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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?

Auras
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  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Mar 24 '22 at 08:59
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    No, because you can compute the total average as a suitably weighted average of the per-color averages. So the total average must lie between the minimum and the maximum per-color average. – MJD Mar 24 '22 at 12:56

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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 $S$:

$$\overline{S} \geq \overline{C_1}$$

Nick
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