# 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.