Margin of Error Calculator

Margin of error is the half-width of an interval produced by a sampling procedure. A poll at 52% plus or minus 3 points gives the interval 49% to 55%. The confidence level describes how often that procedure captures the true population value over repeated samples, not a probability assigned to this one interval. It does not cover survey bias or nonresponse error.

The standard error formula contains p * (1 - p), which is maximized when p = 0.5 (giving 0.25). Any other value of p produces a smaller product and therefore a smaller margin. Plugging in 0.5 gives you the widest possible margin for your sample size, which is the safe choice before you have data or when you want to report a worst-case headline number.

For a two-sided normal interval, 90%, 95%, and 99% use critical values about 1.645, 1.96, and 2.576. Choose the level before seeing the result and follow your field's protocol. Higher confidence gives wider intervals and higher long-run coverage for the same sample; it does not by itself make a study rigorous or safe.

The standard formula assumes you are sampling from an effectively infinite population. Once your sample is more than about 5% of the actual population, that assumption breaks and the real margin is smaller than the formula suggests. The correction factor sqrt((N - n) / (N - 1)) shrinks the margin to reflect that. It matters for small populations like one company's employees or one school's students, and is negligible for national polls.

With p = 0.5 and 95% confidence, a simple random sample of about 1,067 gives a margin near 3 percentage points before design effects. Halving that margin requires roughly four times the sample because margin scales with 1 / sqrt(n). Real polls also depend on weighting, clustering, response bias, and sampling design.