Sometimes, it's easier to do the algebra than wave hands. It has already been argued that a proportion is the mean of a variable that is 1 when the individual has a characteristic and 0 otherwise. The standard deviation of any variable involves the expression .

Let's suppose there are m 1s (and n-m 0s) among the n subjects. Then, and is equal to (1-m/n) for m observations and 0-m/n for (n-m) observations. When these results are combined, the final result is

and the sample variance (square of the SD) of the 0/1 observations is

The sample proportion is the mean of n of these observations, so the standard error of the proportion is calculated like the standard error of the mean, that is, the SD of one of them divided by the square root of the sample size or