standard error of an estimate of fraction of objects with a known binary probability

Question

Lets have a group of N people that can be either men M or women W (or in general objects that can be 0 or 1). For each person $i$ we have an estimate of probability $P_i$ from <0,1> that it is a man. We want to estimate the proportion of men in the group (which must be a number from <0,1> too). The expected value is obviously $$ E(M)=\frac{\sum P_i}{N} $$ However, how to estimate the standard error of E(M)? Using the formula for $\sigma$ of a binomial distribution, I thought it should be $$ \sigma^2=\frac{\sum P_i(1-P_i)}{N} $$ but looking at the $\sigma$ estimates with some simple examples of a few people, I think that is not correct. Any hints?