Margin of Error Calculator

How to use this calculator

Enter the sample proportion (the percentage giving a particular answer), the number of respondents and the confidence level, pre-set to the 95% used by most pollsters. The calculator returns the standard error, the margin of error and the lower and upper bounds of the confidence interval.

It also works from a published result. Enter the reported interval’s bounds and the sample size to find the confidence level that was used, or the proportion and the margin to find how many people must have been asked. To plan a new survey around a target margin, the sample size calculator uses the worst-case proportion of 50%.

The formula

SE = √(p̂ (1 − p̂) / n)

E = z × SE

CI = p̂ ± E

CL: confidence level
z: critical value
p̂: sample proportion
n: sample size
SE: standard error
E: margin of error (±)
L: lower bound
U: upper bound

Worked example

In a poll of 1,000 likely voters, 48% support a ballot measure. What is the margin of error at 95% confidence?

Standard error SE = √(0.48 × 0.52 / 1,000) = 1.58%.
Margin of error E = 1.96 × 1.58% = ±3.10 points.
Interval: 48% ± 3.1 = 44.9% to 51.1%.

Support is somewhere between about 45% and 51% — the poll cannot say whether the measure has majority support.

Assumptions and limits

Simple random sample; weighting and design effects in real polls typically inflate the margin by 10–50%.
Normal (Wald) approximation, reasonable when n·p̂ and n·(1 − p̂) are both above about 10. For small samples or proportions near 0% or 100%, a Wilson interval is better.
Covers sampling error only — not non-response, coverage or question-wording bias.

Frequently asked questions

Why do most national polls quote ±3 points?

With about 1,000 respondents and p̂ near 50%, E = 1.96 × √(0.25 / 1,000) = 3.1 percentage points.

Does the margin apply to the lead between two candidates?

No. The margin on the difference between two shares in the same poll is roughly twice as large, so a 4-point lead with ±3 margin is not clear-cut.

Why does the margin shrink so slowly with sample size?

It scales with 1/√n: to halve the margin you need four times as many respondents.