Consider a population $\mathcal{C}$ of $N$ real numbers, possibly with multiplicities. For an integer $n\leq N$, let $A_n$ be the random variable denoting the mean of $n$ random samples of $\mathcal{C}$ without replacement. Let $B_n$ denote the mean of $n$ random samples with replacement. Let $\mu$ denote the mean of elements in $\mathcal{C}$.
Is it true that the distribution of $A_n$ is no less concentrated about $\mu$ than the distribution of $B_n$?
Precisely, the claim would be that for any $n$ and any real number $t\geq 0$, $\Pr[|A_n-\mu|\geq t]\leq \Pr[|B_n-\mu]\geq t]$
Intuitively, this seems to be true. For example, consider drawing $n-1$ samples without replacement, giving $A_{n-1}$. Suppose $A_{n-1}>\mu$. Then the mean of the remaining samples is less than $\mu$, so the next sample will tend to move $A_n$ back towards $\mu$. In the extreme case $n=N$, then clearly $A_N$ is exactly $\mu$, whereas $B_N$ is still varies about $\mu$.