Updates automatically as you type.
Calculated in your browser; your inputs are not sent to us. Privacy
Calculation details
Review the generated output before relying on it.
This tool runs in your browser. Your calculator inputs and results are never transmitted to us or to ad/affiliate partners. Ads and sponsored links may set third-party cookies to serve and measure them, but they never receive your calculation values. If you explicitly save a scenario, its permitted fields stay in local browser storage until you clear them. See our Privacy Policy.
Use this result
Share the current inputs or ask ChatGPT to explain the calculation in context.
More options
What does this calculator estimate?
A margin of error is the uncertainty band around a survey percentage: the amount the result could shift in either direction, given a different random sample from the same population. A poll of 400 respondents reporting 51% yes with a +/-4.9% margin of error covers roughly 46% to 56% at 95% confidence. Use this calculator before fielding a survey, or when reading one, to see how much the sample size can be trusted at the 90%, 95%, or 99% confidence level.
- MoE = z * sqrt(p(1-p)/n); with the worst-case proportion p = 0.5 this reduces to z * 0.5 / sqrt(n). (Source: BoringToolsKit calculation methodology)
- The calculator's z scores are 1.645 at 90% confidence, 1.96 at 95%, and 2.576 at 99%, from the standard normal distribution. (Source: NIST/SEMATECH e-Handbook of Statistical Methods, standard normal table)
- Once the observed proportion p-hat is known, the tighter form is z * sqrt(p-hat(1-p-hat)/n): a result near 0% or 100% wobbles far less than a 50/50 split, so the half-split is the conservative choice. (Source: BoringToolsKit calculation methodology)
- Halving the margin of error costs four times the sample size, because n scales with the square of the margin: 385 respondents give +/-5% at 95% confidence, while 1,540 are needed for +/-2.5%. (Source: BoringToolsKit calculation methodology)
- A tiny margin of error says nothing about bias: it only bounds random sampling error, not bad question wording, non-response, or an unrepresentative frame. (Source: BoringToolsKit calculation methodology)
Confidence level vs. the margin of error
The confidence level and the margin of error are two ways of quoting the same uncertainty. Pick a confidence level and the calculator returns the z score behind it: 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%. Raising the confidence level widens the band, and lowering it narrows the band. A poll of 400 respondents carries a +/-4.9% margin at 95% confidence but only +/-4.1% at 90%, simply because you accept more risk of being wrong.
The 95% level is the survey default because it is a reasonable tradeoff: you are willing to be wrong about the interval roughly one time in twenty in exchange for a band that stays narrow enough to be useful. Quoting 99% confidence without widening expectations is how survey write-ups quietly overpromise; the tradeoff always has to be paid somewhere.
How sample size changes the margin of error
The margin of error shrinks with the square root of the sample size, so each halving costs four times the respondents. At the worst-case 50/50 split, 100 respondents give +/-9.8% at 95% confidence, 385 give +/-5.0%, and 1,067 give +/-3.0%, which is exactly what the calculator returns for those inputs. Getting to +/-2.0% takes 2,401 respondents, roughly six times the size of a typical national poll.
This is the arithmetic behind why most credible polls cluster near +/-3% or +/-5%: pushing much below ±2% requires thousands of respondents, and beyond that the remaining uncertainty is usually dominated by coverage and non-response problems the sample size cannot fix.
When the margin of error does not fix a bad sample
The margin of error only quantifies the random variation of probability samples taken from a well-defined population. If respondents were self-selected online, recruited by quotas rather than random selection, or drawn from a customer list rather than the voting or buying population, the calculator's band is a lower bound at best, because the biases involved can push results further than the stated margin.
An 8% non-response rate on a random frame still leaves most of the band valid, but a 60% self-selection rate changes what the number means altogether. Read post-election reviews of failed polls for the reminder: most of them missed on skew, not on the arithmetic.
How this calculator works
Formula
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 to verify
- 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.
Frequently asked 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.
Cite this tool
BoringToolsKit. “Margin of Error Calculator.” boringtoolskit.com/margin-of-error-calculator/ (reviewed October 2026). Free to reference in articles, syllabi, and answer posts with a link.
Privacy: Inputs and results stay in this browser. Any future sponsored recommendation or advertisement will be clearly labeled and kept separate from the calculation.