Forms · Glossary
What is margin of error?
Margin of error is the amount a survey result could differ from the true population value purely because a sample was asked instead of everyone. A result of 40 percent with a margin of 5 points at 95 percent confidence means the population figure most likely sits between 35 and 45 percent.
It is the most quoted number in a survey report and the most misread, usually as a guarantee that the result is accurate. It covers one source of error only, and it grows quickly when a result is split into smaller groups.
Nuwan Madhusanka · Co-founder
5 min read · Published
| Completed responses | Margin of error | A 50 percent result could be |
|---|---|---|
| 100 | Plus or minus 9.8 points | 40.2% to 59.8% |
| 200 | Plus or minus 6.9 points | 43.1% to 56.9% |
| 400 | Plus or minus 4.9 points | 45.1% to 54.9% |
| 600 | Plus or minus 4.0 points | 46.0% to 54.0% |
| 1,000 | Plus or minus 3.1 points | 46.9% to 53.1% |
| 2,000 | Plus or minus 2.2 points | 47.8% to 52.2% |
The formula behind the number
For a percentage, the margin is the z score for the chosen confidence level multiplied by the square root of the result times one minus the result, divided by the number of responses. At 95 percent confidence the z score is 1.96. Using a result of 50 percent gives the widest margin, which is why published polls often quote that one figure for the whole survey. Two consequences are worth remembering. Precision improves with the square root of the sample, so quartering the margin needs sixteen times the responses, and going from 400 to 100 responses doubles it from 4.9 to 9.8 points. And when the sample is a large share of a small population, a correction narrows it: 150 answers from a staff of 300 carries a margin of about 5.7 points rather than the 8.0 the basic formula gives.
What the margin does not cover
Pew Research Center describes the margin of sampling error as how close a survey result can reasonably be expected to fall to the true population value. The key word is sampling. The margin measures only the uncertainty from asking a random selection of people instead of everyone. It says nothing about a leading question, a scale that crowds answers at the top, people who declined to take part and think differently, or a list that left out part of the population. Those errors are often larger than the margin, and they do not shrink as more responses arrive. Weighting, which corrects for groups that responded unevenly, also widens the margin, an increase statisticians call the design effect. A result reported with a tidy plus or minus figure is precise about one thing and silent about the rest.
Subgroups and differences
A report usually quotes one margin, calculated for the full sample, and then presents results for smaller groups as though the same margin applied. It does not. Pew's explainer shows a national sample of 1,067 with a margin of 3 points, and a subgroup of about 160 people within it carrying a margin of roughly 8 points. The same article makes a second point that is missed even more often: the margin for the gap between two figures is larger than the margin for either one, about twice as large in its example of two candidates. So when a staff survey shows one team at 62 percent satisfied and another at 55, the question is not whether each figure is precise but whether the seven point gap is bigger than the uncertainty on the gap. With a few dozen people per team, it usually is not.
When a margin should not be printed
The calculation assumes every member of the population had a known chance of being selected. A random sample from a complete customer list meets that assumption. A survey link posted on social media, placed on a website for anybody who wanders past, or forwarded through group chats does not, because nobody knows who had the chance to answer. Quoting a margin for such a survey lends it a precision it does not have. The honest report says how many people answered, how they found the survey, and what kind of people they appear to be. Results from open surveys can still be useful for spotting problems and collecting ideas, provided nobody treats them as a measurement of the whole population.
Calculating it from form responses
Every submission is one row, and every field its own column in the CSV export, with choice questions written out as the option labels people saw. That makes the arithmetic straightforward: count the rows that chose an answer, divide by the rows that answered the question, and apply the formula with that count as the sample size. If a question was optional, use the number who answered it, not the total number of responses, since skipped rows shrink the sample for that item. A rating grid exports as one column of statement and answer pairs, so split it before counting any single statement. Report the number of responses and the margin beside each percentage that matters.
Questions people ask
Why do so many polls say plus or minus 3 points?
Because a sample of about 1,000 gives a margin of about 3 points at 95 percent confidence, and that size is a common balance between cost and precision. Pew's example uses 1,067 responses for exactly that margin. Beyond that, each extra point of precision costs a large number of additional interviews, so many polls stop there.
Does the margin apply to averages as well as percentages?
Yes, with a different formula. For an average, such as a mean rating, the margin is the z score multiplied by the standard deviation of the answers divided by the square root of the number of responses. A rating question where answers are spread widely has a larger margin than one where most people chose the same point.
Is the margin the same for every question in a survey?
No. It depends on the result and on how many people answered that question. A result of 20 percent from 400 answers has a margin of about 3.9 points rather than 4.9, and an optional question answered by half the sample has a wider margin than a required one. The single figure in a report is usually the widest case.
What margin of error is acceptable?
One that is smaller than the difference that would change your decision. If you will act only when support passes a clear majority, 5 points is plenty. If a movement of two points matters, you need thousands of responses. Deciding the smallest meaningful change first is the quickest way to agree on a realistic sample size.
Does a bigger population mean a bigger margin of error?
Hardly at all, once the population is large. A random sample of 1,000 from a town of 50,000 and from a country of millions carries almost the same margin. Population size only makes a real difference when the sample is a sizeable share of it, and then the correction makes the margin smaller, not larger.
How should the margin appear on a chart?
As error bars or a shaded band around each figure, with the number of responses in the caption. If bars for two groups overlap heavily, avoid titling the chart as though one group is ahead. Leaving the uncertainty off a chart is how a difference within the margin becomes a headline in the next meeting.
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Create a form with OneCraftRelated questions
- What sample size do I need for a survey?What sample size do I need for a survey? Work it out from population, confidence level and margin of error, with a table for 90, 95 and 99 percent.
- Confidence level vs confidence intervalConfidence level vs confidence interval: the level is how often the method captures the true value, the interval is the range. A worked survey example.
- What is survey weighting?What is survey weighting? Adjusting responses so the sample matches the population. A raw vs weighted worked example, the main methods, and its limits.
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