, the answer is $1977.5$. 
What is the median of the following list of $4042$ number?  $1$, $2$, $3$, $\cdots $, $2021$, $1^{2}$, $2^{2}$, $3^{2}$, $\cdots $, $2021^{2}$ 

A) $$1974.5$$ 
B) $$1975.5$$ 
C) $$1976.5$$ 
D) $$1977.5$$ 
E) $$1978.5$$ 