First moment

Population: $\Omega = \{ x_1, \dots, x_N \}$
Collection of $n$-samples: $\mathcal{S} = \{ s \in \Omega^n \mid \forall i,j \in s, i \ne j \}$
Collection of $n$-samples containing $x$: $\mathcal{S}_x = \{ s \in \mathcal{S} \mid x \in s \}$
Observe that $|\mathcal{S}_x| = \binom{N-1}{n-1}$.
Let population mean be zero. $\mu = 0$, i.e. $\sum_{i = 1}^N x_i = 0$
Fix an order for $\mathcal{S}$: $\mathcal{S} = \{ s_1, s_2, \dots, s_{|\mathcal{S}|} \}$.
$j$-th $n$-sample mean $m_j = \frac1n \sum_{x \in \mathcal{S}_j} x$
Remark: I don’t use $\sum s_j$ as in $\cup \mathcal{T}$ in topology to avoid misreading the $n$-sample $s_j$ as an element.
mean of $n$-sample mean $m = \frac{1}{|\mathcal{S}|} \sum_{s_j \in \mathcal{S}} m_j$