Descriptive statistics
Margin of Error Calculator
The ± figure quoted beside a poll — how far the estimate could plausibly be from the truth.
How to read this result
The margin of error is half the width of a confidence interval. "48% ± 3 points" means the interval runs from 45% to 51%. Everything true of confidence intervals is true of it.
It covers sampling error and nothing else. A biased sampling frame, people who refuse to answer, a leading question and a mistranslated option are all excluded from this number and are usually larger sources of error than the ± it quotes.
p = 0.5 is the conservative assumption. The margin depends on p(1 − p), which is largest at 0.5. Using it when the true proportion is unknown guarantees the margin is not understated.
Population size barely matters. Sampling 1,000 people from a city gives almost exactly the same precision as 1,000 from a country. The finite population correction only bites when the sample is a sizeable fraction — around 5% or more — of the whole population.
The formula
- 1.96 for 95% confidence
- the expected proportion; 0.5 is the conservative choice
- the sample size and, optionally, the population size
The second square root is the finite population correction, and equals 1 when the population is large relative to the sample.
Worked example
Why national polls use about a thousand people
A poll of 1,067 people reports 48% support. What is the margin of error, and why is 1,067 such a common sample size?
- Use the conservative p = 0.5, which maximises p(1 − p).0.5 × 0.5 = 0.25
- Divide by n and take the square root — the standard error.√(0.25 / 1067) = 0.015307
- Multiply by the 95% z multiplier.1.95996 × 0.015307 = 0.030001
- As percentage points.±3.0 points
1,067 is where the margin hits exactly ±3 points, which is why so many polls land near a thousand respondents. Getting to ±1.5 points would take four times as many people for twice the precision.
Checked against 1.959964 × √(0.25/1067), the standard survey formula.
The calculator above is loaded with these numbers by the Load the worked example button.
Assumptions, and when to use something else
A random sample, independent responses, and — for the proportion version — enough successes and failures for the normal approximation. The margin of error is a statement about sampling variability under those conditions and nothing else.
The assumptions that break in practice are not statistical. Non-response is the big one: if the people who answer differ systematically from those who do not, no sample size fixes it, and the quoted margin becomes an understatement of the real uncertainty.
- You want the interval, not just the marginConfidence interval calculatorThe full interval, with four methods for proportions.
- You are planning the surveySample size calculatorWorks backwards from the margin you want to the n you need.
- You are comparing two survey resultsz-test for proportionsTwo margins overlapping does not mean the difference is insignificant — test it properly.
Questions people ask
What sample size do I need for a ±3% margin of error?
About 1,067 for a large population at 95% confidence, using the conservative p = 0.5. For ±5% you need about 385; for ±1% about 9,604. The relationship is 1/e², so tripling the precision costs nine times the sample.
Does the population size matter?
Hardly ever. Sampling 1,000 from a town of 50,000 gives a margin only 1% narrower than 1,000 from a country of 50 million. The correction matters once the sample is more than about 5% of the population.
Two poll results differ by more than the margin of error. Is the difference significant?
Not necessarily, and this is the classic error. The margin of error applies to each estimate separately; the difference between two estimates has its own, larger, uncertainty. Use the z-test for proportions rather than comparing error bars by eye.
Does the margin of error include non-response bias?
No. It covers random sampling variation only. Response rates below 10% are common in modern polling, and the bias that introduces is typically larger than the ±3 points quoted — which is why polls miss by more than their stated margin more often than chance alone would explain.