# Margin of Error Calculator

> Margin of error calculator: compute survey margin of error from sample size, confidence level, and population size. Free, private, no signup, in your browser.

- **Canonical URL:** https://boringtoolskit.com/margin-of-error-calculator/
- **Category:** Statistics
- **Evidence level:** authored-methodology
- **Reviewed:** October 2026
- **Last verified:** 2026-10-12
- **Privacy:** Inputs and results stay in this browser unless the page clearly labels an external request.

## How it works

MoE = z * sqrt(p(1-p)/n) for proportions, or z * s / sqrt(n) for means; the calculator uses the conservative worst-case p = 0.5, giving z * 0.5 / sqrt(n), with z = 1.645 at 90%, 1.96 at 95%, and 2.576 at 99% confidence.

## Worked example

n = 385 at 95% confidence: MoE = 1.96 * sqrt(0.25/385) = +/-5.0%. A poll of 400 respondents gives +/-4.9%, and reading a reported 51% means the true value plausibly lies anywhere from roughly 46% to 56%.

## Assumptions

- Results use the maximum-variance proportion p = 0.5, the conservative choice when the true split is unknown; real results near 0% or 100% have a smaller true margin.
- The sample is treated as a simple random sample from the stated population; weights, clustering, and unequal probabilities are not modeled.
- Finite-population correction for small sampling fractions below 5% of the population is ignored, so results are conservative in those cases.

## Common questions

### What does the margin of error mean for a percentage result?

It is how far the reported percentage could shift in either direction in 95 or 100 repetitions of the same survey. A result of 51% with a +/-3% margin of error covers roughly 48% to 54% at the stated confidence level.

### How is the margin of error different for a mean versus a proportion?

For a proportion the margin is z * sqrt(p(1-p)/n) and caps out at 0.5 in the exponent. For a mean it is z * s / sqrt(n), using the standard deviation of whatever quantity is being measured rather than a 50/50 split.

### How does sample size change the margin of error?

It shrinks with the square root of the sample size, so halving the margin requires four times as many respondents. 385 respondents gives +/-5% at 95% confidence; 1,540 are needed for +/-2.5%.

### What is the margin of error at 100 respondents?

At 95% confidence with a 50/50 split, 100 respondents give a margin of about +/-9.8%, so a reported 55% is really anywhere from roughly 45% to 65%.

### Does a smaller margin of error mean the poll is more accurate?

Not by itself. The margin only bounds random sampling error, so a poll with heavy non-response or a skewed sample can have a tiny stated margin and still be systematically wrong.

## Related tools

- /sample-size-calculator/
- /confidence-interval-calculator/
- /standard-deviation-calculator/

## Sources

- [e-Handbook of Statistical Methods, cumulative distribution function of the standard normal distribution](https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm)
- [Calculation methodology and verification standard](/methodology/)

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