Updated June 17, 2026
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The margin of error is calculated as the critical z-value multiplied by the standard error of the proportion: MOE = z * sqrt(p * (1 - p) / n). For a 95% confidence level, z = 1.96. A smaller sample produces a larger margin of error.
The formula
MOE = z * sqrt( p * (1 - p) / n )
Where:
- z is the critical value for your confidence level (1.96 for 95%, 2.576 for 99%)
- p is the estimated proportion (use 0.5 if unknown, which gives the widest, most conservative margin)
- n is the sample size
Worked example: finding the margin of error for 95% confidence
You survey 400 people; 52% say yes. What is the margin of error at 95% confidence?
| Step | Calculation | Value |
|---|---|---|
| Identify z for 95% | Standard table | 1.96 |
| p = 0.52, n = 400 | p*(1-p) = 0.52 * 0.48 | 0.2496 |
| Divide by n | 0.2496 / 400 | 0.000624 |
| Take square root | sqrt(0.000624) | 0.02498 |
| Multiply by z | 1.96 * 0.02498 | 0.049 (4.9%) |
The margin of error is plus or minus 4.9 percentage points. The true population proportion is likely between 47.1% and 56.9%.
What does a 2% margin of error mean?
A 2% margin of error means the survey result could be up to 2 percentage points above or below the true value. If 55% of respondents favor a policy, the actual figure in the full population is likely between 53% and 57% at whatever confidence level you chose.
What does 0.05 margin of error mean?
A margin of error of 0.05 (5 percentage points) is common in general-population surveys. To achieve a 5% margin at 95% confidence with p = 0.5, you need n = (1.96^2 * 0.25) / 0.05^2 = 384 people. Achieving a tighter margin requires a larger sample.
What is a good margin of error?
For national opinion polls, plus or minus 3% is standard. For internal business surveys, plus or minus 5% is usually acceptable. Scientific studies targeting real-world decisions often aim for 1% to 2%. The tighter the margin you need, the larger the sample you must collect. Cross-link your result with the confidence interval to understand the full range.
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