Statistics
Calculate margin of error for population mean or proportion
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Margin of Error
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Z-Score
1.96
How it works
Quantifies the sampling uncertainty around a survey result, giving the plus-or-minus band usually reported alongside a poll. It captures random sampling error only, not the many other ways a survey can be wrong.
Formula
MOE = z × √( p(1 − p) / n )
Variables
zcritical value for the confidence level (1.96 for 95%)pobserved sample proportionnsample size
Worked example
Inputs: 52% support in a sample of 1,000, at 95% confidence
- p(1 − p) = 0.52 × 0.48 = 0.2496
- 0.2496 / 1,000 = 0.0002496
- √0.0002496 = 0.015799
- 1.96 × 0.015799 = 0.03097
Result: ±3.1 points, so the true value plausibly lies between 48.9% and 55.1%
Notes
- A 52-48 result with a ±3.1 point margin is a statistical tie, because both figures fall inside each other range.
- The margin on the gap between two candidates is roughly double the margin on either one, so a lead needs to exceed about twice the quoted figure to be meaningful.
- Quoted margins almost always assume p = 0.5, which is the maximum. At p = 0.10 the same 1,000-person sample has a margin of only ±1.9 points.
- This measures random sampling error alone. Question wording, coverage gaps, and non-response can each shift a result by more than the stated margin and are not captured at all.
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Frequently Asked Questions
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