When a pollster says "48% of voters support Candidate A, with a margin of error of ±3%", what does that actually mean? It means that if you repeated the survey hundreds of times with the same sample size, the true share of support in the whole population would fall inside roughly 45% to 51% about 95% of the time. This margin of error calculator turns three numbers — your sample size, the proportion you observed, and your chosen confidence level — into that ± range and the full confidence interval, so you can read poll headlines, A/B test results, and survey findings with a clear head.
The math is the standard proportion confidence interval taught in every introductory statistics course: MOE = z × √(p(1−p)/n). Slide the inputs and the result card updates instantly.
How to Use
- Enter the Sample Size (n) — the number of people who actually answered the question. A national poll of 1,000 likely voters is the classic benchmark.
- Enter the Observed Proportion (p) as a percentage. If 48% picked "Yes", type 48. If you do not yet know the result, leave it at 50%: that is the worst-case (widest) margin and the figure pollsters plan around.
- Choose the Confidence Level. 95% is the industry default for polling and A/B tests. Pick 90% for a tighter, less conservative interval, or 99% when you need extra certainty and can tolerate a wider range.
- Read the Margin of Error (±) on the result card. Add and subtract it from your observed proportion to get the confidence interval: the lower and upper bounds.
A few practical notes. The formula assumes a simple random sample from a large population and that p is not extremely close to 0% or 100%. If your population is small (say, all 500 employees at a company), the margin is actually a little smaller than this calculator reports because of the finite population correction — for typical survey sizes of 1,000+ that correction is negligible.
The Margin of Error Formula
The margin of error for a single proportion is:
MOE = z × √(p × (1 − p) / n)
Where:
- p is the observed proportion (between 0 and 1).
- n is the sample size.
- z is the critical value from the standard normal distribution for your chosen confidence level: 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%.
Worked example on a poll of 1,000 people where 50% support a candidate: p = 0.5, n = 1,000, z = 1.96. The standard error is √(0.5 × 0.5 / 1,000) = √0.00025 = 0.01581, and the margin of error is 1.96 × 0.01581 = 0.0310, or ±3.1%. The 95% confidence interval is therefore 46.9% to 53.1% — which is exactly why a "tie" in a 1,000-person poll (49% to 47%) is genuinely too close to call.
Notice that p = 0.5 always gives the largest possible margin, because p(1−p) peaks at p = 0.5. That is why sample-size planners always assume 50/50 when designing a survey: it is the conservative, worst-case choice.
Why Sample Size Matters
Doubling the sample size does not halve the margin of error — it shrinks it by the square root. Going from n = 500 to n = 2,000 quadruples the respondents but only cuts the margin roughly in half. Going from 1,000 to 10,000 cuts it by about 3.16×. This is the reason every serious national poll stops at around 1,000 respondents: beyond that, each additional respondent buys very little precision.
Conversely, a sample of 100 gives a margin of about ±10% at 95% confidence — wide enough that a 45% result is statistically indistinguishable from a 55% result. That is why small-sample surveys on social media are so often meaningless.
Examples
Example 1 — The classic national poll. A survey of 1,000 likely voters finds Candidate A at 48% and Candidate B at 45%. With a 95% confidence level, the margin of error is ±3.1%. Candidate A's interval is 44.9%–51.1% and Candidate B's is 41.9%–48.1%. Because the intervals overlap heavily, the race is a statistical dead heat — a headline that says "Candidate A Leads by 3" is misleading.
Example 2 — An A/B test on a website. You show a new button to 5,000 visitors and 4.2% click it, versus 3.9% on the old button. With p = 4.2% and n = 5,000 at 95% confidence, the margin is about ±0.6 percentage points. The new button's interval is 3.6%–4.8% and the old one's (if it also had n = 5,000 at 3.9%) is 3.3%–4.5%. The intervals overlap, so you cannot yet declare the new button a winner.
Example 3 — A large-sample census-style survey. A market research firm samples 1,000,000 customers and finds 62% prefer a new packaging design. At 95% confidence the margin is only ±0.1% — a razor-thin interval. That is why huge-sample surveys can reliably detect differences as small as 0.2–0.3%, but most of the time that level of precision is unnecessary.
Example 4 — A tiny internal pulse survey. You ask 80 employees whether they want remote Fridays and 60% say yes. At 95% confidence the margin is about ±10.8%. The true company-wide view could be anywhere from 49% to 71% — a wide range, which is why a single 80-person pulse is only a directional signal.
FAQ
What is a good margin of error for a survey?
For most public-opinion and marketing surveys, a margin of error of ±3% at 95% confidence is the industry standard, and it requires roughly 1,000 respondents when the true proportion is near 50/50. A margin of ±5% is acceptable for exploratory work, and ±10% or wider is only suitable for internal directional checks. Tighter than ±2% usually requires sample sizes above 2,500, which rapidly becomes expensive.
How do I interpret a margin of error in a news poll?
Think of it as a range around the headline number. If a poll says Candidate A is at 52% with a ±3% margin, the true level of support is plausibly anywhere from 49% to 55%. Two candidates are statistically "ahead" only if their intervals do not overlap. A 2-point lead in a poll with a ±3% margin is not a real lead — it is within the noise.
Why is 95% the default confidence level?
The 95% level comes from the normal distribution: about 95% of samples fall within 1.96 standard errors of the true value. It was popularized by Ronald Fisher in the early 1900s and became the convention in polling, science, and A/B testing. You can switch this calculator to 90% for a narrower (but less reliable) interval, or 99% for a wider (but more conservative) one.
Does the margin of error depend on population size?
Almost never, for large populations. A sample of 1,000 gives roughly the same ±3% margin whether the population is 100,000 or 100 million. The population only matters when it is small relative to the sample (say, sampling 300 out of 500 employees), in which case a finite-population correction shaves a little off the margin. This calculator assumes a large population.
How many respondents do I need for a ±5% margin?
Plug in a target margin of 5%, a confidence level of 95%, and p = 50% (the worst case). Rearranging the formula gives n ≈ z² × p(1−p) / MOE² = 1.96² × 0.25 / 0.05² = 384 respondents. That is where the familiar "about 400 people" rule of thumb comes from. For ±3% you need roughly 1,067 respondents.
What happens when the proportion is 0% or 100%?
The standard formula collapses to a zero margin at p = 0% or p = 100%, which is mathematically correct in the continuous formula but misleading in practice — a sample where 0 out of 1,000 people chose an answer does not mean the true population rate is exactly zero. For extreme proportions, statisticians use adjusted intervals like the Wilson score or Clopper–Pearson instead. This calculator clamps the bounds to 0% and 100% so the displayed interval never leaves the plausible range.
Disclaimer
This margin of error calculator is provided for informational purposes only and does not constitute statistical, survey-methodology, legal, or financial advice. Results are based on the normal-approximation proportion interval and assume a simple random sample, a large population, and proportions not extremely close to 0% or 100%. Real polls and A/B tests must also account for sampling design, weighting, non-response bias, and survey-mode effects that this simplified tool does not model. Always consult a qualified statistician or methodologist before making a major decision based on a survey result.