When a poll reports that 54% of voters support a measure “with a margin of error of ±2.8 points”, the margin of error is the half-width of a confidence interval around the sample percentage. It describes the sampling error only, meaning the uncertainty that comes from interviewing a random sample instead of everyone, and not other problems such as biased questions or people who refuse to answer.
Three things drive it. A larger sample shrinks it in proportion to 1/√n, so going from 400 to 1,600 respondents halves the margin. A higher confidence level enlarges it through the critical value z* (1.645 at 90%, 1.96 at 95%, 2.576 at 99%). And the proportion itself matters: p̂(1 − p̂) is largest at 50%, which is why pollsters quote the “maximum” margin of error for p = 50%.
When the sample is a sizeable share of a finite population, say 400 out of 5,000 employees, each respondent tells you more, and the finite population correction √((N − n)/(N − 1)) reduces the margin. It is usually applied when the sample exceeds about 5% of the population. Turn it on below and enter the population size.
How to use the Margin of Error Calculator
- Proportion given as
- Sample proportion (p̂): Use 50% for the maximum (worst-case) margin of error
- Number of people who gave the answer
- Sample size (n)
- Confidence level
- Custom confidence level
- Apply the finite population correction?
- Population size (N)
Results update as you type, and everything is calculated in your browser.
How it works
Compute p̂ (from the percentage, or from the count divided by n), the standard error √(p̂(1 − p̂)/n), multiply by z* for the confidence level and, if selected, by the finite population correction. The interval is p̂ ± MOE.
Worked example
A poll of n = 1,200 voters finds 648 (54%) in favour. At 95% confidence z* = 1.96 and the standard error is √(0.54 × 0.46 ÷ 1,200) = 1.439 points, so the margin of error is ±2.82 percentage points and the support is estimated at 51.18% to 56.82%. If the 1,200 voters came from a town of only 5,000, the finite population correction √(3,800 ÷ 4,999) = 0.872 reduces the margin to ±2.46 points.
Margin of error at p = 50%
| Sample size | 90% | 95% | 99% |
|---|---|---|---|
| 100 | ±8.2 | ±9.8 | ±12.9 |
| 400 | ±4.1 | ±4.9 | ±6.4 |
| 1,000 | ±2.6 | ±3.1 | ±4.1 |
| 2,500 | ±1.6 | ±2.0 | ±2.6 |
Inputs explained
| Field | What to enter |
|---|---|
| Proportion given as | Choose one: Percentage of the sample, Number of people who gave the answer |
| Sample proportion (p̂) | In %. Use 50% for the maximum (worst-case) margin of error |
| Number of people who gave the answer | A number |
| Sample size (n) | A number |
| Confidence level | Choose one: 70%, 75%, 80%, 85% … |
| Custom confidence level | In % |
| Apply the finite population correction? | Choose one: No (the population is large), Yes (enter the population size) |
| Population size (N) | A number |
What each result means
- Critical value z* = ninv(1 – (1 – [Confidence])÷2)
- Margin of error = [Critical value z×]×sqrt([p̂]×(1 – [p̂])÷[Sample size (n)])×[Finite population correction factor]×100
- Range from = max(0, [p̂]×100 – [Margin of error])
- Range to = min(100, [p̂]×100 + [Margin of error])
- Sample proportion used = [p̂]×100
- Standard error = sqrt([p̂]×(1 – [p̂])÷[Sample size (n)])×[Finite population correction factor]×100
- Maximum margin of error (p = 50%) = [Critical value z×]×sqrt(0.25÷[Sample size (n)])×[Finite population correction factor]×100
- Sample as a share of the population = ([Sample size (n)]÷[Population size (N)])×100
Formula
MOE = z* × √(p̂(1 − p̂) ÷ n) · with FPC: × √((N − n) ÷ (N − 1)) · p̂ = k ÷ n
Worked example
Inputs: Proportion given as: Percentage of the sample; Sample proportion (p̂): 54 %; Sample size (n): 1,200; Confidence level: 95%; Apply the finite population correction?: No – the population is large.
Results: Critical value z* = 1.96; Margin of error = 2.82 ± percentage points; Range from = 51.18 %; Range to = 56.82 %; Sample proportion used = 54 %; Standard error = 1.439 percentage points; Maximum margin of error (p = 50%) = 2.83 ± percentage points; Sample size that would halve this margin = 4,800.
Key terms
- Margin of error
- Half the width of the confidence interval around an estimate.
- Finite population correction
- Factor √((N − n)/(N − 1)) that shrinks the error when the sample is a large part of the population.
- Critical value z*
- Number of standard errors that gives the chosen confidence level.
Frequently asked questions
Why do polls of about 1,000 people have a margin of error of ±3%?
At 95% confidence and p = 50%, MOE = 1.96 × √(0.25 ÷ 1,000) = 3.1 percentage points.
How do I calculate the margin of error?
MOE = z* × √(p̂(1 − p̂)/n). For 54% of 1,200 at 95%: 1.96 × √(0.54 × 0.46 ÷ 1,200) = 2.82 points.
When should I use the finite population correction?
When the sample is more than about 5% of the population. Surveying 1,200 of 5,000 people cuts the margin from 2.82 to 2.46 points.
Does a bigger population need a bigger sample?
Hardly, once the population is large. A sample of 1,000 gives about ±3.1% whether the population is 100,000 or 300 million.
What does the margin of error not include?
Non-sampling errors: biased question wording, non-response, people answering untruthfully or a sample that is not random.
Is the margin of error the same for every answer in a poll?
No. It depends on each percentage. It is largest for answers near 50% and smaller for answers near 0% or 100%.
How is the margin of error for the difference between two candidates calculated?
It is larger, roughly 1.7 to 2 times the margin for a single percentage, because both percentages are uncertain.
Sources
Guides on this topic
- How to Calculate Sample Size for a Survey (Formula + Examples): 385 responses give ±5% at 95% confidence for almost any large population. Here is the formula behind that number and how to adjust it for your survey.