Margin of Error Calculator
Calculate the margin of error for surveys, polls, and research studies. The margin of error tells you how much the survey results might differ from the true population value. This calculator also determines the required sample size to achieve a target margin of error. See also our Confidence Interval Calculator, Sample Size Calculator, and Z-Score Calculator.
How to Use the Margin of Error Calculator
The margin of error (MOE) quantifies the uncertainty in survey or poll results. When a poll reports that 60% of respondents favor a candidate with a margin of error of ±3%, the true population proportion is likely between 57% and 63%. The margin of error depends on three factors: the confidence level, the sample size, and the variability in the population (estimated by the proportion). This calculator helps you understand the precision of your survey results and plan future studies.
Enter your confidence level (typically 95% for most surveys), sample size, and the estimated population proportion. If you don't know the proportion, use 0.5 (50%) as this gives the maximum (most conservative) margin of error. The calculator also computes the required sample size to achieve a target margin of error, which is essential for survey planning and budgeting. A smaller target MOE requires a larger sample size, with the relationship being quadratic — halving the MOE requires quadrupling the sample size.
The margin of error applies to simple random samples and assumes the sample is representative of the population. In practice, non-response bias, question wording, and sampling frame issues can introduce additional error not captured by the margin of error. For finite populations, a finite population correction factor can be applied to reduce the margin of error when the sample is a significant fraction of the population.
Margin of Error Formulas
Margin of Error (Proportion):
E = z* × √(p(1-p)/n)
Margin of Error (Mean):
E = z* × (σ/√n)
Required Sample Size:
n = (z*/E)² × p(1-p)
n = (z* × σ / E)² (for means)
With Finite Population Correction:
E_adj = E × √((N-n)/(N-1))
n_adj = n / (1 + (n-1)/N)
Confidence Interval:
CI = p̂ ± E
Width = 2 × E
Example Calculation
A political poll surveys 400 likely voters and finds that 52% support Candidate A. Calculate the margin of error at 95% confidence.
Given: n = 400, p = 0.52, confidence = 95%
z* for 95% confidence = 1.960
SE = √(p(1-p)/n) = √(0.52 × 0.48 / 400)
SE = √(0.2496/400) = √0.000624 = 0.02498
MOE = z* × SE = 1.960 × 0.02498 = 0.04896
Margin of Error: ±4.90%
95% CI: (47.1%, 56.9%)
Since the CI includes 50%, the lead is not
statistically significant at the 95% level.
Margin of Error by Sample Size (p=0.5)
| Sample Size (n) | MOE (95% CI) | MOE (99% CI) |
|---|---|---|
| 100 | ±9.80% | ±12.88% |
| 200 | ±6.93% | ±9.11% |
| 400 | ±4.90% | ±6.44% |
| 600 | ±4.00% | ±5.26% |
| 1000 | ±3.10% | ±4.07% |
| 1500 | ±2.53% | ±3.33% |
| 2000 | ±2.19% | ±2.88% |
| 5000 | ±1.39% | ±1.82% |
Frequently Asked Questions
What is a good margin of error for a survey?
For most surveys, a margin of error of ±3% to ±5% at 95% confidence is considered acceptable. Political polls typically aim for ±3% or less. Academic research may require ±2% or less for precise estimates. The acceptable MOE depends on the stakes of the decision being made — higher-stakes decisions warrant smaller margins of error and thus larger sample sizes.
Why use p = 0.5 when the proportion is unknown?
The value p = 0.5 maximizes the product p(1-p) = 0.25, which gives the largest possible margin of error for any given sample size. This is the most conservative estimate — the actual MOE will be equal to or smaller than this value regardless of the true proportion. Using p = 0.5 ensures your sample size is adequate even in the worst case scenario.
How does population size affect the margin of error?
For large populations, population size has virtually no effect on the margin of error — only sample size matters. The finite population correction (FPC) only becomes relevant when sampling more than about 5% of the population. For example, a sample of 1,000 from a population of 10,000 would have a slightly smaller MOE than a sample of 1,000 from a population of 10 million, but the difference is negligible for the larger population.
Does the margin of error account for all sources of error?
No. The margin of error only accounts for sampling error — the random variation due to surveying a sample rather than the entire population. It does not account for non-response bias, measurement error, question wording effects, interviewer effects, or coverage error. These non-sampling errors can be larger than the margin of error and are much harder to quantify.
How do I reduce the margin of error?
The most effective way to reduce the margin of error is to increase the sample size. However, the relationship is not linear — to halve the MOE, you need to quadruple the sample size (since MOE is proportional to 1/√n). You can also reduce the confidence level, but this means accepting a higher risk that the interval doesn't contain the true value. There are diminishing returns to increasing sample size beyond a certain point.
What is the difference between margin of error and standard error?
Standard error (SE) measures the variability of the sampling distribution — how much the sample statistic would vary across repeated samples. Margin of error (MOE) equals the critical value times the standard error: MOE = z* × SE. The margin of error is always larger than the standard error because it incorporates the desired confidence level. At 95% confidence, MOE ≈ 1.96 × SE.