Margin of Error Calculator
Calculate the margin of error for surveys and polls. Enter sample size and confidence level to get margin of error and confidence interval.
Results
Margin of Error
The margin of error (MOE) tells you how much your survey result might differ from the true population value. A poll reporting "48% ± 3%" means the true proportion is likely between 45% and 51%, at the stated confidence level.
The Formula
MOE = z* × √(p(1−p)/n), where z* is the critical z-value for your confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%), p is the sample proportion, and n is the sample size.
Sample Size and MOE
MOE is proportional to 1/√n. To cut the margin of error in half, you need four times as many respondents. This is why most national polls use ~1,000 respondents — it gives ±3% at 95% confidence, which is considered acceptable for most political polling.
Worked Example
A political pollster surveys 1,067 likely voters and finds 52% support Candidate A. Using 95% confidence (z*=1.96) and p=0.52:
MOE = 1.96 × √(0.52 × 0.48 / 1067) = 1.96 × √(0.000234) = 1.96 × 0.01530 ≈ ±3.0%
So the reported result is 52% ± 3%, meaning the true support is likely between 49% and 55%. Since this range includes 50%, the race is still within the margin of error and cannot be called a definitive lead.
How Confidence Level Affects MOE
| Confidence Level | z* Value | MOE (n=1000, p=0.5) |
|---|---|---|
| 90% | 1.645 | ±2.60% |
| 95% | 1.960 | ±3.10% |
| 99% | 2.576 | ±4.07% |
A higher confidence level produces a wider margin of error. The tradeoff is between certainty and precision: the 99% CI is wider than the 95% CI because you need a larger range to be 99% confident it contains the true value.
Frequently Asked Questions
If a poll shows 52% support with ±3% MOE at 95% confidence, the true support is likely between 49% and 55%. The margin represents sampling uncertainty — you surveyed a sample, not everyone.
p(1−p) is maximized at p=0.5. Using 0.5 gives the largest possible MOE for a given n, so your sample will be sufficient regardless of the actual population proportion.
MOE is proportional to 1/√n, not 1/n. Doubling n multiplies 1/√n by 1/√2 ≈ 0.707 — about a 29% reduction, not 50%. To halve MOE, you need 4× the sample size.
CI = p̂ ± MOE. The confidence interval is the sample proportion plus and minus the margin of error. If 53% support a policy and MOE = 3%, the 95% CI is [50%, 56%].
No. Margin of error calculations assume a simple random sample from the population. Online opt-in polls, convenience samples, and self-selected surveys do not have a valid margin of error. The MOE only measures random sampling error — it does not account for biases in who chose to respond.
Using p=0.5 and 95% confidence: n = z*² × p(1−p) / MOE² = 1.96² × 0.25 / 0.02² = 3.8416 × 0.25 / 0.0004 = 2,401 respondents. For ±1% MOE you'd need approximately 9,604 respondents — demonstrating why very tight margins require enormous samples.
When your sample is a large fraction of a small population, the standard MOE formula overestimates the true error. The corrected MOE = MOE × √((N−n)/(N−1)), where N is the total population size and n is the sample size. For example, if you survey 200 out of 500 employees, applying this correction reduces the MOE by about 28% compared to the standard formula.