๐ Margin of Error Calculator for Surveys and Polls
By ToolNimba Editorial Team ยท Updated 2026-06-20
Leave at 50% for the worst-case (largest) margin of error.
Add this to apply the finite population correction.
- z-score
- -
- Standard error
- -
- FPC applied
- -
Margin of error = z * sqrt(p(1 - p) / n), shown as a percentage.
The margin of error (MOE) tells you how much a survey or poll result could differ from the true value in the whole population, just because you measured a sample instead of everyone. Enter your sample size, choose a confidence level, and this calculator returns the margin of error as a plus-or-minus percentage along with the resulting confidence interval. Leave the proportion at 50% and you get the worst-case (largest) margin of error, which is what most published polls report.
What is the Margin of Error Calculator?
Every time you survey a sample rather than an entire population, your result is an estimate, not the exact truth. The margin of error puts a number on that uncertainty. It is the half-width of a confidence interval: if a poll says 52% support with a margin of error of plus or minus 3 percentage points at 95% confidence, the honest reading is that the true level of support is very likely somewhere between 49% and 55%. The margin of error answers the question "how far off could this sample be?" and it shrinks as your sample size grows.
The formula for a proportion is MOE = z sqrt(p (1 - p) / n). Here n is the sample size, p is the sample proportion (written as a decimal between 0 and 1), and z is the critical value from the normal distribution that matches your chosen confidence level. The most common z-scores are 1.645 for 90% confidence, 1.96 for 95% confidence, and 2.576 for 99% confidence. Higher confidence means a larger z, which makes the interval wider: you trade precision for certainty. The piece sqrt(p * (1 - p) / n) is the standard error of the proportion, and multiplying it by z scales it up to the confidence level you want.
The proportion p matters because the quantity p * (1 - p) is largest when p is 0.5. That is why pollsters who do not yet know the result, or who want a single conservative number, set p to 50%. At p = 0.5 the margin of error is at its maximum, so the reported figure is guaranteed to be wide enough no matter how the answers actually split. If you already know your result is lopsided (say 90% to 10%), plugging in the real proportion gives a smaller, more accurate margin of error.
One refinement matters when your sample is a large slice of a small population. The basic formula quietly assumes the population is effectively infinite. When you sample, for example, 500 people out of a club of 2,000, you can apply the finite population correction, which multiplies the standard error by sqrt((N - n) / (N - 1)), where N is the population size. This shrinks the margin of error because sampling a big fraction of a small group leaves less room for error. The correction is negligible when n is tiny relative to N, so it is usually skipped for national polls but worth using for surveys of a known, limited group.
When to use it
- Reporting the plus-or-minus figure for a political poll, customer survey, or market research study at a stated confidence level.
- Checking whether two survey results are far enough apart to be a real difference rather than sampling noise.
- Deciding how large a sample you need by seeing how the margin of error shrinks as the sample size grows.
- Adjusting the margin of error with the finite population correction when you survey a big share of a small, known group.
How to use the Margin of Error Calculator
- Enter your sample size (n), the number of people or items you actually surveyed.
- Pick the confidence level you want to report at: 90%, 95%, or 99%.
- Set the sample proportion, or leave it at 50% for the conservative worst-case margin of error.
- Optionally add the total population size to apply the finite population correction, then read off the margin of error and confidence interval.
Formula & method
Worked examples
A poll surveys 1,000 people at 95% confidence, using the worst-case proportion of 50%.
- Set p = 0.5, n = 1000, and z = 1.96 for 95% confidence
- Compute p(1 - p) = 0.5 x 0.5 = 0.25
- Standard error = sqrt(0.25 / 1000) = sqrt(0.00025) = 0.0158
- MOE = 1.96 x 0.0158 = 0.0310, which is 3.10%
Result: Margin of error is about +/- 3.10 percentage points (true value roughly 46.9% to 53.1% around a 50% result).
A survey of 400 customers finds 70% satisfaction and reports it at 99% confidence.
- Set p = 0.70, n = 400, and z = 2.576 for 99% confidence
- Compute p(1 - p) = 0.70 x 0.30 = 0.21
- Standard error = sqrt(0.21 / 400) = sqrt(0.000525) = 0.0229
- MOE = 2.576 x 0.0229 = 0.0590, which is 5.90%
Result: Margin of error is about +/- 5.90 percentage points, so the confidence interval is roughly 64.1% to 75.9%.
Worst-case margin of error (p = 50%) by sample size and confidence level
| Sample size (n) | 90% (z=1.645) | 95% (z=1.96) | 99% (z=2.576) |
|---|---|---|---|
| 100 | +/- 8.23% | +/- 9.80% | +/- 12.88% |
| 250 | +/- 5.20% | +/- 6.20% | +/- 8.15% |
| 500 | +/- 3.68% | +/- 4.38% | +/- 5.76% |
| 1000 | +/- 2.60% | +/- 3.10% | +/- 4.07% |
| 2000 | +/- 1.84% | +/- 2.19% | +/- 2.88% |
| 5000 | +/- 1.16% | +/- 1.39% | +/- 1.82% |
Critical z-scores for common confidence levels
| Confidence level | z-score | Meaning |
|---|---|---|
| 90% | 1.645 | Narrower interval, less certainty. |
| 95% | 1.960 | The most widely reported default for polls. |
| 99% | 2.576 | Wider interval, more certainty. |
Common mistakes to avoid
- Confusing the margin of error with the confidence level. The confidence level (such as 95%) is how often the method captures the true value over many samples; the margin of error is the plus-or-minus width of one interval. They are different numbers, and raising the confidence level actually widens the margin of error.
- Forgetting that p = 50% gives the largest MOE. Because p(1 - p) peaks at p = 0.5, using 50% always gives the biggest, most conservative margin of error. If you plug in a very lopsided proportion you will get a smaller MOE, which is correct but only valid for that specific result.
- Treating the margin of error as the only source of error. This formula covers sampling error only. Biased question wording, non-response, a non-random sample, or coverage gaps add error that the margin of error does not measure at all. A tight MOE does not make a flawed survey accurate.
- Ignoring the finite population correction for small groups. When your sample is a large fraction of a small, known population, the plain formula overstates the margin of error. Applying the finite population correction with the real population size N gives a smaller, more honest figure.
Glossary
- Margin of error (MOE)
- The plus-or-minus amount that a sample result could differ from the true population value due to sampling, at a stated confidence level.
- Confidence level
- The percentage of samples (such as 95%) for which the method would capture the true value; it sets the z-score used in the formula.
- z-score
- The critical value from the normal distribution for a confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%.
- Sample proportion (p)
- The fraction of the sample with the trait of interest, written between 0 and 1. Setting p to 0.5 gives the worst-case margin of error.
- Standard error
- The quantity sqrt(p(1 - p) / n), the typical sampling variability of a proportion before scaling by the z-score.
- Finite population correction (FPC)
- The factor sqrt((N - n) / (N - 1)) that shrinks the margin of error when the sample is a large share of a small, known population N.
Frequently asked questions
What is a margin of error in a survey?
The margin of error is the plus-or-minus range around a survey result that accounts for the fact that you measured a sample instead of the whole population. A result of 52% with a margin of error of plus or minus 3 points means the true value is likely between 49% and 55% at the stated confidence level.
How do you calculate the margin of error?
Use MOE = z * sqrt(p(1 - p) / n). Pick the z-score for your confidence level (1.96 for 95%), put in your sample proportion p as a decimal and your sample size n, then multiply by 100 to read the answer as a percentage. This calculator does all of that for you.
What sample size do I need for a 3% margin of error?
At 95% confidence with the worst-case proportion of 50%, you need about 1,067 responses for a margin of error of plus or minus 3 percentage points. Around 1,000 gets you close to 3.1%, and roughly 1,100 brings you just under 3%. Use this tool to test exact sizes.
Why do pollsters use 50% for the proportion?
The term p(1 - p) is largest when p is 0.5, so using 50% produces the biggest possible margin of error. That makes the reported figure conservative: it is wide enough no matter how the actual answers split, which is why most published polls state the margin of error at p = 50%.
Does a higher confidence level increase the margin of error?
Yes. A higher confidence level uses a larger z-score (1.96 at 95% versus 2.576 at 99%), which widens the interval. You are buying more certainty that the interval contains the true value, and the price is a larger margin of error for the same sample size.
When should I use the finite population correction?
Use it when your sample is a meaningful fraction of a small, known population, for example surveying 500 of a 2,000-member group. The correction shrinks the margin of error. For large populations where the sample is a tiny slice, the adjustment is negligible and can be skipped.