Use this margin of error calculator to estimate survey precision from sample size, confidence level, and expected proportion.
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Use this margin of error calculator to estimate poll or survey precision from sample size,
confidence level, and expected proportion, then see how finite-population correction or
design-effect inflation changes the interval you report.
Best use of this margin of error calculator Start with the conservative 50% proportion when the true rate is unknown, then add a known
population size or a design effect only when your survey design justifies it.
Confidence level
Quick assumptions
Use 50% when you do not have a prior estimate. That gives the largest plausible margin of
error for a proportion survey at the chosen sample size and confidence level.
Add population size only for closed populations where the sample is a meaningful share of
the total. Add design effect only if weighting, clustering, or complex sample design means
your raw sample behaves like a smaller effective sample.
Survey precision sheet
±4.99%
95% confidence interval:
45.01% to 54.99%.
The result uses the standard simple-random-sample proportion margin-of-error formula.
±4.99%
Margin of error
45.01% – 54.99%
Confidence interval
385
Effective sample size
1.96
Critical z-value
Finite-population effect
Not applied
Leave population blank when the population is very large or unknown. That keeps the
estimate slightly conservative.
Design effect
SRS
Under simple random sampling, the effective sample size matches the raw sample size.
Interpretation
50% ± 4.99%
If a poll result is reported at 50%, this setup
implies a likely interval from 45.01% to
54.99% under the stated assumptions.
Confidence-level comparison
Holding the same sample settings constant, higher confidence widens the reported interval.
Level
MOE
Interval
z
80%
±3.27%
46.73% – 53.27%
1.28
85%
±3.67%
46.33% – 53.67%
1.44
90%
±4.19%
45.81% – 54.19%
1.64
95% (selected)
±5%
45.01% – 54.99%
1.96
99%
±6.56%
43.44% – 56.56%
2.58
Sample-size sensitivity
Margin of error shrinks with the square root of n, which is why doubling the sample does
not cut the uncertainty in half.
Sample size
Effective n
MOE
Interval
193
193
±7.05%
42.95% – 57.05%
385 (current)
385
±5%
45.01% – 54.99%
770
770
±3.53%
46.47% – 53.53%
1,540
1,540
±2.5%
47.5% – 52.5%
Target-margin planning
Use these rows when the real question is how many completed responses you need for a
target survey margin of error under the same confidence, proportion, population, and
design-effect assumptions.
Target MOE
Completes needed
Effective n
Achieved MOE
Current sample
±10%
97
97
±9.95%
Meets target
±5%
385
385
±5%
Meets target
±3%
1,068
1,068
±3%
Needs more completes
±2%
2,401
2,401
±2%
Needs more completes
Method caution The reported margin of error covers sampling variability only. It does not fix bias from poor sampling frames, low response rates, question wording, or weighting choices.
Margin of error calculator — survey precision, interval width, and reporting context
A margin of error calculator turns a sample size, confidence level, and expected proportion into the plus-or-minus range around a survey result.
What a margin of error actually means
If a poll reports 52% support with a margin of error of ±3% at 95% confidence, the corresponding interval is 49% to 55%. That is the reporting shorthand most people recognise from election polling and survey dashboards.
The important point is that the margin of error is the half-width of the confidence interval, not a guarantee that the poll is right by exactly that amount. It is a statement about sampling variability under the method assumptions, not a catch-all measure of every kind of survey error.
The standard formula behind a margin of error calculator
For a simple random sample estimating a proportion, the classic formula is z multiplied by the standard error of the proportion. The standard error shrinks as the sample gets larger, which is why larger surveys produce narrower intervals.
This page uses the usual proportion workflow because that is the dominant search intent for poll, survey, customer feedback, and opinion-share questions. The result is also shown as a confidence interval because most readers want to interpret the estimated range, not only the headline plus-or-minus number.
MOE = z × √(p̂(1 − p̂) / n)
Simple-random-sample margin of error for a proportion.
CI = [p̂ − MOE, p̂ + MOE]
Confidence interval implied by the same sample proportion and margin of error.
Why 50% is the conservative default
When the true proportion is unknown, many survey planners use 50% as the default. That is not arbitrary. The product p × (1 − p) is largest at 0.5, so 50% produces the maximum margin of error for a given sample size and confidence level.
That conservative default is useful because it avoids claiming more precision than the survey design can support. If you already know the result will be closer to 20% or 80%, the margin of error will be smaller than the 50% case.
How sample size changes precision
Margin of error shrinks with the square root of sample size. That means bigger samples help, but with diminishing returns. Doubling the sample does not halve the margin of error. You usually need about four times the sample size to cut the margin roughly in half.
That tradeoff matters in practice because many readers really want a planning answer rather than a textbook answer. A survey with around 385 completed responses gives about ±5% precision at 95% confidence under the conservative 50% assumption for a very large population. Around 1,000 completed responses produces a margin of error close to ±3%.
Finite population correction: when population size actually matters
Many calculators ignore population size because, for very large populations, it barely changes the answer. But if your sample is drawn from a smaller closed population and represents a meaningful share of that whole group, finite population correction can reduce the margin of error materially.
This matters for internal surveys, professional associations, course cohorts, and customer lists where you might be sampling a large fraction of everyone who exists in the target population. In those situations, using the large-population formula without correction is conservative, but it overstates uncertainty.
MOE_FPC = MOE × √((N − n) / (N − 1))
Finite population correction for a sample that is a non-trivial share of the whole population.
Design effect: why weighting and clustering can make the real MOE larger
A simple random sample formula assumes every observation contributes equally and independently. Real surveys often break that assumption through weighting, clustering, stratification, or other design choices. Those choices can change the variance of the estimate relative to a simple random sample.
When weighting or clustering inflates variance, pollsters describe that inflation with a design effect. A design effect above 1 means the effective sample size is smaller than the raw respondent count, so the margin of error should be wider than the simple textbook formula suggests. This is one reason a headline sample size can look large while subgroup estimates still feel noisy.
Why subgroup margins of error are always larger
The reported margin of error for a survey usually refers to the full sample. But people often want to interpret smaller segments such as one age group, one region, or one customer tier. Those subgroups have fewer cases than the full sample, which means their margin of error is larger.
That is why a poll with an overall margin of error near ±3% can still have subgroup margins that are far wider. If a full sample of about 1,000 yields roughly ±3%, a subgroup with only 250 respondents may land closer to ±6% under the same assumptions.
What margin of error does not cover
The margin of error covers sampling variability only. It does not solve bias from a bad sampling frame, nonresponse, poor wording, mode effects, coverage problems, or mistakes in weighting assumptions. A technically correct margin of error on a biased survey can still produce a misleading conclusion.
This is where many public explanations go wrong. The poll may have a mathematically correct ±3% sampling margin of error and still miss the true value by much more because of systematic error. That is why poll method disclosures and weighting details matter alongside the MOE itself.
Worked example: 385 respondents at 95% confidence
Suppose a survey has 385 completed responses, uses the conservative 50% proportion assumption, and reports at 95% confidence. The standard simple-random-sample margin of error is about ±5.0%. If the observed result is 52%, the implied interval is roughly 47.0% to 57.0%.
Now imagine the same survey comes from a known population of 1,000 people instead of an effectively infinite population. The finite population correction trims that margin of error. If the survey also uses heavy weighting and the design effect is 1.5, the uncertainty widens again because the effective sample behaves more like a smaller survey.
Why this page now shows confidence-level and sample-size comparison rows
Most margin-of-error searches are not only about one finished result. They are planning questions. People want to know what happens if they report at 90% instead of 95%, or if they grow the sample from 400 to 800, or whether a known population changes the estimate enough to matter.
That is why the live calculator now includes comparison rows across common confidence levels and sample-size scenarios. Those rows answer the practical follow-up question directly instead of forcing the reader to rerun the calculator repeatedly to build the same intuition.
Using target-margin planning rows
The target-margin planning rows answer the reverse question behind many survey searches: how many completed responses would be needed for a target margin of error such as ±10%, ±5%, ±3%, or ±2%. That makes the page useful both after a survey is finished and before fieldwork starts.
Those rows use the same confidence level, expected proportion, optional finite population correction, and optional design effect as the headline margin of error result. If your current sample already meets a target, the row says so. If it does not, the row shows the completed-response threshold that would be needed before you can responsibly report that tighter survey precision.
Frequently asked questions
What is the difference between margin of error and confidence interval?
The margin of error is the plus-or-minus half-width of the confidence interval. If a result is 52% with a margin of error of 3 percentage points, the confidence interval is 49% to 55%.
Why do margin of error calculators default to 95% confidence?
Because 95% confidence is the most common reporting standard in polling and survey research. It balances caution and precision more effectively than 90% or 99% for many general reporting situations.
Why is 50% used when the true proportion is unknown?
Because 50% gives the largest possible margin of error for a proportion estimate at a fixed sample size and confidence level. It is the conservative planning default.
How many responses do I need for about ±5% margin of error?
Under the standard large-population simple-random-sample formula at 95% confidence with the conservative 50% assumption, the usual answer is about 385 completed responses.
How many responses do I need for about ±3% margin of error?
A common rule-of-thumb answer is around 1,000 completed responses at 95% confidence under the conservative 50% assumption for a large population.
When does population size affect margin of error?
Population size matters when the sample is a meaningful fraction of the total population. If you are surveying a very large population, the finite population correction is usually negligible. If you are surveying a small closed group, it can reduce the margin of error materially.
What is design effect in a margin of error calculation?
Design effect measures how much a complex survey design changes the variance relative to a simple random sample. A design effect above 1 inflates the margin of error and reduces the effective sample size.
Why is the subgroup margin of error larger than the overall margin?
Because subgroup estimates are based on fewer cases than the full sample. Fewer cases mean a larger standard error and therefore a wider margin of error.
Does margin of error include nonresponse bias or question wording bias?
No. Margin of error covers random sampling variability only. It does not account for systematic error such as nonresponse bias, poor wording, coverage gaps, or flawed weighting assumptions.
Why can a poll miss by more than its reported margin of error?
Because the reported margin of error usually covers sampling error only. Real polling error can also come from turnout modelling, weighting, response patterns, and other non-sampling problems.
Can I use this page to plan sample size for a target margin of error?
Yes. The target-margin planning rows show the completed responses needed for common precision targets under the same confidence level, expected proportion, population size, and design-effect assumptions. For a fuller fieldwork plan that also models response rate, completion rate, and subgroup targets, use the dedicated sample size calculator.
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