If I average two independent realizations of $N(0,1)$, I get a random variable with distribution $N(0,1/2)$.

Now suppose $X_1,\ldots,X_n$ are independent realizations of $N(0,1)$. Sort them in ascending order, and of the $n-1$ pairs of adjacent realizations, randomly select $k\leq n-1$ of them without replacement, and average each pair to form the random variables $Y_1,\ldots,Y_k$.

Are these new random variables independent realizations of $N(0,\sigma^2)$ for some $\sigma$?