Margin of error calculator
How far a survey result could sit from the real answer, for an ordinary percentage or a Net Promoter Score. It shows the likely range, how big a change between two surveys has to be before it means anything, and the formula behind each number.
A result of 50% from 400 responses means the real figure is likely between 45.1% and 54.9%, at 95% confidence.
Comparing with an earlier survey of the same size? A change needs to be bigger than about 6.9 points before it is more than noise.
Margin = 1.96 × √(p(1 − p) ÷ n), where 1.96 is the z-value for 95% confidence and p = 0.5. Leave the result at 50% for the most cautious margin, which is the one published "±X%" figures usually quote.
SurveyService states the number of responses behind every figure in its AI summaries, so readers can see how much weight each one carries. See a sample report →
What does margin of error mean?
A survey asks some people and uses their answers to estimate what everyone thinks. The margin of error is how far that estimate could be from the truth by chance alone. If 62% of respondents agree and the margin is ±5 points at 95% confidence, the real figure is very likely between 57% and 67%: run the same survey again and again, and about 95 times in 100 the range would contain the true value.
n is the number of responses, p is the result as a fraction, and z is 1.645, 1.96 or 2.576 for 90%, 95% or 99% confidence. If you surveyed a large share of a small group, multiply by √((N − n) ÷ (N − 1)), where N is the group size; the calculator does this when you enter a population.
Why is the margin of error for NPS so much wider?
An NPS is the share of promoters minus the share of detractors, on a scale from −100 to +100. Both shares carry their own uncertainty, and the scale is twice as wide as a percentage, so the same number of responses gives a much less precise score:
| Responses | Percentage result | NPS |
|---|---|---|
| 50 | ±13.9 points | ±22.9 points |
| 100 | ±9.8 points | ±15.9 points |
| 200 | ±6.9 points | ±11.3 points |
| 400 | ±4.9 points | ±8.0 points |
| 1,000 | ±3.1 points | ±5.1 points |
At 95% confidence, a percentage near 50% and an NPS with responses split evenly into thirds. The demo survey behind our sample report is a real example: 144 responses give an NPS of -33 with a margin of ±12.6, so the score is likely somewhere between -45 and -20. Quoting it as exactly -33 suggests more precision than the data has.
Has my score really changed?
When you compare two surveys, both results carry a margin of error, so the change between them carries a larger one. Combine them by squaring each margin, adding them and taking the square root.
Two quarterly NPS surveys of 200 responses each, both split evenly, have a margin of about ±11.3 each. A change needs to be bigger than about 16 points before it is more than noise. A score moving from +10 to +18 between those quarters may well be nothing.
This is a rule of thumb rather than a formal significance test, but it stops the most common mistake in survey reporting: celebrating, or worrying about, a change that chance alone could explain.
What doesn't the margin of error cover?
Only the chance variation from asking a sample. It does not cover:
- Who chose to answer. If unhappy customers ignored the survey, the result leans positive however many responses you have. See survey response rate.
- How the question was asked. A leading question produces a precise, wrong answer. See how to write survey questions.
- Who was asked at all. A survey of newsletter subscribers describes newsletter subscribers, not every customer.
Frequently asked questions
What is a good margin of error for a survey?
Plus or minus 5 percentage points at 95% confidence is the common standard, and needs about 385 responses from a large audience. For decisions where a few points matter, aim for 3. For a rough read of opinion, 10 can be enough, as long as you say so when you report the results.
What is the margin of error for an NPS?
Wider than for an ordinary percentage, because an NPS runs from −100 to +100 and depends on two shares at once. With 200 responses split evenly between promoters, passives and detractors, it is about ±11 points at 95% confidence, so a score of 0 could really be anywhere from about -11 to +11.
How do I know if a change between two surveys is real?
Combine the two margins: square each, add them, and take the square root. A change smaller than that is within the noise. Two surveys with a ±10 point margin each need a change of about 14 points before it is more than chance.
Does margin of error cover all survey error?
No. It only covers the chance variation from asking a sample instead of everyone. It says nothing about bias from who chose to answer, leading question wording, or an audience that does not match the people you want to describe. A survey can have a small margin of error and still be wrong.
- Sample size calculator · Choose a margin of error first, then see how many responses it needs
- NPS calculator · Work out the score itself from your responses
- Response rate calculator · How many people to invite to get the responses you need
- NPS vs CSAT · What each score measures and how it is calculated
- Survey report example · How to show the base behind every figure in a report