The maximum margin of error for a particular sample size is obtained when p = 0.5 (equivalent to 50%). The margin of error is proportional to the square root of p* (1-p).

The equation for the standard error of the mean is the sample standard deviation divided by the square root of the sample size.

The margin of error is a way of calculating how big that swing is likely to be. The larger the sample, the smaller the error margin.