Margin of Error Calculator
Calculate the margin of error for a survey or poll instantly. Enter your sample percentage, sample size, and confidence level to get the margin of error, confidence interval, and standard error. Turn on Finite Population Correction when your sample is a large slice of a known population. Flip to Reverse-solve mode to find the sample size that achieves a target margin of error.
Formula
Worked example
With a sample percentage of 50%, n = 1000 and 95% confidence: p = 0.5, standard error = sqrt(0.5 * 0.5 / 1000) = 0.0158, margin of error = 1.96 * 0.0158 = 0.031, about +/-3.1 percentage points. With FPC and a population of 5000: FPC = sqrt((5000-1000)/(5000-1)) = 0.894, giving MOE = 0.894 * 3.1 = +/-2.8 pp.
What the margin of error measures
The margin of error tells you how far a sample result is likely to sit from the true value in the whole population. When a poll reports that 52% of voters favour a candidate "with a margin of error of plus or minus three points", it means the real figure is plausibly anywhere from 49% to 55%. The margin exists because you surveyed a sample rather than every person, and different random samples would have produced slightly different percentages. Reporting the margin alongside the headline number is what separates an honest estimate from a misleading one, because it shows the precision of the result rather than implying false certainty.
How the formula is built
The margin of error for a proportion equals the critical value z multiplied by the standard error, where the standard error is the square root of p(1-p)/n. Here p is the observed sample proportion (the percentage as a decimal) and n is the sample size. The critical value comes from the standard normal distribution: 1.282 for 80%, 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%. Because n sits under a square root, the margin falls slowly as the sample grows; quadrupling the number of respondents only halves the margin, which is why very precise polls require very large samples.
Finite population correction (FPC)
The standard formula assumes you are sampling from an infinite population. When your sample makes up 5% or more of the total population, the finite population correction factor reduces the standard error. The FPC factor is the square root of (N-n)/(N-1), where N is the population size and n is the sample size. As n approaches N the factor approaches zero, reflecting that a census has no sampling error. Enable the FPC toggle and enter your population size when surveying a known, bounded group such as employees at a company, students at a school, or households in a small town.
Reverse-solve: finding the sample size you need
Flip the calculator to Reverse-solve mode to answer the planning question: how many people do I need to survey to achieve a margin of error no larger than X percent? The formula inverts the standard MOE equation to n = z^2 * p(1-p) / e^2, where e is your target margin as a decimal. Because the result must be a whole number of respondents, the calculator rounds up to the next integer. When finite population correction is enabled, the finite-population adjusted formula n = n0 * N / (n0 + N - 1) is applied, giving a smaller required sample because sampling a large fraction of the population is inherently more informative.
Why 50% gives the widest margin
The quantity p(1-p) is largest exactly when p equals 0.5, so a result of fifty-fifty produces the biggest standard error and therefore the widest margin of error for a given sample size. As the percentage moves toward 0% or 100%, that product shrinks and the estimate becomes tighter. This is why survey designers often plug in 50% when planning a sample size: it gives the most conservative, worst-case margin and guarantees the real margin for any other result will be no larger. This calculator uses the normal approximation, which is reliable when both n*p and n*(1-p) are at least about 5 to 10.
z critical values by confidence level
| Confidence level | z critical value | Tail area (each side) |
|---|---|---|
| 70% | 1.036 | 0.150 |
| 75% | 1.150 | 0.125 |
| 80% | 1.282 | 0.100 |
| 85% | 1.440 | 0.075 |
| 90% | 1.645 | 0.050 |
| 92% | 1.751 | 0.040 |
| 95% | 1.960 | 0.025 |
| 96% | 2.054 | 0.020 |
| 98% | 2.326 | 0.010 |
| 99% | 2.576 | 0.005 |
Two-sided critical values from the standard normal distribution. All ten levels are selectable in this calculator.
Frequently asked questions
What does a margin of error of plus or minus 3% mean?
It means the true population value is likely within 3 percentage points of your sample result, at the stated confidence level. If a poll reports 52% with a +/-3% margin at 95% confidence, the real figure is plausibly between 49% and 55%, and intervals built this way capture the true value about 95% of the time.
How do I make the margin of error smaller?
Increase the sample size. Because the margin falls with the square root of n, you need roughly four times as many respondents to cut it in half. Lowering the confidence level also shrinks the margin by using a smaller z value, but that makes you less certain the interval contains the true value. When sampling a large fraction of a known population, enabling finite population correction will also reduce the margin.
When should I use the finite population correction?
Use FPC when your sample is 5% or more of the total known population. For example, if you are surveying 200 employees at a company of 1000 people, your sampling fraction is 20%, and ignoring FPC overstates the uncertainty. For large populations (millions of people), FPC makes a negligible difference and is usually omitted.
Why does this calculator use 50% as the default percentage?
The product p(1-p) is largest at 50%, which produces the widest possible margin of error for a given sample size. Using 50% gives the most conservative estimate, so if you do not yet know the result, planning around it guarantees the real margin will be no larger than the figure you calculate.
How do I find the sample size needed for a desired margin of error?
Switch to Reverse-solve mode, enter the desired margin of error and confidence level, and the calculator returns the minimum sample size. The formula is n = z^2 * p(1-p) / e^2 rounded up to the next integer. For a 95% confidence level and a +/-3% margin with p = 50%, the formula gives about 1068 respondents. If you know the population size, toggle on FPC to get a more efficient (smaller) sample size.
What confidence level should I use?
Academic research and medical studies typically use 95%, which means 1 in 20 confidence intervals will miss the true value. Market research and quick polls often use 90% to keep sample sizes manageable. Use 99% only when the stakes of being wrong are very high, since it requires a substantially larger sample. The ten confidence levels available here cover the full range of common practice.