What this calculator does
Margin of error is the confidence multiplier times the standard error of a proportion. A survey of 400 people splitting 50-50 has a margin of error of 4.90% at 95% confidence, giving an interval from 45.1% to 54.9%.
The square root relationship is what makes polling expensive. Halving the margin needs four times the sample: 400 respondents give 4.90% and 1,600 give 2.45%. This is why most published polls cluster around a thousand respondents and a margin near 3%, which is roughly where the cost curve turns sharply upward.
The formula
The standard error of a proportion is the square root of p(1−p)/n, and the margin of error multiplies it by the z-score for the chosen confidence level: 1.645 for 90%, 1.96 for 95% and 2.576 for 99%. The formula assumes simple random sampling and ignores non-response bias, question wording and every other source of error that real surveys face.
| Term | Meaning |
|---|---|
| Margin of error | The half-width of the confidence interval around the sample proportion. |
| Confidence level | How often intervals built this way contain the true value, across repeated samples. |
| Standard error | √(p(1−p)/n), the sampling variability of the proportion. |
| Sampling error | The only error this captures. Non-response and question bias are not included. |
The inputs explained
| Field | What to enter |
|---|---|
| Sample proportion (%) | Sample proportion as a percentage. If unknown, use 50%, which gives the largest and therefore most conservative margin. |
| Sample size | Sample size, meaning completed responses rather than people contacted. |
| Confidence level | Confidence level. 95% is the convention in published polling. |
When to use it
Reporting a survey result
A proportion without a margin of error is incomplete, since it gives no sense of how precise the estimate is.
Judging whether a lead is real
Two candidates within the margin of error are not distinguishable, which is the point most often missed in reporting.
Planning a sample size
Working backwards from a target margin gives the sample needed, and shows how steeply the cost rises.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the proportion affect the margin?
The same sample size across a range of sample proportions.
| Sample proportion | Margin of error | Confidence interval | Standard error |
|---|---|---|---|
| 10% | 2.94% | 7.06% to 12.9% | 0.0150 |
| 30% | 4.49% | 25.5% to 34.5% | 0.0229 |
| 50% | 4.90% | 45.1% to 54.9% | 0.0250 |
| 70% | 4.49% | 65.5% to 74.5% | 0.0229 |
| 90% | 2.94% | 87.1% to 92.9% | 0.0150 |
Questions
Why do most polls use about 1,000 people?
Because that gives a margin of error near 3% at 95% confidence, which is generally considered good enough. Halving it to 1.5% would need 4,000 respondents, roughly quadrupling the cost for a improvement most users of the poll would not act on differently.
What does 95% confidence actually mean?
That if you repeated the survey many times, about 95% of the intervals constructed this way would contain the true population value. It does not mean there is a 95% probability that the true value lies in this particular interval, which is a different and much-abused claim.
Does the population size matter?
Barely, unless your sample is a large share of it. Sampling 1,000 people gives essentially the same margin whether the population is 100,000 or 100 million. Only when the sample exceeds about 5% of the population does the finite population correction make a visible difference.
Is the margin of error the whole error?
No, and this is the most important caveat. It captures sampling variability only. Non-response bias, question wording, sampling frame problems and respondents misreporting are all excluded, and in practice they frequently dominate the stated margin.
For sample size planning, see the sample size calculator. For an interval around a mean instead, see the confidence interval calculator.