OneTaskBy Tim King · Updated
Margin of Error Calculator for Surveys and Polls
Survey Margin of Error Formula, With an Optional Population Size Correction
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How the Margin of Error Calculator works
This margin of error calculator tells you how far a survey or poll result could be from the true figure. Enter the number of responses and the percentage that gave an answer, choose a confidence level and optionally the population size, and it returns the plus-or-minus and the likely range.
When a poll reports 52% support "plus or minus 3 points", the 3 is the margin of error. It comes from sampling alone: ask a different thousand people and the figure would wobble. It does not cover badly worded questions, people who refuse to answer or a sample that is not random.
How the margin of error calculator works
The formula is MOE = z × √(p(1 − p) ÷ n), where p is the result as a proportion, n the sample size and z the critical value for the confidence level, 1.96 for 95%. When you give a population size N, it multiplies by the finite population correction √((N − n) ÷ (N − 1)).
First example. A poll of 1,000 with a 50% result has MOE = 1.96 × √(0.25 ÷ 1000) ≈ ±3.1 points.
Second example. A staff survey gets 200 replies from 400 employees with 70% satisfied. The correction shrinks the margin from ±6.4 to about ±4.5 points.
How many responses give ±5%? About 385 from a large population, at 95% confidence.
Margin of error for a 50% result at 90%, 95% and 99% confidence by sample size
| Sample size | 90% | 95% | 99% |
|---|---|---|---|
| 50 | ±11.63% | ±13.86% | ±18.21% |
| 100 | ±8.22% | ±9.80% | ±12.88% |
| 150 | ±6.72% | ±8.00% | ±10.52% |
| 200 | ±5.82% | ±6.93% | ±9.11% |
| 250 | ±5.20% | ±6.20% | ±8.15% |
| 300 | ±4.75% | ±5.66% | ±7.44% |
| 400 | ±4.11% | ±4.90% | ±6.44% |
| 500 | ±3.68% | ±4.38% | ±5.76% |
| 600 | ±3.36% | ±4.00% | ±5.26% |
| 750 | ±3.00% | ±3.58% | ±4.70% |
| 800 | ±2.91% | ±3.46% | ±4.55% |
| 1,000 | ±2.60% | ±3.10% | ±4.07% |
| 1,200 | ±2.37% | ±2.83% | ±3.72% |
| 1,500 | ±2.12% | ±2.53% | ±3.33% |
| 2,000 | ±1.84% | ±2.19% | ±2.88% |
| 2,500 | ±1.64% | ±1.96% | ±2.58% |
| 3,000 | ±1.50% | ±1.79% | ±2.35% |
| 4,000 | ±1.30% | ±1.55% | ±2.04% |
| 5,000 | ±1.16% | ±1.39% | ±1.82% |
| 7,500 | ±0.95% | ±1.13% | ±1.49% |
| 10,000 | ±0.82% | ±0.98% | ±1.29% |
| 20,000 | ±0.58% | ±0.69% | ±0.91% |
| 50,000 | ±0.37% | ±0.44% | ±0.58% |
| 100,000 | ±0.26% | ±0.31% | ±0.41% |
Tips
- 50% is the worst case. Results near 0% or 100% have smaller margins, so 50% gives the safe upper bound.
- Subgroups have bigger margins. A 1,000-person poll split by region may leave only 100 per region, about ±10 points.
- Leave population blank for national polls; the correction only matters when you sample a sizeable share.
FAQ
What is a good margin of error for a survey?
For most decisions ±5 points at 95% confidence is acceptable, and published opinion polls usually aim for ±3. Below ±2 the sample sizes become expensive, needing over 2,400 responses, so the extra precision is only worth paying for when results are expected to be very close.
Why does the margin of error not depend on population size?
For large populations it barely changes, because what matters is how many people you asked, not what fraction. A random sample of 1,000 gives about ±3 points for a town or a whole country. Only when the sample is a large share of the population does the correction narrow it.
Does a bigger margin of error mean the poll is wrong?
No. It means the result is less precise. The true value is likely to be anywhere within the range, so two candidates on 48% and 52% with a ±3 margin are statistically tied, even though the headline puts one of them ahead of the other.
See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.
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