Margin of Error Calculator
Determine how precise your poll or survey results are. The margin of error tells you how far the true population proportion could differ from your sample proportion at a given confidence level.
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Formula
MoE = z* × √[ p(1−p) / n ] × 100
The margin of error for a proportion uses the critical z-value (z*), the estimated proportion p (expressed as a decimal), and the sample size n. The term p(1−p) is maximized when p = 0.5, which is why 50% is used as a conservative default when the true proportion is unknown. Multiply by 100 to express the result as a percentage.
How to use the Margin of Error Calculator
- 1
Enter your sample size (n)
- 2
Enter your confidence level
- 3
Enter your proportion (%)
Value should be in %.
- 4
Read your results instantly
Results update in real time as you type.
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What margin of error means in polls
When a poll reports a candidate has 52% support with a margin of error of ±3%, that means the true proportion is likely between 49% and 55%. A race where both candidates are within the margin of error of 50% is considered a statistical tie — you cannot conclude which candidate is truly ahead based on that sample alone.
The margin of error applies to each individual estimate, not to the difference between two estimates. Comparing two candidates' numbers requires a separate calculation that accounts for the correlation between the two proportions.
The 50% rule and conservative estimates
The margin of error is largest when the proportion is 50% and shrinks toward the extremes. A poll measuring a 10% or 90% proportion has a smaller margin of error than one measuring a 50% proportion, all else equal.
When you do not know the true proportion in advance (which is almost always the case in exploratory surveys), use 50% as your proportion to get the most conservative (widest) margin of error. This ensures you do not underestimate uncertainty.
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Sample size and diminishing returns
Increasing sample size reduces margin of error, but with diminishing returns. Going from n=100 to n=400 cuts the margin of error in half. Going from n=400 to n=1600 cuts it in half again. But going from n=1,000 to n=10,000 only reduces it by about 68% — a tenfold increase in cost for a modest improvement in precision.
For most opinion polling, sample sizes between 400 and 1,500 offer a practical balance between cost and precision. A well-designed random sample of 1,000 gives a margin of error of about ±3% at 95% confidence.
Tips & Insights
Assume 50% if proportion is unknown
Using p=50% gives the largest possible margin of error for a given sample size, making it the safest conservative estimate for planning surveys.
Random sampling is required
The margin of error formula assumes a simple random sample. Convenience samples, opt-in surveys, and self-selected polls do not have a statistically valid margin of error regardless of their size.
Reported MoE is often for 95% confidence
News reports almost always state margin of error at the 95% confidence level unless they specify otherwise. When comparing polls, confirm they use the same confidence level.
Worked Examples
National opinion poll
Margin of error: ±3.10%. The poll result is accurate to within about 3 percentage points at 95% confidence — typical for national polls.
Small business customer survey
Margin of error: ±7.33%. With only 150 respondents, there is substantial uncertainty — results could be off by more than 7 percentage points.
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Frequently Asked Questions
What is margin of error?
Margin of error is the range within which the true population value is likely to fall. It quantifies the uncertainty in a sample-based estimate at a given confidence level.
How do I reduce margin of error?
Increase your sample size. The margin of error is inversely proportional to the square root of the sample size, so quadrupling n halves the margin of error.
Why is 50% the worst case for margin of error?
The formula includes p(1−p), which is maximized at p=0.5. Values closer to 0 or 1 produce a smaller product, resulting in a smaller margin of error.
Does margin of error account for all sources of error?
No. Margin of error only reflects sampling variability. It does not account for response bias, question wording effects, nonresponse bias, or other systematic errors in survey design.
What margin of error is acceptable?
It depends on the purpose. Political polls typically accept ±3-5%. Medical studies may require ±1-2%. Business surveys often tolerate ±5-10%. Acceptable precision depends on the stakes of the decision.
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