= How do I combine proportions? =

Proportions which measure the same underlying concept, such as attention, over blocks of results can be pooled by simply obtaining an aggregate sum of the hit rates.

Pooled estimate = $$\frac{\mbox{total number of correct responses}}{\mbox{Total number of possible correct responses}}$$

So for example if 5 out of the 16 questions were correctly answered in the first session and 17 out of 25 in the second session the pooled estimate would be (5+17)/(16+25)= 22/41. This pooled proportion is a sum of observed frequencies.

__Note__ Do not sum percentages to obtain a pooled estimate because a percentage does not take into account the sample sizes (total possible number correct). e.g. 1 correct out of 2 and 500 out of 1000 both have 50% correct but we would have more confidence in the latter as it is based on a far larger sample size. This is why for analysis purposes, such as chi-square tests, the observed frequencies are used rather than percentages.