Margin of error calculator
How precise your survey is, given the sample you actually collected.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- How precise your survey is, given the sample you actually collected.
- Formula used
e = z · √( p(1−p) / n )- Example
- Sample of 400 at 95% confidence and 50% proportion:
Margin of error calculator
This is the inverse of the sample size calculation: you already have the responses and want to know how precisely you can talk about them.
It is the “±3.1 points” you see at the foot of any published poll.
How it works
The standard error of the proportion is multiplied by the z value for your confidence level. If you supply the population, the finite population correction is applied as well.
Formula
e = z · √( p(1−p) / n )
With a finite population:
e = e · √((N − n) / (N − 1))
Worked example
Sample of 400 at 95% confidence and 50% proportion:
Standard error: √(0.25/400) = 0.025
e = 1.96 × 0.025 = 4.9%
The result reads as 50% ± 4.9.
Explanation
Why a thousand respondents is enough for a whole country
It is the most counter-intuitive result in statistics: the margin of error depends on the size of the sample, not of the population. A thousand people give roughly the same margin for a city of a hundred thousand as for a country of fifty million. The reason is that what reduces uncertainty is the amount of information collected, not the proportion it represents.
Quadruple to halve
The margin falls with the square root of sample size. To go from three per cent to one and a half you must multiply the number of interviews by four, not two. That is what makes sharpening polls beyond a certain point so expensive.
The error nobody measures
The margin of error covers only sampling randomness. It says nothing about selection bias, badly worded questions, people who refuse to answer or people who lie. In practice those errors usually exceed the published margin, and they appear in no figure at all.
Statistical ties
When two options are separated by less than twice the margin, the difference is not conclusive. Reporting that one leads by two points with a margin of three is reading noise as signal.
Frequently asked questions
Why does the margin not depend on population size?
Because precision is driven by how many cases you observed, not by how big the universe is, unless your sample is a large fraction of it.
Does it apply to each question separately?
Yes. The margin depends on the observed proportion, so a 90% answer carries a smaller margin than a 50% one.
Do I need more interviews if the population is huge?
Practically no. Beyond a certain size the margin depends only on the sample: a thousand people give a similar margin for a city or a country.
What is a statistical tie?
When the difference between two options is smaller than the combined margin of error, so it cannot be said which is ahead.
Need to calculate something else?
These tools are often used alongside this calculator.
Sample size
How many responses you need to estimate a percentage or a mean, with finite population, design effect and…
Confidence interval
The range the true mean is likely to fall in at your chosen confidence level.
Z score
How many standard deviations a value sits from the mean and which percentile it occupies.
Standard deviation
Sample and population deviation, variance and coefficient of variation.
Pearson correlation
Pearson’s r between two variables, with p value, confidence interval, regression line and scatter plot.
t-test
Compare two means (independent or paired) or one mean with a value, with p value, confidence interval and…