Margin of Error Calculator
The ± figure quoted after every poll — what it means, how sample size drives it, and why the difference between two candidates has a bigger one.
For a proportion, the margin of error is z × √(p(1 − p)/n).
How the margin of error calculator works
For a proportion, the margin of error is z × √(p(1 − p)/n). At 95% confidence, p = 50% and n = 1,000, that is about ±3.1 points — which is why almost every national poll quotes ±3.
The number applies to a single proportion. The margin on the *difference* between two candidates is roughly twice as large, which is why "within the margin of error" is misused constantly.
Formula: MoE = z √(p(1 − p) / n)
Worked examples
| Inputs | Margin of error | Note |
|---|---|---|
| A poll of 1,000 | 3.099% | ±3.1 points |
| A poll of 400 | 4.8999% | ±4.9 points |
| A poll of 4,000 | 1.5495% | ±1.55 points — four times the cost, half the margin |
FAQFrequently asked questions
Why is the margin of error almost always ±3?
Because almost every national poll samples about 1,000 people, and 1,000 at 95% confidence gives ±3.1 points.
Does the population size matter?
Barely. Polling 1,000 people gives the same precision in a country of 5 million as in one of 300 million — a fact that surprises almost everyone.
What does "within the margin of error" mean?
Usually it is misused. The quoted margin applies to one share; the margin on the *gap* between two candidates is about twice as large, so a 4-point lead in a ±3 poll is not outside it.
Does it cover every kind of error?
No — only random sampling error. Non-response bias, question wording and bad weighting are not in the number, and they are usually the bigger problem.
How big a sample do I need for ±1 point?
About 9,600 at 95% confidence. Precision is expensive: the cost rises with the square of the precision.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.