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.
Best Answer
You are right that there are two approaches. Your first solution would be the population proportion. This would answer the question "What proportion of valid responses were correct". The second approach is the mean sample proportion. This answers the question "What was the average proportion of valid responses that were correct per subject". The methods will produce different results in most cases. Both are valid, but just have different interpretations.
I do think your larger formula contains an error. The n * should be in the denominator not the numerator. The denominator should be n * Valid1 Valid2...Validn. You'll find that the expression is then equivalent to adding the subject proportions and dividing by the number of subjects as follows: ((Correct1/Valid1)+(Correct2/Valid2)+...(Correctn/Validn))/n