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Margin of Error Calculator

Find the ± margin of error of a survey or poll result. Enter the sample size, confidence level and observed percentage (and the population size for small populations) to get the margin, the confidence interval and how the margin changes with more respondents.

Last updated:

Estimate
%

Use 50% if unknown (worst case)

For small populations; leave empty if very large

In % for proportions, in data units for means: shows the sample size needed

Enter your values to see the result.
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How to use the Margin of Error Calculator

  1. Enter the sample size: how many people answered.
  2. Choose the confidence level (95% is standard for polls).
  3. Enter the observed percentage, or 50% for the most cautious margin.
  4. Add the population size if you surveyed a sizeable share of a small group, such as a company's staff.
  5. Read the ± margin and the interval; optionally enter a target margin to see the sample size required.

Margin of error formula

MOE = z × √(p(1 − p) ÷ n) × √((N − n) ÷ (N − 1))

  • z: critical value for the confidence level (see the table).
  • p: observed proportion as a decimal (0.5 if unknown).
  • n: sample size; N: population size. Leave out the last factor for a large population.

Example: 1,000 respondents, 50%, 95% confidence: 1.96 × √(0.25 ÷ 1000) = 1.96 × 0.01581 = 0.031, so ±3.1 percentage points.

Confidence levelz
80%1.282
90%1.645
95%1.960
98%2.326
99%2.576
99.9%3.291

Margin of error at 95% confidence (p = 50%)

Sample sizeLarge populationPopulation 5,000Population 1,000
100±9.80%±9.70%±9.30%
200±6.93%±6.79%±6.20%
400±4.90%±4.70%±3.80%
1,000±3.10%±2.77%—
2,000±2.19%±1.70%—

Frequently asked questions

How is the margin of error calculated?

For a proportion, MOE = z × √(p(1 − p) ÷ n), where z is the critical value for the confidence level (1.96 for 95%), p the observed proportion and n the sample size. For a mean it is z × σ ÷ √n.

What is the margin of error for 1,000 respondents?

About ±3.1 percentage points at 95% confidence when the result is near 50%. With 400 respondents it is about ±4.9 points and with 2,000 about ±2.2 points.

Why use 50% for the proportion?

p(1 − p) is largest at p = 0.5, so 50% gives the widest, most conservative margin. Polls usually quote this worst-case figure so it applies to every question in the survey.

What is the finite population correction?

When the sample is a large share of the population (roughly more than 5%), the margin shrinks by √((N − n) ÷ (N − 1)). For 300 responses from 1,200 employees the factor is about 0.87, so ±5.4% becomes ±4.7%.

Does a bigger population need a bigger sample?

Barely. Without the correction, the margin depends on the sample size, not the population size. A sample of 1,000 is about as precise for a city as for a whole country, provided it is random.

What does the margin of error not cover?

Only random sampling error. It does not account for biased samples, non-response, badly worded questions or people answering untruthfully, which can be larger than the margin itself.

How does the confidence level change the margin?

Higher confidence needs a wider margin. With n = 1,000 and p = 50%, the margin is ±2.6% at 90%, ±3.1% at 95% and ±4.1% at 99% confidence.

Are all the tools on FreeToolCabin tested for quality and accuracy?

Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.

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