Margin of Error Calculator
Calculate sampling error margins, z-critical scores, and confidence interval bounds for survey polling, academic research, and quality inspection.
| Desired MOE | 90% Confidence | 95% Confidence | 99% Confidence |
|---|---|---|---|
| ±1.0% | 6,765 | 9,604 | 16,587 |
| ±2.0% | 1,691 | 2,401 | 4,147 |
| ±3.0% | 752 | 1,067 | 1,843 |
| ±5.0% | 271 | 384 | 664 |
How Margin of Error is Calculated
The margin of error measures the range within which the true population parameter is expected to fall with a specified degree of statistical confidence. The mathematical formula for sample proportion surveys is:
For deeper analysis and related planning, you can also explore our Area Calculator and Average Calculator.
$$\text{MOE} = z^* \times \sqrt{\frac{p(1 - p)}{n}} \times \text{FPC}$$
Where:
- $z^*$ is the critical value from the standard normal distribution corresponding to the chosen confidence level ($\alpha = 1 - \text{Confidence}$).
- $p$ is the estimated proportion of the attribute in the sample (expressed as a decimal from 0 to 1). When unknown, $p = 0.5$ is assumed for maximum variance.
- $n$ is the total sample size (number of respondents).
- $\text{FPC} = \sqrt{\frac{N - n}{N - 1}}$ is the Finite Population Correction factor, applied when sampling without replacement from a known population of size $N$.
Frequently Asked Questions
What is margin of error in statistics?
Margin of error expresses the maximum expected difference between the results found in a survey sample and the true values of the entire population, at a specified confidence level.
Why is 50% commonly used as the sample proportion?
Using a proportion of p = 0.50 maximizes the product p(1-p) = 0.25, generating the most conservative (largest) margin of error estimate when the actual population split is unknown.
When should finite population correction (FPC) be applied?
FPC should be applied when the sample size represents more than 5% of the total population (n/N > 0.05). For large populations where n/N is very small, FPC approaches 1.0 and has negligible effect.
What is the difference between a 95% and 99% confidence level?
A 95% confidence level means that if the survey were repeated 100 times under identical methodology, 95 of those samples would capture the true population value within the margin of error. Increasing confidence to 99% widens the interval (increases the margin of error) unless you increase sample size.
How does sample size affect the margin of error?
Because sample size $n$ appears in the denominator under a square root, quadrupling your sample size cuts the margin of error in half. For example, moving from 100 to 400 respondents halves the margin of error from approximately ±9.8% to ±4.9% at 95% confidence.