# Probability: Binomial data

If the true success probability for binomial data is close to 0.50, why would you expect to have less certainty with your mean parameter estimate than if the true success probability were closer to 0 or 1.

• The variance of a binomial $\mathcal B(n,p)$ is $np(1-p)$ which is maximal at $p=0.5$ for a fixed $n$. – winperikle Sep 1 at 11:37
• @winperikle got the math, but I have a feeling you want more of an intuition about why we should be more confident when the observed probability is lower or higher than $0.5$. – Dave Sep 1 at 12:10
• The discussion given in the answer here attempts to provide some intuition as to why you expect to see spread (as measured by standard deviation, say) decrease with bounded variables, as the mean goes closer to a bound. The question was about correlation, but the reasoning is the same. There are other answers on site with similar discussions. – Glen_b Sep 2 at 8:27

When $$p=0.5$$, each single experiment, say coin toss, has greater uncertainty than any other $$p$$. For example, if $$p$$ was $$0$$, all coin tosses would turn up Tails, and there'd be no uncertainty over the results. So, if a single experiment result is more uncertain for $$p=0.5$$ compared to other $$p$$, we'd also expect the mean of multiple experiments to be more uncertain. Here, I assumed the uncertainty is defined by the entropy (or the variance).