A poll that reports "47% approve, margin of error ±3 points" is really making two claims: a best guess and an honesty zone around it. That zone is the confidence interval — the range where the true value probably sits. A 95% confidence interval means that if you repeated the poll 100 times, about 95 of those intervals would capture the real number. It does not mean there is a 95% chance the truth is inside this particular interval, though that is how most people read it.

The width comes from three ingredients: the sample mean, the sample standard deviation, and the sample size, combined as x̄ ± z* × (s / √n). Bigger samples shrink the interval — quadrupling respondents halves the margin — while more confidence (99% instead of 95%) widens it, because you need a bigger net to be surer of the catch.

This calculator builds the interval from your four inputs: the Sample Mean (x̄), the Sample Standard Deviation (s), the Sample Size (n), and your chosen Z* critical value (1.96 for 95% confidence, 2.576 for 99%). A restaurant owner who surveyed 64 customers with an average rating of 4.2 and a standard deviation of 0.8 gets 4.2 ± 0.196 — a true rating plausibly between 4.0 and 4.4.

With the pre-filled defaults — Sample Mean (x̄) 50, Sample Standard Deviation (s) 10, Sample Size (n) 30, Z* (1.96 for 95%) 1.96 — the Confidence Interval Calculator returns Margin of Error 3.5785, Lower Bound 46.4215, Upper Bound 53.5785, Interval Width 7.1569. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.