Free Margin of Error Calculator
Calculate the margin of error for a survey or poll from your confidence level, sample size, and the observed proportion, using the standard normal-approximation formula MoE = Z·√(p(1−p)/n).
Enter your confidence level, sample size, and observed proportion to find the survey margin of error.
Quick answer
The margin of error is MoE = Z·√(p(1−p)/n), where Z is the critical value for your confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%), p is the sample proportion, and n is the sample size. For a 95% confidence level with n = 1000 and the worst-case p = 0.5, MoE = 1.96·√(0.5·0.5/1000) ≈ 0.03099, or about ±3.10 percentage points. Larger samples shrink the margin of error in proportion to 1/√n.
Formula & method
Margin of error (proportion)
MoE = Z · √( p(1 − p) / n )
- MoE — Margin of error (half-width of the confidence interval)
- Z — Critical value of the standard normal distribution for the chosen confidence level
- p — Sample proportion (as a fraction between 0 and 1)
- n — Sample size (number of respondents)
Result is on the proportion scale (0–1); multiply by 100 to express it as a percentage. Use the worst-case p = 0.5 when no estimate is available, since p(1 − p) is maximized there.
Examples
- Input
- Confidence = 95% (Z = 1.96), n = 1000, p = 50%
- Result
- MoE ≈ 3.099%
- Why
- p = 0.5, so p(1−p) = 0.25. Then √(0.25/1000) = √0.00025 = 0.015811. MoE = 1.96 × 0.015811 = 0.030990, i.e. about ±3.10 percentage points.
- Input
- Confidence = 90% (Z = 1.645), n = 600, p = 50%
- Result
- MoE ≈ 3.358%
- Why
- p(1−p) = 0.25, so √(0.25/600) = √0.000416667 = 0.020412. MoE = 1.645 × 0.020412 = 0.033578, or about ±3.36 percentage points. The smaller n and lower confidence trade precision for a wider net.
- Input
- Confidence = 99% (Z = 2.576), n = 1500, p = 40%
- Result
- MoE ≈ 3.258%
- Why
- p = 0.4, so p(1−p) = 0.24. √(0.24/1500) = √0.00016 = 0.012649. MoE = 2.576 × 0.012649 = 0.032584, i.e. about ±3.26 percentage points. Note that p away from 0.5 lowers p(1−p), partly offsetting the larger Z.
When to use this tool
- Reporting the ± figure for a survey, poll, or A/B test result expressed as a percentage or proportion.
- Checking whether a sample is large enough to detect a difference before you collect more data.
- Comparing how confidence level, sample size, and proportion each affect the precision of an estimate.
- Quickly sanity-checking a published poll's stated margin of error against its sample size.
Common mistakes
- Entering the proportion p as a raw fraction (0.5) instead of a percent (50) — this field expects a percentage, so 50 means 50%, not 5000%.
- Forgetting that the margin of error is a half-width: the full 95% confidence interval is p ± MoE, so the interval is 2 × MoE wide, not MoE wide.
- Assuming a bigger sample halves the error linearly. Because MoE scales with 1/√n, you must quadruple the sample size to halve the margin of error.
- Using this normal-approximation formula for very small samples or proportions near 0% or 100%, where it becomes inaccurate and an exact (e.g. Wilson) interval is preferable.
Frequently asked questions
What is a good margin of error for a survey?
For most market research and political polling, a margin of error of ±3% to ±5% at 95% confidence is considered acceptable. Below ±3% usually requires more than 1,000 respondents, while a margin above ±5% signals the sample may be too small to draw firm conclusions. The right target depends on how small a difference you need to detect.
Why is the margin of error largest when the proportion is 50%?
The formula contains p(1 − p), which reaches its maximum value of 0.25 exactly at p = 0.5. Any proportion further from 50% — say 20% or 80% — produces a smaller p(1 − p) and therefore a smaller margin of error. Because of this, pollsters often use p = 0.5 as a conservative, worst-case assumption when the true proportion is unknown.
How does sample size affect the margin of error?
The margin of error is proportional to 1/√n, so it shrinks slowly as the sample grows. Doubling the sample size reduces the margin only by a factor of √2 (about 29%), and you must quadruple n to cut the margin in half. This is why polls rarely exceed a few thousand respondents — the precision gains become expensive.
What Z-score should I use for 90%, 95%, and 99% confidence?
The critical Z-values are 1.645 for 90% confidence, 1.96 for 95% confidence, and 2.576 for 99% confidence. These correspond to the two-sided cutoffs of the standard normal distribution that leave 10%, 5%, and 1% of the probability in the tails, respectively. This calculator selects the correct Z automatically from the confidence level you choose.
Is margin of error the same as a confidence interval?
They are closely related but not identical. The margin of error is the half-width of the interval, while the confidence interval is the full range p − MoE to p + MoE. So if a poll reports 52% support with a ±3% margin of error at 95% confidence, the confidence interval is 49% to 55%.
Does this calculator account for population size?
No — it uses the standard formula that assumes the population is large relative to the sample, which is true for almost all real surveys. If you are sampling a meaningful fraction of a small, finite population, you would apply a finite population correction factor, which slightly reduces the margin of error.
Sources & references
External references open in a new tab. We are independent and not affiliated with these organizations.
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Provided “as is” for general information only — results may be inaccurate, so verify before you rely on them. No warranty; use at your own risk.
Built and reviewed by HIFreeTools against the formula shown above and any authoritative references cited on this page. See our methodology and editorial standards.
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