StatReason

Margin of Error Calculator

Given a sample size, find the margin of error for estimating a proportion, or check how precise a poll or survey result actually is.

Use .5 (the default) if you have no prior estimate; it gives the widest, most conservative margin.

What this answers

This calculator answers "given my sample size, how precise is my estimate of a proportion?" It is the plus-or-minus figure commonly quoted alongside poll results (for example, "plus or minus 3 percentage points"), computed directly from your sample size and confidence level rather than from an observed count.

How it is calculated

The margin of error is the z critical value for your confidence level, multiplied by the standard error of a proportion, which itself depends on the assumed proportion p and the sample size n. Using p equal to .5 gives the largest possible margin of error for a given n, the standard conservative assumption used when you do not already know roughly what the true proportion is. An optional finite-population correction narrows the margin further when you are sampling a meaningful fraction of a known, bounded population rather than an effectively infinite one.

Worked example

For a sample size of 100, the conservative assumption p equals .5, and 95% confidence: the margin of error works out to almost exactly 9.8 percentage points, a well-known reference figure for this specific combination. Larger samples shrink this margin, but only as the square root of the sample size, so quadrupling your sample size only halves your margin of error, not eliminates it.

Assumption audit

Calculated from your data: the exact margin of error given your stated sample size, assumed proportion, and confidence level.
Evidence to review: whether your assumed proportion (.5 by default) is actually a reasonable estimate for your situation. A markedly different true proportion would give a smaller true margin of error than this conservative calculation shows.
You must verify: that your sample was drawn through simple random sampling. This margin of error omits non-sampling error entirely: it says nothing about nonresponse bias, question wording effects, or a non-representative sampling frame.

What this result does not mean

A margin of error describes sampling precision alone. It does not account for non-sampling error such as a biased sample, leading survey questions, or systematic nonresponse, all of which can make an actual estimate far less trustworthy than its stated margin of error implies.

Limitations

This margin of error applies specifically to a single proportion estimated from a simple random sample. It does not directly apply to more complex survey designs (stratified or clustered samples) without further adjustment.