What it is and when to use it
The margin of error tells you how far a survey or poll result could plausibly sit from the true value for the whole population, purely because you asked a sample instead of everyone. If a poll of 1,000 people says 52% support a proposal and the margin of error is plus or minus 3 points, the honest reading is that support in the full population is probably somewhere between 49% and 55%, not exactly 52%.
Use this calculator when you have a sample size and want to know how precise a proportion-based result is at 90%, 95% and 99% confidence, or when you are judging whether a reported poll gap is bigger than sampling noise. It only needs the sample size because it assumes the most cautious 50/50 response split, which gives the widest possible margin for that sample. It covers random sampling error only, not bias from poor question wording or who chose to respond.
The formula and its variables
For a proportion, the margin of error is MOE = z × √( p(1 − p) / n ), then multiplied by 100 to express it in percentage points.
- n: the number of completed responses in your sample (the calculator rounds to a whole number).
- p: the expected share answering a given way. The calculator fixes p at 0.5 because p(1 − p) peaks there, so the result is a worst-case margin.
- z: the critical value for your confidence level: 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%.
Because n sits under a square root, precision improves slowly. Quadrupling the sample size only halves the margin of error.
Worked example: a poll of 1,000 people
Enter n = 1,000. First compute the standard error at p = 0.5: √(0.25 / 1000) = √0.00025 = 0.015811.
At 95% confidence: 1.96 × 0.015811 = 0.03099, which the calculator shows as ±3.10%. At 90% it is 1.645 × 0.015811 = ±2.60%, and at 99% it is 2.576 × 0.015811 = ±4.07%.
To go the other way, the sample needed for ±3% at 95% is n = 1.96² × 0.25 / 0.03² = 1,067.1, so you would round up to 1,068 responses.
Common mistakes and how to interpret the result
- Treating the margin as a guarantee. At 95% confidence, about 1 poll in 20 will miss by more than the stated margin even when everything is done correctly.
- Applying the margin to a difference between two groups. The margin for the gap between two candidates, or between two subgroups, is larger than the margin for a single percentage, and subgroups have smaller n than the whole sample.
- Ignoring non-sampling error. Low response rates, leading questions and unrepresentative panels can shift results by much more than ±3 points, and the formula cannot see any of that.
- Using the formula for tiny populations. If the sample is a large fraction of a small population (over roughly 5% of it), a finite population correction makes the true margin smaller than shown.