Calculate a total percentage based on individual percentages but without original values.

Question

(I hope the question/title made sense.)

Let's say I have the following list:

+----------+--------------+------------+-----------+
|   Item   |   Expected   |   Actual   |     %     |
+----------+--------------+------------+-----------+
| Item A   |       141068 |      45714 |    32.41% |
| Item B   |        44942 |      20844 |    46.38% |
| Item C   |       193052 |      50597 |    26.21% |
| Item D   |        67152 |      27084 |    40.33% |
+----------+--------------+------------+-----------+
| Total    |       746214 |     144239 |    19.33% |
+----------+--------------+------------+-----------+

The way for calculating $total\ \%$ is pretty straight forward.

But is it possible to figure out the $total\ \%$ without knowing the $Expected$ And $Actual$ values?

ie. is there a formula for "summing" up the percentages, without actually adding them together?

Visual illustration:

+----------+--------------+------------+-----------+
|   Item   |   Expected   |   Actual   |     %     |
+----------+--------------+------------+-----------+
| Item A   |           ?? |         ?? |    32.41% |
| Item B   |           ?? |         ?? |    46.38% |
| Item C   |           ?? |         ?? |    26.21% |
| Item D   |           ?? |         ?? |    40.33% |
+----------+--------------+------------+-----------+
| Total    |           ?? |         ?? |    19.33% | <-- Can I somehow find this number?
+----------+--------------+------------+-----------+

Answer 1

No, there is not. To see this, consider a simpler example. Suppose that there are two items, each at $50\%$. The following three arrays show clearly that the total percentage depends heavily on the amounts: it’s a weighted average of the individual percentages, so it depends both on those percentages and on their weights, which in turn depend on the absolute quantities involved.

$$\begin{array}{rrr} 100&10&10.00\%\\ 100&90&90.00\%\\ \hline 200&100&50.00\% \end{array}$$

$$\begin{array}{rrr} 100&10&10.00\%\\ 500&450&90.00\%\\ \hline 600&460&76.67\% \end{array}$$

$$\begin{array}{rrr} 100&10&10.00\%\\ 1000&900&90.00\%\\ \hline 1100&910&82.73\% \end{array}$$

There is one situation in which you can infer the total percentage: if the individual percentages are all the same, the total percentage will agree with them. It’s also the case that the overall percentage must lie between the smallest and the largest of the individual percentages, so if that range is narrow, you do have a good estimate of the overall percentage.

Answer 2

No, this is impossible. You would need the "expected" or the "actual" entry in every row in order to know how much weight each percentage has to have in the "total percentage."

Extreme example: percentage A = 0%, percentage B = 100%. If Actual A=1 and Actual B=0 then total percentage=0%. If Actual A=0 and Actual B=1 then total percentage=100%.