A poll result of 42 percent sounds precise until you are told the margin of error is plus or minus 4 points. This margin of error calculator works out the uncertainty band around a mean or a proportion, at the confidence level you choose, and also tells you the sample size you would need to reach a target margin in the first place.
What Is the Margin of Error Calculator?
The margin of error is the plus or minus band that should be attached to a survey result so that readers understand the precision of the estimate. It is not a statement about the chance that any particular answer is wrong, it is a statement about how far the true value is likely to sit from the estimate.
This calculator handles the two standard cases: estimating a mean, where the sample standard deviation drives the uncertainty, and estimating a proportion, where the observed percentage does.
How the Formulas Work
For a mean: margin of error = z × s ÷ √n
For a proportion: margin of error = z × √(p(1 − p) ÷ n)
The critical value z is 1.645 for 90 percent confidence, 1.96 for 95 percent and 2.576 for 99 percent. For a proportion the variance is largest when p is 0.5, which is why a poll showing 50 percent always has a wider margin than one showing 5 percent on the same sample size.
Sample Size and Precision
Solving the same formulas backwards gives the sample size needed for a target margin. For a proportion the requirement is n = p(1 minus p) divided by the square of the desired margin, and for a mean it is n = (z times s divided by the margin) squared.
The relationship is always the square root, so precision is expensive. Going from a margin of 5 points to 2 points needs more than six times as many responses, and going from 2 points to 1 point needs four times as many again.
Reporting Results Honestly
The standard errors above assume simple random sampling. If the sample is stratified, clustered or quota-based, the true error is usually larger, because the design breaks the independence assumption. In those cases the published figure needs a design effect applied on top.
It also assumes there is no non-response bias, which is usually the largest source of error in modern polling and is not captured by any margin of error calculation.
Frequently Asked Questions
What is a good margin of error for a poll?
Most published polling uses a 95 percent confidence level with a margin of error of plus or minus 2 to 4 percentage points. A margin wider than 5 points is usually disclosed clearly, and some reputable pollsters now publish the full interval rather than a single figure.
Why does a larger sample reduce the margin of error?
Because the standard error falls with the square root of the sample size, not linearly. Quadrupling the sample halves the margin of error, which is why precision gains get expensive quickly.
Does a 99 percent confidence level give twice the margin of 95 percent?
Not quite. The critical value goes from 1.96 to 2.576, so the margin is about 31 percent wider. That extra width buys a lot less additional certainty than the doubling of the confidence percentage suggests.
What sample size do I need for a margin of 5 percent?
For a proportion near 50 percent at 95 percent confidence, 385 responses give a margin of about 5 points, and about 1,000 gives roughly 3 points. The calculator returns the exact figure for the percentage you enter.
Does the margin of error account for non-response bias?
No. It measures sampling error only. If the people who did not respond differ systematically from those who did, the estimate can be badly biased and no margin of error will reveal that.