Margin of Error
Margin of error for a survey from sample size, proportion and confidence.
Margin of error
± 3.1%
95% confidence
Sample size
1,000
z-value
1.96
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How the Margin of Error works
This calculator computes the margin of error for a survey or poll given your sample size, population proportion, and desired confidence level — giving researchers, journalists, students, and decision-makers an instant read on how precisely their sample reflects the true population.
The margin of error (MOE) quantifies the uncertainty inherent in any survey that samples a subset of a larger population. Because you can never interview everyone, your sample result will differ slightly from the true population value. The MOE defines a symmetric band around your sample estimate within which the true population parameter is expected to fall — with a specified level of statistical confidence. For example, if a poll reports 54% support with a ±3% margin of error at 95% confidence, it means you can be 95% confident the true support level lies between 51% and 57%.
Three inputs drive the calculation. First, sample size (n) is the number of completed, valid responses — larger samples shrink the MOE because more data means less uncertainty. Second, the population proportion (p) represents your best estimate of the true rate of the characteristic you're measuring, often set to 0.5 (50%) when unknown, because that value maximizes variance and therefore yields the most conservative (widest) margin of error. Third, the confidence level — typically 90%, 95%, or 99% — controls the critical z-value used in the formula; higher confidence demands a wider interval, all else being equal. Understanding how these three levers interact is key to designing surveys that are both accurate and cost-effective.
A common mistake is conflating margin of error with total survey error. The MOE only captures random sampling error — the statistical variation expected from drawing a random sample. It does not account for non-response bias (when certain groups systematically opt out), question wording effects, response bias, or coverage error (when some segments of the population have no chance of being sampled). A methodologically flawed poll can have an impressively small MOE yet still produce wildly inaccurate results. Always treat MOE as a floor on uncertainty, not a ceiling.
For practical survey design, work the formula in reverse: decide on an acceptable MOE first, then solve for the required sample size. Most public opinion polls target a ±3% MOE at 95% confidence, which requires roughly 1,067 respondents assuming p = 0.5. If your population is small and you're sampling a substantial fraction of it (say, more than 5%), apply the finite population correction (FPC) factor to get a tighter, more accurate MOE. Subgroup analyses — breaking results down by age, region, or party affiliation — effectively reduce sample size for each subgroup and can dramatically inflate their individual margins of error, a nuance frequently overlooked in media reporting.
Formula
MoE = z · √(p(1−p) / n)
Pro tips
- Always use p = 0.5 when you have no prior estimate of the proportion — it produces the widest (most conservative) margin of error and protects you from underestimating uncertainty.
- To halve your margin of error, you must quadruple your sample size, because MOE shrinks with the square root of n — budget for this non-linear relationship before fieldwork begins.
- Report MOE for every subgroup separately; a poll of 1,000 people may have a ±3% overall MOE but a ±9% MOE for a 120-person subgroup, making subgroup comparisons unreliable.
- If your survey targets a small, well-defined population (e.g., employees at a single company), apply the finite population correction — you may need far fewer interviews than a standard calculator suggests.
- Pair your MOE with the confidence level every time you report results; saying '±4%' without specifying '95% confidence' omits critical context that changes how readers should interpret your data.
Key terms
- Margin of Error (MOE)
- — The maximum expected difference, at a given confidence level, between a survey's sample estimate and the true population parameter.
- Confidence Level
- — The probability — expressed as a percentage such as 95% — that the calculated confidence interval contains the true population value if the same survey were repeated many times.
- Critical Z-Value
- — The number of standard deviations from the mean corresponding to a chosen confidence level; common values are 1.645 (90%), 1.960 (95%), and 2.576 (99%).
- Population Proportion (p)
- — The estimated fraction of the population that has the characteristic of interest; when unknown, 0.5 is used to produce the most conservative margin of error.
- Sample Size (n)
- — The count of valid survey respondents; larger sample sizes reduce the margin of error by decreasing the standard error of the proportion.
- Finite Population Correction (FPC)
- — An adjustment factor applied when the sample represents more than roughly 5% of a finite population, which reduces the margin of error to reflect the lower remaining uncertainty.
