Margin of Error Calculator
How far a poll or survey result could be from the truth, at any confidence level, with the correction for sampling much of a small population.
Result
Margin of error: ±3.099%
Sampling error only. It says nothing about a biased sample or a badly worded question.
- Margin as a number
- 0.03099
- Range, lower
- 46.901%
- Range, upper
- 53.099%
- Critical value
- 1.959964
https://calcgallery.com/statistics/margin-of-error-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Margin of Error Calculator
- Choose a margin for a proportion, such as a poll result, or for a mean.
- Enter the confidence level and the sample size. For a proportion, leave the expected proportion at 50% unless you know better.
- If the sample is a noticeable share of a small group, enter the size of the group too.
The formulas
proportion MoE = z* × √(p(1 − p) ÷ n)mean MoE = t* × s ÷ √nsmall population, multiply by √((N − n) ÷ (N − 1))z* for 95% confidence is 1.960, for 90% 1.645 and for 99% 2.576. The page calculates it for any level you choose.
Sample size and the square root
The margin shrinks with the square root of the sample size. Four times as many people halve it. A poll of 400 has a 95% margin of about ±4.9 points, 1,000 about ±3.1, and 1,600 about ±2.5. Diminishing returns set in quickly, which is why national polls rarely go far beyond a thousand or two.
The finite population correction
The usual formula assumes the population is effectively unlimited. When the sample takes in a large share of it, there is less left unmeasured and so less uncertainty. The correction factor √((N − n) ÷ (N − 1)) is close to 1 for a large population and falls towards 0 as the sample approaches the whole group. The page applies it only when you enter a population size, and says so when it has.
To find the sample needed for a given margin, use the sample size calculator. It works through the same formula in reverse.
With your numbers
- Margin = 1.959964 × √(0.5 × (1 − 0.5) ÷ 1,000)
- Margin of error = ±3.099%
Quick reference
Sampling error only. It says nothing about a biased sample or a badly worded question.
| Example | Margin of error |
|---|---|
| Poll of 1,000 at 95% | ±3.099% |
| Poll of 1,000 at 99% | ±4.0727% |
| Survey of 400 at 95% | ±4.8999% |
| 10% of 1,000 at 95% | ±1.8594% |
| 400 of a staff of 500 | ±2.1935% |
| Mean, s 12, n 36 at 95% | ±4.060216 |
Common questions about the Margin of Error Calculator
- Why is the expected proportion set to 50%?
- Because p(1 − p) is largest at 50%, so it gives the widest margin, the cautious one. Published poll margins normally assume it. If a result is far from 50%, the true margin is smaller: at 10% with 1,000 people it is about ±1.9 points, against ±3.1 at 50%.
- Does a bigger population need a bigger sample?
- Hardly, once the population is large. A poll of 1,000 has about the same margin whether it samples a city or a country. What matters is the sample size, not the population size. The exception is when the sample is a large share of the population: surveying 400 of 500 staff leaves little room for error, and the finite population correction reduces the margin from ±4.9 to about ±2.2 points.
- What does ±3% actually mean?
- That at the stated confidence level, usually 95%, the true figure is expected to lie within 3 percentage points of the sample result. A poll at 52% with a ±3% margin is consistent with anything from 49% to 55%, so it does not show a majority either way.
- Does the margin of error cover all mistakes in a survey?
- No. It covers only the randomness of which people happened to be sampled. It says nothing about a sample that is not representative, people who refused to answer, leading questions or people misreporting. Those errors can be much larger than the margin and do not shrink with a bigger sample.