Margin of Error Calculator
Find the margin of error of a survey proportion.
Results update as you type.
About this calculator
The margin of error tells you how far a survey result is likely to be from the true population value — the "plus or minus" figure quoted next to poll results. For a proportion it is z*·√(p(1 − p) / n), where p is the sample proportion, n the sample size and z* the critical value for your confidence level (1.96 for 95%).
It reports the margin, the confidence-interval bounds around your proportion and the full interval width, and a table shows how the margin shrinks as the sample grows. The default 50% proportion gives the most conservative (largest) margin, because the quantity p(1 − p) peaks at p = 0.5, which is why pollsters often quote it.
Worked example: with 1,000 respondents split 50/50 at 95% confidence, the margin is 1.96 × √(0.5 × 0.5 ÷ 1000) ≈ 1.96 × 0.0158 ≈ 3.1 percentage points. So a poll result of 50% really means somewhere between about 46.9% and 53.1%. To halve that margin you would need roughly four times the sample.
Frequently asked questions
Why does a bigger sample lower the margin of error?
The margin shrinks with the square root of the sample size, so quadrupling n halves the margin of error. This diminishing return is why very large samples are needed to push the margin below one or two percentage points.
Why is 50% the default proportion?
The quantity p(1 − p) is largest at p = 0.5, so assuming a 50/50 split produces the widest, most conservative margin of error. Using it guarantees the true margin is no larger than reported.
How do I calculate the margin of error?
Multiply the critical value z* for your confidence level by the square root of p(1 − p) ÷ n. For p = 0.5, n = 1000 and 95% confidence, that is 1.96 × √(0.25 ÷ 1000) ≈ 3.1 percentage points.
What is the difference between the margin of error and the confidence interval?
The margin of error is the half-width; the confidence interval is the full range you get by adding and subtracting the margin from your result. A 48% result with a 3-point margin gives a confidence interval of 45% to 51%.
Does a bigger population require a bigger margin of error?
No. Beyond a few thousand people the population size barely affects the margin — what matters is the sample size. That is why a national poll and a city poll can share the same margin with the same number of respondents.
How does the confidence level change the margin of error?
A higher confidence level uses a larger z*, which widens the margin. Moving from 95% (z* = 1.96) to 99% (z* = 2.576) increases the margin by about 31% for the same sample.
API — use this calculator from code
Call this calculator as a free JSON endpoint — no key required. Send the field values below as query parameters or JSON. Anything you omit uses the same default this page is pre-filled with; an unknown parameter is a 400, never a silent zero. Read the full API docs →
Endpoint
GET https://calculator.free/api/v1/margin-of-error/curl
curl "https://calculator.free/api/v1/margin-of-error/?n=1000&p=50&conf=1.96"JavaScript fetch()
const r = await fetch(
"https://calculator.free/api/v1/margin-of-error/?" + new URLSearchParams({
"n": "1000",
"p": "50",
"conf": "1.96"
}));
const data = await r.json();
console.log(data.results);Results are estimates for general guidance only, not financial, medical or tax advice.