Margin of Error Calculator
Find the ± margin of error for a survey proportion at your chosen confidence level.
95% confidence interval: 46.9% to 53.1% (50% ± 3.1%).
The margin of error for a proportion is MOE = z × √(p(1 − p) ÷ n). For a poll of n = 1000 with p = 0.5 at 95% confidence (z = 1.96), MOE = 1.96 × √(0.25 ÷ 1000) = 1.96 × 0.0158 = ±3.1%, so the result is 50% ± 3.1%.
What the margin of error measures
The margin of error is the ± cushion around a survey percentage that accounts for random sampling. Because you polled a sample rather than everyone, the true population figure is likely — but not certain — to sit within that band. A reported “52% ± 3%” means the plausible range is roughly 49% to 55% at the stated confidence level. It captures sampling error only; it does not fix a biased sample, leading questions, or non-response.
critical z times the standard error of the proportion
Worked example
A poll of 1000 people where 50% pick option A, reported at 95% confidence.
- 1 Pick the critical z for your confidence level. At 95% confidence, z = 1.96 (the two-tailed normal critical value).
- 2 Write the proportion as a decimal. p = 50% ÷ 100 = 0.5, so p(1 − p) = 0.5 × 0.5 = 0.25.
- 3 Divide by the sample size and take the root. √(0.25 ÷ 1000) = √0.00025 = 0.0158 — the standard error.
- 4 Multiply by z and convert to a percentage. 1.96 × 0.0158 = 0.031 = ±3.1%, giving an interval of 46.9% to 53.1%.
Critical z-values by confidence level
The two-tailed normal critical value used in the margin of error.
| Confidence | Critical z |
|---|---|
| 80% | 1.282 |
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
Margin of error at 95% confidence (p = 0.5)
How the ± margin shrinks as the sample grows — with diminishing returns.
| Sample size (n) | Margin of error |
|---|---|
| 100 | ±9.8% |
| 400 | ±4.9% |
| 1000 | ±3.1% |
| 2000 | ±2.2% |
Reading the result and its limits
Bigger samples help, but with diminishing returns. Because the margin scales with 1 ÷ √n, quadrupling the sample only halves the margin — going from n = 400 to n = 1600 takes you from ±4.9% to ±2.5%, not to zero. Doubling from 1000 to 2000 barely moves it from ±3.1% to ±2.2%.
p = 0.5 is the worst case. The quantity p(1 − p) peaks at 0.25 when p = 0.5, so a 50/50 split produces the widest margin. Lopsided results (say p = 0.1) have a smaller margin, which is why 0.5 is the safe, conservative default when you do not yet know the proportion.
Finite populations. When your sample is a large fraction of a small population, you can multiply the margin by the finite population correction √((N − n) ÷ (N − 1)), which shrinks it. For big populations relative to the sample the correction is near 1 and can be ignored.
Planning ahead. To find the sample size that hits a target margin before you collect data, rearrange the formula or use the sample size calculator.