In averages of averages, what are the individual calculation representative of?

Question

I have read Why is an average of an average usually incorrect?

What I am trying to understand is what the individual calculations show me. From the question:

$$\frac{n_1}{n_1+n_2+n_3}a_1+\frac{n_2}{n_1+n_2+n_3}a_2+\frac{n_3}{n_1+n_2+n_3}a_3$$

The average of the averages is:

$$\frac{1}{3}a_1 + \frac{1}{3}a_2 + \frac{1}{3}a_3$$

What does the individual number mean?

$$\frac{n_1}{n_1+n_2+n_3}a_1$$

Is this number useful outside of the calculation of the whole? For example, if I have this,

+-----+--------------+-------------+
| avg | total in set |  weighted?  |
+-----+--------------+-------------+
|  50 |         1500 | 31.91489362 |
|   8 |          700 | 2.382978723 |
|   3 |          150 | 0.191489362 |
|     |              |             |
|  61 |         2350 | 34.4893617  |
+-----+--------------+-------------+

What does 31.91 mean or represent, even though 34.4 is correct?

Answer 1

The term itself is a bit confusing. Rather, we can think of the correct average as

$\frac{n_1a_1+n_2a_2+n_3a_3}{n_1+n_2+n_3}$.

Now, we can ask ourselves what is the term $n_1a_1$? This can be thought of as the entire 'amount' in group 1. So, if we were trying to determine the average wealth and had the wealth from 3 income brackets, with the first being the highest, we could think of $n_1a_1$ as being the total wealth of the people with the highest incomes.