Margin of Error Calculator
The ± figure quoted with a survey result, from its sample size and confidence level.
Last updated
Your details
50% is the most cautious assumption and gives the widest margin.
Optional — applies a finite-population correction.
Result
- Margin of error
- ± 3.099%
- Interval
- 46.9% to 53.1%
- Critical z
- 1.95996
- To halve this margin
- 4,000 responses
At 95% confidence
Precision improves with the square root of the sample
1,000 responses at 95% confidence…
MoE = z × √(p(1−p) ÷ n)
p(1−p) = 0.5 × 0.5 = 0.25
√(0.25 ÷ 1,000) = 0.015811
1.96 × 0.015811
MoE = ± 3.099%
Formula — written step by step
1,000 responses at 95% confidence…
MoE = z × √(p(1−p) ÷ n)
p(1−p) = 0.5 × 0.5 = 0.25
√(0.25 ÷ 1000) = 0.015811
1.95996 × 0.015811
MoE = ± 3.099%
p is the observed proportion and n the sample size. Fifty percent is used when the result is unknown because p(1−p) is largest there — it gives the widest, most cautious margin.
A worked example
Is a two-point lead real?
- Poll
- Candidate A 51%, B 49%
- Sample
- 1,000 respondents
- Margin of error
- ± 3.1 points
- A ranges from 47.9% to 54.1%
- B ranges from 45.9% to 52.1%
- The two intervals overlap substantially
- The lead is smaller than the margin
A statistical tie — the lead is not meaningful
A lead is only reliable when it exceeds roughly twice the margin of error, because both figures carry uncertainty. Reporting a two-point lead from a poll with a three-point margin as though one candidate is ahead is the most common misuse of survey data.
How to use this calculator
- Enter how many people responded — the achieved sample, not the number invited.
- Enter the observed proportion, or leave it at 50% for the most cautious margin.
- Add the population size if it is small relative to your sample; below about twenty times the sample it narrows the margin meaningfully.
- Remember this figure covers sampling error alone. It says nothing about who you failed to reach.
What the ± figure covers
The margin of error quantifies sampling error: the variation you would see if you drew a different random sample of the same size from the same population. It is derived from the sample size, the observed proportion and the confidence level, and nothing else.
That last point is the important one. It covers only random sampling variation. It makes no allowance for people who refused to answer, for questions that led respondents, for a sampling frame that missed part of the population, or for anyone who answered untruthfully. Those errors are frequently larger, and no formula estimates them.
Population size hardly matters
The most counter-intuitive result in survey statistics is that the population size is almost irrelevant. A sample of a thousand gives about a three-point margin whether you are polling a city or a continent.
The finite-population correction only bites when the sample is a substantial fraction of the whole — surveying 300 of a 1,000-member organisation, for example, narrows the margin appreciably. Once the population exceeds roughly twenty times the sample, the correction is negligible and the calculator’s uncorrected figure applies.
Comparing two numbers
The margin applies to each figure separately. When comparing two — a lead between candidates, or a change since last month — the uncertainty in the difference is larger than the uncertainty in either number alone.
A workable rule is that a difference should exceed about twice the margin before it is treated as real. A two-point lead in a poll with a three-point margin is not a lead; it is noise, and reporting it as a lead is the most common way survey results are misread.
Formula & methodology
What this calculator assumes
- Covers random sampling error only. Non-response, leading questions and an unrepresentative sampling frame are frequently larger sources of error and no formula estimates them.
- Assumes simple random sampling. Clustered or stratified designs have a different effective sample size.
- p = 50% is the default because p(1−p) is largest there, giving the most cautious margin.
Sources
- Lohr, Sampling: Design and Analysis, 3rd edition, CRC Press, 2021 — sampling error and finite-population correction.
Margin of error at 95% confidence, p = 50%
| Sample size | Margin of error |
|---|---|
| 100 | ± 9.80% |
| 250 | ± 6.20% |
| 500 | ± 4.38% |
| 1,000 | ± 3.10% |
| 2,000 | ± 2.19% |
| 5,000 | ± 1.39% |
| 10,000 | ± 0.98% |
The population size barely matters for large populations. A sample of 1,000 gives roughly a three-point margin whether the population is fifty thousand or fifty million — a result that surprises almost everyone.
Frequently asked questions
Why is 50% used as the default proportion?
Because p(1−p) is at its maximum there, giving the widest margin. Using it means the quoted margin is the most cautious one for that sample size.
Does a bigger population need a bigger sample?
Barely. Precision depends almost entirely on the sample size, not the population. A thousand responses gives about the same margin for a city as for a country.
Does the margin of error cover bias?
No. It measures random sampling variation only. Non-response, leading questions and an unrepresentative frame are usually larger problems, and no formula captures them.
How do I halve my margin of error?
Quadruple the sample. Precision improves with the square root of n, so each halving costs four times the data.