First, yes 44s are indeed correct. (Better yet, look for yourself.) But what do the 44s refer to? 44 is the average of all the responses. When Perry writes “Republicans estimated the share at 46 percent,” (we have similar language in the paper, which is regrettable as it can be easily misunderstood), it doesn’t mean that every Republican thinks so. It may not even mean that the median Republican thinks so. See OA 1.7 for medians, OA 1.8 for distributions, but see also OA 2.8.1, Table OA 2.18, OA 2.8.2, OA 2.11 and Table OA 2.23.
Key points =
1. Large majorities overestimate the share of party-stereotypical groups in the party, except for Evangelicals and Southerners.
2. Compared to what people think is the share of a group in the population, people still think the share of the group in the stereotyped party is greater. (But how much more varies a fair bit.)
3. People also generally underestimate the share of counter-stereotypical groups in the party.