margin of error

Half the width of a confidence interval, e.g. about 1.96 x SE for 95%.

The margin of error is the ± that comes with an estimate from a sample: "42% said yes, ±5 percentage points". It is half the width of the confidence interval, so the true value is likely between 37% and 47%.

For a share it is roughly $q \times \sqrt{\frac{p(1-p)}{n}}$ , where $q$ is 1.96 for 95% confidence, $p$ the share and $n$ the number of answers. Two things follow. More answers shrink it, but slowly: halving the margin takes about four times as many. And it is widest at $p = 50\%$ , which is why survey planners use 50% when they don't know what to expect.

When the whole group is small and you reach a large part of it, the finite population correction makes the margin smaller: asking 400 of 2,000 people tells you more than asking 400 of two million.

The margin covers one thing only: chance in who ended up in the sample. It says nothing about people who didn't answer, a biased invitation list or a leading question. Those errors don't shrink with more answers.

Example

CatChow surveys 400 randomly chosen subscribers about an evening delivery slot, and 50% say they'd use it.

$\sqrt{\frac{0.5 \times 0.5}{400}} = 0.025$ .
At 95%: $1.96 \times 0.025 \approx 0.049$ , a margin of about ±4.9 percentage points.
The true share among all subscribers is likely between 45.1% and 54.9%.

If CatChow has only 2,000 subscribers, the finite population correction $\sqrt{\frac{2000-400}{2000-1}} \approx 0.89$ narrows the margin to about ±4.4 points.

Common mistakes

Applying the survey's margin to a segment. If 60 of the 400 answers come from annual-plan customers, their margin is the margin for 60 answers, about ±12 points, not ±4.9.
Reading it as the whole error. Non-response, self-selected answers and badly worded questions are outside the margin, and a bigger sample makes a biased number look more precise, not more correct.
Mixing up percent and percentage points. ±5 means five points around the share (40% → 35–45%), not 5% of 40%.
Comparing two margins to judge a difference. Two overlapping ranges can still hide a significant difference; test the difference itself.

Learn it in the course

95% confidence intervals · Building a 95% confidence interval from its formula and margin of error, so you report a range of plausible values instead of one number.
Course recap: choosing the right test · A decision guide that ties the whole course together.