Let's say I have various tanks that have different mixtures of water, acid and debris. Let's say I find the percentages of each tank and come up with the following:
Tank #1: 50% water, 25% acid, 25% debris
Tank #2: 45% water, 30% acid, 25% debris
Tank #3 50% water, 20% acid, 30% debris
How do I find the mean distribution of the tanks knowing that each tanks proportions tally up to 100%?
Here's what I have so far. If I take the mean of each proportion individually, I obtain: 48% water, 25% acid, 27% debris. This adds up to 100%, which is good. But this approach seems a touch naive and I'm wondering if this method worked here but will not necessarily work in all situations.
Is there a different way I should be looking for the mean of proportions?
If we have several sets $S^{j}\,:=\,\left\{ a_{1}^{j},\dots,a_{N}^{j}\right\}$ for $j=1,\dots,K$ where the $a_{i}^{j}$ denote amounts of a certain kind "$i$" in set $j$, we can define an average amount of kind "$i$" as $$M^{i}\,:=\,\frac{\sum_{j=1}^{K}a_{i}^{j}}{\sum_{j=1}^{K}\sum_{i=1}^{N}a_{i}^{j}}$$ This does not specify any properties of the "mean", it is just taking the "vanilla" standard average, which might or might not be sufficient in your case.
Since in your case for every $j\in\left\{ 1,\dots,K\right\}$ we have $\sum_{i=1}^{N}a_{i}^{j}=1$ (or $100\%$) this simplifies to $$M^{i}\,:=\,\frac{\sum_{j=1}^{K}a_{i}^{j}}{K}$$
which is the calculation you made.
Note that we can assume $N$ to be the same for every set $j$, as we can set the $a_{i}^{j}$ we do not need to $0$.