Margin of Error Calculator
How precise is that survey percentage?
Margin of Error Calculator
MOE = z × √(p(1 − p) / n), reported in percentage points
Calculated instantly
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What This Calculator Does
The Margin of Error Calculator turns a survey sample size and proportion into a plus-or-minus range at your chosen confidence level. It reports the margin of error in percentage points and the lower and upper bounds of the confidence interval.
How It Works
For a proportion, the standard error is the square root of p(1 − p) divided by n. Multiply that by the critical value for your confidence level to get the margin of error. The interval is the sample proportion plus or minus that margin. This is the same arithmetic behind the “±3 points” line in a published poll.
Example
A poll of 400 people finds 50% support. At 95% confidence the margin of error is 1.96 × √(0.5 × 0.5 / 400) = 4.9 percentage points, so the interval runs from 45.1% to 54.9%. Halving the margin to about 2.5 points would need roughly 1,600 respondents.
Formula
MOE = z × √(p(1 − p) / n)Multiply by 100 to express the result in percentage points.
CI = p ± MOEn = z² × p(1 − p) / MOE²Reading the Results
Margin of error
The plus-or-minus figure in percentage points. It covers random sampling error at the stated confidence level, not bias from who answered or how the question was asked.
Lower and upper bounds
The ends of the confidence interval around your sample proportion. For a 95% interval, about 95% of samples drawn the same way would produce an interval containing the true value.
Sources and References
NIST/SEMATECH e-Handbook of Statistical Methods
Standard error of a proportion
The handbook gives the large-sample normal approximation used here.
American Association for Public Opinion Research — margin of error
MOE describes sampling error only
AAPOR stresses that margin of error excludes non-response, question wording, and coverage error.
References identify the method or data behind this calculator. Where a standard or tariff changes, the current edition applies, and the assumptions stated above take precedence over a remembered figure.
Frequently Asked Questions
What does a 3% margin of error mean?
If the poll reports 50% with a ±3 point margin at 95% confidence, the true share is plausibly between 47% and 53%. It does not mean the poll is 3% wrong, and it says nothing about systematic bias.
Does margin of error shrink with population size?
Barely. It depends mainly on sample size. A poll of 1,000 people has roughly the same margin whether the population is a city or a country, which is why national polls can be that small.
Why is the margin widest at 50%?
The term p(1 − p) peaks at p = 0.5. A race that is close to a tie therefore has the largest margin of error; a 90/10 result has a narrower one with the same sample.
What is not included in the margin of error?
Non-response bias, question wording, order effects, and coverage gaps. Those are usually larger than sampling error in real surveys, which is why a small margin does not guarantee an accurate poll.