Calculate margin of error from sample data or find required sample size for a desired MOE. Supports proportions and means with finite population correction.
The margin of error (MOE) quantifies the uncertainty in a survey or poll result. When a poll reports "52% ± 3%," the 3% is the margin of error, meaning the true population value is likely between 49% and 55%. Understanding and calculating MOE is essential for interpreting any survey, poll, or sample-based estimate.
This calculator works in two modes: (1) calculate the margin of error from a given sample, or (2) determine the sample size needed to achieve a desired margin of error. It handles both proportion estimates (surveys, polls) and mean estimates (continuous measurements), with optional finite population correction for sampling from known-size populations.
The calculator also shows how MOE varies with confidence level and sample size, making it easy to explore trade-offs between precision, confidence, and data collection cost. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
Every sample-based estimate has uncertainty. The margin of error makes that uncertainty concrete and interpretable. This calculator handles both directions — from sample to MOE, and from desired MOE to required sample size — saving time in study planning. The interactive tables let you see how changing confidence level or sample size affects precision.
Margin of Error (Proportion): MOE = z* × √(p̂(1−p̂)/n) Margin of Error (Mean): MOE = z* × (s/√n) With Finite Population Correction: MOE_adj = MOE × √((N−n)/(N−1)) Required Sample Size: n = (z*/MOE)² × p̂(1−p̂) With FPC: n_adj = n / (1 + (n−1)/N) Where z* is the critical value for the confidence level
Result: MOE = ±3.10%
A survey of 1,000 people finding 52% support has a margin of error of ±3.1% at 95% confidence. The 95% CI is [48.9%, 55.1%]. Since this interval includes 50%, the lead is not statistically significant.
When media reports a poll with "margin of error ±3%," this means at the stated confidence level (usually 95%), the true population proportion is expected to fall within 3 percentage points of the reported figure. If Candidate A polls at 48% ± 3%, the true support is likely between 45% and 51%. If the race is within the margin, it's a statistical tie.
There's a diminishing returns relationship between sample size and precision. Going from n=100 to n=400 cuts MOE in half. But going from n=1,000 to n=4,000 also only cuts MOE in half. Most surveys find n=1,000-1,500 a practical sweet spot, yielding ±3% at 95% confidence. Beyond that, additional precision comes at steep cost.
The margin of error increases for subgroups within a sample. If you survey 1,000 people but analyze only the 200 who are age 18-24, the MOE for that subgroup is much larger (based on n=200, not n=1,000). Always check whether subgroup analyses have adequate sample sizes.
It's the maximum expected difference between the sample statistic and the true population parameter at a given confidence level. A MOE of ±3% means the true value is likely within 3 percentage points of the sample estimate.
For large populations, MOE depends almost entirely on sample size, not population size. Polling 1,000 people gives the same accuracy whether the population is 1 million or 100 million. FPC only matters for small populations.
95% is standard and widely expected. Use 99% for high-stakes decisions (medical, legal). Use 90% for exploratory research where slightly more risk is acceptable. Higher confidence = wider MOE.
Three ways: increase sample size (most common), lower the confidence level (trade-off), or reduce variability in the population (often not controllable). Doubling the sample roughly reduces MOE by 29%.
When you sample a substantial fraction of the total population, the standard MOE formula overestimates uncertainty. The FPC factor √((N−n)/(N−1)) reduces the MOE. It's negligible when n < 0.05×N.
No. MOE only captures random sampling error — the variability from taking a random sample. It does not account for non-response bias, question wording effects, interviewer bias, or coverage error, which can be much larger.