# Average proportion for proportions with different denominators

Say I have an experiment in which subjects are asked to respond to some stimulus. Their responses are transcribed and coded, first as "Valid/Invalid", and then for the valid responses, "Correct/Incorrect". I would like to compute the average of the proportion, $\frac{\text{number correct}}{\text{number valid}}$.

There seem to be two different approaches to this. The first is I could use the total proportion correct, $\frac{\sum{Correct_n}}{\sum{Valid_n}}$.

The second would be to find the actual average of the proportions. I cannot straightforwardly compute the average proportion by adding the subject proportions and dividing by the number of subjects, as each subject had a different number of valid responses, and therefore each proportion has a different denominator. So, I end up the the following expression:

$\frac{n*[(Valid_2 \times Valid_3 \times...Valid_n)Correct_1 + (Valid_1 \times Valid_3 \times ... Valid_n)Correct_2... ]}{Valid_1 \times Valid_2 \times...Valid_n}$

While the two approaches seem similar, I can't figure out if they are going to end up being the same thing, and further I can't decided which one makes more sense, or even if either of them do.

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