Survey Margin of Error Calculator

A survey margin of error calculator works out the range within which a poll’s true result likely falls, given your sample size, population size (optional), and confidence level. You enter the number of people surveyed, the confidence level (usually 95%), and the percentage picking an answer (often 50% for the worst case), and the calculator returns the margin of error as a plus-or-minus percentage point figure.

Formula

The margin of error (MOE) for a proportion is calculated as:

MOE = z × √(p(1 − p) / n)

Where:

• z is the z-score corresponding to your confidence level (1.96 for 95%, 1.645 for 90%, 2.576 for 99%)
• p is the proportion (as a decimal) of respondents who chose a particular answer. If you don’t know it, use 0.50 (50%) for the largest possible margin
• n is the sample size (number of people surveyed)

The result is expressed as a percentage, e.g., ±3.1%. It means that if you repeated the survey many times, the true population value would fall within that range around your observed percentage, about 95 times out of 100 (for a 95% confidence level).

Worked Example

Using the default values from the calculator: sample size = 1000, response percentage = 50%, confidence level = 95%.

1. Convert the response percentage to a decimal: p = 0.50
2. Calculate p(1 − p) = 0.50 × 0.50 = 0.25
3. Divide by n: 0.25 / 1000 = 0.00025
4. Take the square root: √0.00025 = 0.0158
5. Multiply by the z-score for 95% (1.96): 1.96 × 0.0158 = 0.0310
6. Convert to a percentage: 3.1%

So the margin of error is ±3.1%. If 55% of respondents said “yes,� the true population figure is likely between 51.9% and 58.1% (95% of the time).

When to Use This Calculator

Use this calculator when you are planning a survey or interpreting poll results. It helps you decide how many people you need to survey to get a desired precision. For example, if you want a margin of error of ±5% at 95% confidence, you need at least 385 respondents (assuming p = 0.5). The calculator also helps you evaluate the reliability of published polls: a poll of 500 people has a margin of error of about ±4.4%, while one of 2,000 has ±2.2%.

This calculator uses the standard formula for a simple random sample. It assumes that the sample is representative of the population and that responses are independent. It does not account for design effects (like clustering or weighting) that can increase the actual margin of error in complex surveys.

How to Choose the Right Sample Size

To determine the sample size needed for a target margin of error, you can rearrange the formula:

n = (z² × p(1 − p)) / MOE²

For example, to achieve a ±3% margin of error at 95% confidence with p = 0.5:

• n = (1.96² × 0.25) / 0.03² = (3.8416 × 0.25) / 0.0009 = 0.9604 / 0.0009 = 1067.1
• Round up to 1068 respondents.

If you have a small population (e.g., all employees in a company of 500), you can apply the finite population correction (FPC):

MOE_adjusted = MOE × √((N − n) / (N − 1))

Where N is the population size. For a population of 500 and a sample of 200, the correction factor is √((500−200)/(500−1)) = √(300/499) ≈ 0.775, so the margin of error is reduced by about 22.5%. Most survey calculators ignore the FPC when the population is large (over 20,000) because the effect is negligible.

Common Mistakes and Misconceptions

One common mistake is using the margin of error as if it applies to the difference between two candidates. If Candidate A has 52% and Candidate B has 48% with a margin of error of ±3%, the difference (4 points) is larger than the margin of error, but the uncertainty around the difference is actually about ±6 points (roughly double), so the race may be a statistical tie. To properly test a difference, you need a two-proportion test.

Another misconception is that the margin of error is the maximum error for any answer. In fact, the margin of error is largest when p = 50% and smaller when p is closer to 0% or 100%. For example, with a sample of 1000, the margin of error for p = 50% is ±3.1%, but for p = 10% it is only ±1.9%.

Also, the margin of error only accounts for sampling error, not for bias from question wording, nonresponse, or poor sampling frames. A well-conducted survey with a small margin of error can still be wildly inaccurate if the sample is not representative.

Limits of the Formula

The formula assumes a simple random sample with replacement (or sampling without replacement from a large population). It also assumes the sample proportion is not extremely close to 0 or 1; when p is very small (e.g., 1%), the normal approximation may be poor, and you should use exact binomial methods. Additionally, the formula gives the margin of error for each individual percentage, not for the difference between two percentages. For subgroup analyses (e.g., by age or gender), the margin of error is larger because the effective sample size is smaller.

Finally, the margin of error does not tell you the probability that a particular confidence interval contains the true value. That probability is the confidence level (e.g., 95%), which applies to the method over many repetitions, not to any single interval.

Frequently Asked Questions

What is a good margin of error for a survey?

It depends on the purpose. For political polls, ±3% is common. For academic research, ±5% is often acceptable. For quality control in manufacturing, you might need ±1% or less. The smaller the margin of error, the larger the sample size required.

Why do pollsters often report a margin of error of ±3%?

Because a sample of about 1,067 respondents at 95% confidence yields a margin of error of ±3% when p = 50%. Many national polls use samples of 1,000–1,500, which gives margins of error between ±2.5% and ±3.1%.

Does population size matter?

Only for small populations. For a population of 10,000 or more, the margin of error depends almost entirely on sample size, not population size. For example, a sample of 1,000 from a population of 20,000 has a margin of error of ±3.1%, essentially the same as from a population of 200 million.

What if I don’t know the response percentage?

Use 50% (0.5). This gives the largest possible margin of error, so you’ll be conservative. If your actual response percentage is lower or higher, the true margin of error will be smaller than your calculated value.

How do I account for a finite population?

Apply the finite population correction (FPC) as shown above. Many online calculators have an option for population size; if not, you can manually multiply the margin of error by √((N−n)/(N−1)). For populations over 20,000, the correction is negligible.

Can I use this calculator for non-random samples?

No. The formula assumes a probability sample where every member of the population has a known chance of being selected. If you use convenience sampling, quota sampling, or self-selection (like online polls), the margin of error is not meaningful because the sample may be biased.

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in n | Sample size | people | 1000 | 1
in p | Response percentage | % | 50 | 0
in c | Confidence level | % | 95 | 0
out moe | Margin of error | ±% | (1.96 * sqrt(p*(100-p)/n)) | 1
out se | Standard error | % | sqrt(p*(100-p)/n) | 1