What the margin of error means
When a poll says a candidate has 52% ± 3 points, the ± part is the margin of error. It measures how much the result could differ from the true value in the whole population just because only a sample was asked. At the usual 95% confidence level, the true value is expected to fall within the margin in 95 polls out of 100 conducted the same way.
The formulas
- Standard error of a proportion: SE = √(p × (1 − p) ÷ n)
- Margin of error: E = z* × SE, with z* = 1.96 at 95%
- Finite population correction when the sample is a large share of the population: multiply by √((N − n) ÷ (N − 1))
The margin is largest when p = 50%. Pollsters often publish this maximum margin for the whole survey.
Worked example
A national poll of 1,000 likely voters finds 52% support for a ballot measure.
- SE = √(0.52 × 0.48 ÷ 1,000) = 1.58 points.
- Margin of error at 95%: 1.96 × 1.58 = ±3.1 points.
- The true support is likely between 48.9% and 55.1%.
- Since the interval includes 50%, the poll cannot say with 95% confidence that the measure has majority support.
Margin of error by sample size
At 95% confidence and p = 50%:
| Sample size | Margin of error |
|---|---|
| 100 | ±9.8 points |
| 400 | ±4.9 points |
| 600 | ±4.0 points |
| 1,000 | ±3.1 points |
| 1,500 | ±2.5 points |
| 2,000 | ±2.2 points |
| 5,000 | ±1.4 points |
Precision improves with the square root of the sample: to halve the margin, you need four times as many respondents. That is why most national polls stop around 1,000 to 1,500 interviews.
Reading a horse race correctly
- The lead has a bigger margin. In a two-way race, the margin of error on the gap between the candidates is about twice the margin on each number. A 52 to 48 lead in a poll of 1,000 is not statistically clear.
- Subgroups are much less precise. If only 200 respondents are under 30, their result has a margin of about ±7 points.
- Averages beat single polls. Combining several polls reduces random error, which is why poll aggregators track averages.
What the margin does not cover
The margin of error only measures random sampling error. It says nothing about a sample that does not represent the population, people who refuse to answer, leading questions or voters who change their minds. In practice, total survey error is often larger than the published margin. Opt-in online panels do not have a true margin of error at all; reputable pollsters report a “credibility interval” or explain their weighting instead.