Minimum standard deviation of the sum over partitions of a set when the partitions can be changed

Question

Let's say I have a set of numbers $S = \{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9\}$. I have three partitions over the set.

$P1 = \{x_1, x_2, x_3\}$

$P2 = \{x_4, x_5, x_6, x_7\}$

$P3 = \{x_8, x_9\}$

The number of partitions is fixed but I can change the members of a partition. For example, I can take $x_6$ from $P_2$ and put it in $P_3$.

Is there any way I can calculate the minimum standard definition possible for $\sum P_1$, $\sum P_2$, $\sum P_3$ considering all the different combination of members possible.