Standard error of a proportion

This is the standard deviation of the sampling distribution of $\hat{p}$ (p-hat, the sample proportion), so it tells you how far a sample proportion typically lands from the true proportion. For a confidence interval the formula is $SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$; a significance test puts the null value $p_0$ where p-hat sits, because the test assumes the null hypothesis is true. For example, with p-hat = 0.40 and n = 400, the standard error is $\sqrt{0.40(0.60)/400} = \sqrt{0.0006} = 0.0245$. The standard error is largest when the proportion is near 0.5 and shrinks as it moves toward 0 or 1.