Margin of Error Calculator
Enter sample size, proportion, and confidence level to find the margin of error and confidence interval for survey results — with finite population correction and interpretation of your survey's precision.
Enter your values above to see the results.
Tips & Notes
- ✓Use p=0.50 (50%) when you do not know the expected proportion — this gives the maximum (most conservative) margin of error. Any other p produces a smaller ME, so p=0.50 is the safe default.
- ✓To halve the margin of error, you must quadruple the sample size. ME ∝ 1/√n. Cutting ±4% to ±2% requires going from n=600 to n=2,400 — four times as many respondents.
- ✓Apply finite population correction (FPC) when your sample is more than 5% of the total population. Surveying 200 from 500 people (40%) significantly overstates the error without FPC.
- ✓The margin of error reported in polls typically assumes 95% confidence and p=50%. If the actual proportion is far from 50% (say, 20%), the true ME is smaller than the reported one.
- ✓Margin of error only accounts for random sampling error — not response bias, question wording, nonresponse, or coverage errors. A biased survey with small ME can still be very wrong.
Common Mistakes
- ✗Thinking margin of error accounts for all types of error. ME measures only random sampling variability. It does not account for biased questions, self-selection, nonresponse, or poor sample frames — which can cause errors far larger than the statistical ME.
- ✗Using p=0 or p=1 instead of p=0.5 for unknown proportions. When the true proportion is unknown, p=0.5 maximizes ME and gives the most conservative (safest) estimate. Using the observed proportion only works if the sample is very large.
- ✗Forgetting to apply finite population correction for small populations. Surveying 300 people from a company of 500 samples 60% of the population — standard ME formulas overstate uncertainty by ignoring the FPC adjustment.
- ✗Interpreting margin of error as a hard boundary. ME=±3% means the true value is likely (not certainly) within 3 percentage points. There is still a 5% chance (at 95% confidence) the true value lies outside the interval.
- ✗Assuming that a smaller margin of error means the survey is more accurate. A survey with ME=±1% that uses a biased sample is far less accurate than one with ME=±3% using a representative random sample.
Margin of Error Calculator Overview
The margin of error tells you how close a survey result is likely to be to the true population value. When a poll reports "52% support the candidate, ±3%", the ±3% is the margin of error — the true value likely falls between 49% and 55%. The margin of error quantifies the inherent imprecision of sampling: you surveyed some people, not everyone, so your estimate has uncertainty. Understanding margin of error separates meaningful findings from statistical noise.
Margin of error for a proportion:
ME = z* × √[p(1−p) / n]
EX: 1,000 people surveyed, 52% support a candidate, 95% confidence → ME = 1.96 × √(0.52×0.48/1000) = 1.96 × 0.01580 = ±3.1% → CI: [48.9%, 55.1%]Conservative margin of error (maximum, when p is unknown):
ME_max = z* × √(0.25 / n) = z* / (2√n)
EX: n=1,000, 95% confidence → ME_max = 1.96 / (2×√1000) = 1.96/63.25 = ±3.1% — maximum ME regardless of p, occurs when p=50%Finite population correction (when sampling >5% of population):
ME_adjusted = ME × √[(N−n) / (N−1)]
EX: n=200 from N=1,000 population (20% sampled) → FPC = √(800/999) = 0.894 → ME_adjusted = 0.894 × original ME → 10.6% smaller marginSample size vs. margin of error at 95% confidence:
| Sample Size | Margin of Error (p=50%) | Typical Use |
|---|---|---|
| 100 | ±9.8% | Preliminary/pilot surveys |
| 400 | ±4.9% | Regional polls |
| 1,000 | ±3.1% | National polls (standard) |
| 1,600 | ±2.5% | High-precision surveys |
| 2,500 | ±2.0% | Medical/policy research |
Frequently Asked Questions
Margin of error (ME) is the maximum likely difference between a survey result and the true population value. Poll result: 52% support ±3% ME at 95% confidence → the true support is likely between 49% and 55%. If the interval [49%, 55%] does not include 50%, you can say support is likely above 50%. The ME assumes random sampling — biased samples can be wrong by far more than the stated ME.
With n=1,000, the margin of error at 95% confidence is ME = 1.96/√1000 × 0.5 ≈ ±3.1%. This precision is sufficient for most political, social, and market research. Doubling to n=2,000 only improves precision to ±2.2% — modest gain for double the cost. Quadrupling to n=4,000 gives ±1.55%. The diminishing returns of larger samples make 1,000–1,500 the standard cost-effective choice.
Higher confidence requires a larger critical value z*, which widens the margin of error. At 95% confidence (z*=1.96) with n=1,000: ME=±3.1%. At 99% confidence (z*=2.576): ME=±4.1%. At 90% confidence (z*=1.645): ME=±2.6%. The trade-off: 99% confidence is much harder to achieve falsely (fewer false positives), but the wider interval is less precise about where the true value lies.
The FPC = √[(N−n)/(N−1)] reduces ME when you survey a substantial fraction of the population. If you survey 200 of 500 employees (40% of the population), the standard ME formula overestimates error. FPC ≈ √(300/499) ≈ 0.775 → ME reduces by 22.5%. Apply FPC when n/N > 5% (more than 5% of population sampled). For large populations (N > 100,000), FPC is negligible and can be ignored.
The formula ME = z*×√[p(1−p)/n] is maximized when p=0.5, since p(1−p) = 0.25 is the maximum (at p=0.5 it equals 0.25; at p=0.1 it equals 0.09; at p=0.9 it equals 0.09). Using p=0.5 when the true proportion is unknown guarantees the computed ME is at least as large as the true ME for any p — it is the conservative, safe choice that cannot understate the uncertainty.
No. ME applies specifically to proportions from random probability samples. It does not apply to: convenience samples (self-selected respondents), qualitative questions, multi-item scales, or averages (which use a different formula). A poll that lets people opt in to answer online has no valid margin of error — the sample is not random, so there is no basis for computing a statistical ME.