Margin of Error and Sample Size

Margin of error measures how far your sample proportion is likely to be from the true population proportion. It equals a critical value, z-star, times the standard error of p-hat, which is √(p-hat(1-p-hat)/n). The n sits inside a square root, not by itself. That square root is the whole story: if you want the margin of error to drop by a factor of 2, you need n inside the root to grow by a factor of 4, because √4 = 2. Shrinking error is expensive. Cut it to a third, and sample size jumps by 9.

To find the sample size needed for a target margin of error, start from the formula m = z-star times √(p-hat(1-p-hat)/n) and solve for n: n = (z-star/m)² times p-hat(1-p-hat). Since you rarely know p-hat in advance, use the conservative guess p-hat = 0.5, because p(1-p) is largest there, giving the biggest possible n and guaranteeing your interval is at least as narrow as planned. Always round n up, since rounding down would leave your margin of error too wide.

Notice population size never enters this formula. Margin of error depends on how spread out responses are and how many people you sampled, not on how large the group you are generalizing to is. A poll of 1000 people has nearly the same margin of error whether it describes a town of 50000 or a country of 300 million. This sets up the next step: using margin of error and a sample estimate together to build and interpret a full confidence interval.