Can the average of a set be lower than all of the averages of subsets? Let's imagine there are marbles of different diameter and color. Can the average diameter of all the marbles be lower than all of the average diameter of marbles per color?

Let $S \subseteq \mathbb{R}$ be a finite set. Let $P = \{C_1,C_2,…,C_n\}$ be a partition of $S$. This partition represents the color groupings of the marbles. Also assume that $\overline{C_i} \leq \overline{C_{i+1}}$ for $1 \leq i < n$, where $\overline{C_i}$ denotes the average of $C_i$.
Notice:
$$\overline{S} = \frac{|C_1|\overline{C_1}    + |C_2|\overline{C_2} + … + |C_n|\overline{C_n}}{|S|}$$
And since $\overline{C_i} \geq \overline{C_1}$ for all $1 \leq i < n$ we have:
$$\overline{S} \geq \frac{|C_1|\overline{C_1}    + |C_2|\overline{C_1} + … + |C_n|\overline{C_1}}{|S|}$$
Now factoring out $\overline{C_1}$ and using the fact that $P$ partitions $S$:
$$\overline{S} \geq \overline{C_1}$$
