This margin of error calculator finds the ± figure behind every poll and survey: given a sample size, observed proportion and confidence level, it computes the margin of error — and the confidence interval around your result.
What Is the Margin of Error?
The margin of error is the radius of uncertainty around a survey result. A poll reporting “52% ± 3%” claims the true value plausibly lies between 49% and 55% — the ±3% is the margin of error, and the range is the confidence interval. It quantifies sampling luck only: bigger samples wobble less. Planning the sample in the first place is the job of the Sample Size Calculator; both live in our Statistics Calculators collection.
The Margin of Error Formula
ME = Z × √(p(1 − p) ÷ n), with Z from the confidence level (1.96 for 95%) and p the observed proportion.
Worked example (verified): a poll of 1,000 people at 50% support, 95% confidence: ME = 1.96 × √(0.25 ÷ 1000) = ±3.10% — the number behind nearly every “margin of error ±3 points” news story you have ever read. At n = 385 (the classic sample size), the margin is exactly ±5.00%.
What the Margin of Error Does NOT Cover
- Biased sampling: polling only landlines or only volunteers skews results no matter the sample size.
- Lying, refusal and wording effects: the formula assumes honest answers to neutral questions from a random sample.
- Subgroups: a poll of 1,000 has ±3.1% overall, but a subgroup of 100 inside it has roughly ±9.8% — headlines rarely mention that distinction.
- It is a probability statement: at 95% confidence, about 1 interval in 20 still misses the truth. The Z values come from the same normal curve as our Z-Score Calculator.
How Sample Size Drives the Margin
| Sample size | Margin (95%, p = 50%) |
|---|---|
| 100 | ±9.80% |
| 385 | ±5.00% |
| 1,000 | ±3.10% |
| 2,401 | ±2.00% |
| 9,604 | ±1.00% |
The inverse-square law again: to halve the margin, quadruple the sample. That is why serious polls cluster around 400–1,100 respondents — beyond that, precision gets expensive fast.
Related Tools
- Sample Size Calculator — plan before collecting
- Z-Score Calculator
- Statistics Calculators — the full collection
Frequently Asked Questions
What is a good margin of error?
For published polls, ±3–5% at 95% confidence is standard. Tighter requires much larger samples; looser makes results hard to act on.
Why is the margin largest at 50%?
Because p(1−p) peaks at p = 0.5. A lopsided result (90/10) has a smaller margin — about ±1.9% at n = 1,000.
Does the margin of error account for bias?
No. It covers random sampling variation only, assuming a properly drawn random sample and honest responses.
What does “95% confidence” really mean?
If the same survey were repeated many times, about 95% of the intervals built this way would contain the true value. It is a statement about the method’s long-run reliability.
How do I reduce the margin of error?
Increase the sample size (at inverse-square cost), or accept a lower confidence level. There is no third lever.
Is this margin of error calculator free?
Yes — free, instant, no sign-up.