Where the margin comes from
A poll reports a share, say 48 percent for a candidate. The NIST/SEMATECH e-Handbook gives the interval most often used for a share: the observed proportion plus or minus z times the square root of p(1 − p) divided by n, where n is the number of interviews and z is 1.96 for 95 percent confidence. The margin of error is the plus-or-minus part.
Pollsters quote one margin for the whole poll, and they get it by setting p to 50 percent, because p × (1 − p) is largest there and every other share gives a smaller margin. The sample size calculator page shows how that formula is rearranged for a sample size and when a small population changes it; the confidence interval calculator gives the interval for a share you already have.
| Share | Margin, n = 1,000 |
|---|---|
| 50% | ±3.1 pts |
| 40% | ±3.0 pts |
| 30% | ±2.8 pts |
| 20% | ±2.5 pts |
| 10% | ±1.9 pts |
Margin by sample size
At 95 percent confidence and p = 50 percent, the sample size alone sets the margin:
| Interviews | Margin of error |
|---|---|
| 100 | ±9.8 pts |
| 400 | ±4.9 pts |
| 600 | ±4.0 pts |
| 1,000 | ±3.1 pts |
| 1,500 | ±2.5 pts |
| 2,000 | ±2.2 pts |
| 5,000 | ±1.4 pts |
Two things stand out. The population size is not in the table, because it is not in the formula: 1,000 interviews give ±3.1 points for a state or a country alike. And the margin falls with the square root of n, so going from 400 to 2,000 interviews, five times the work, only brings the margin from ±4.9 to ±2.2.
Interviews needed for a given margin
Turned around, the formula says how many interviews a target margin costs: n = 1.96² × 0.25 ÷ E², rounded up.
| Target margin | Interviews needed |
|---|---|
| ±1 pts | 9,604 |
| ±2 pts | 2,401 |
| ±3 pts | 1,068 |
| ±5 pts | 385 |
Halving the margin quadruples the sample, because E is squared. That is why ±3 is the usual compromise: 1,068 interviews reach it, and the next point of precision costs more than twice as many. Pew Research Center's example poll of 1,067 people carries a margin of about ±3 points, and the formula gives ±3.0 for that n.
Two candidates at 48 and 45
A poll of 1,067 people shows candidate A at 48 percent and candidate B at 45 percent, with a margin of ±3 points. The lead is 3 points. Read it in three steps.
- Each share has its own range. A is likely between 45 and 51 percent, B between 42 and 48. The ranges overlap, so the poll cannot rule out that B is ahead.
- The gap has a larger margin than either share. Pew notes that the margin for the difference between two candidates is generally about twice the margin for one, since both shares can move in opposite directions. Twice ±3.0 is about ±6.0 points, so the 3-point lead is consistent with anything from a 3-point deficit for A to a 9-point lead.
- Inside the margin is not a tie. The lead is the single most likely value, and A is more likely ahead than behind. What the poll cannot do is distinguish a 3-point lead from a dead heat with 95 percent confidence. The honest summary is "A leads narrowly, within the margin", not "the race is tied".
The same logic applies to a change between two polls: a lead that grows from 3 to 6 points has moved by less than the margin of the gap, so it may be no change at all. Whether a gap is larger than chance would produce is the question a significance test answers; the p-value guide covers how to read one, and the A/B test significance calculator runs the same comparison for two conversion rates.
Subgroups
The quoted margin belongs to the full sample. Any slice of it is a smaller poll with a wider margin. Pew's example takes a group that is about 15 percent of the sample: of 1,067 respondents that is roughly 160 people, and the formula gives ±7.7 points for one share and about ±15.5 for the gap between two candidates inside that group. Results for women under 30, or for one state in a national poll, carry margins of that size even though the headline says ±3.
What the margin does not cover
The margin of error describes sampling error only: how far a result can land from the truth purely because the poll reached one random set of 1,000 people and not everyone. Pew Research Center is explicit that the other sources of error are harder to quantify and are rarely reported. Its list:
- Coverage. People the poll could not reach, such as those without internet access in an online poll.
- Nonresponse. People who were reached and declined; if they differ from those who answered, no sample size fixes it.
- Measurement. Respondents who did not answer candidly.
- Likely-voter models. The pollster's guess at who will actually vote.
- Weighting. Adjusting the sample to match the population enlarges the effective margin through the design effect, which a careful pollster folds into the reported number.
So a ±3 margin is a floor, not a ceiling, on how wrong a poll can be. It is also an approximation at the edges: the NIST handbook points out that the simple interval can produce a lower limit below zero for a small share, and recommends the Wilson or an exact method when the sample or the number of positive answers is small. For a national poll near 50 percent those cases do not arise.
Reading a poll in 30 seconds
- Find n and the confidence level. About 1,000 at 95 percent means ±3; 400 means ±5; 2,000 means ±2.
- Add and subtract the margin from each share. If the ranges overlap, the leader is not clearly ahead.
- Double the margin before judging a lead or a change between polls.
- Treat any subgroup number as a much smaller poll.
- Remember that the margin counts sampling error only; who was reached and who answered can matter more.
Frequently asked questions
What does a margin of error of plus or minus 3 points mean?
If the poll says 48 percent and the margin is ±3 points at 95 percent confidence, the share in the whole population is likely between 45 and 51 percent. Likely means that if the same poll were run many times, about 95 of every 100 results would land within 3 points of the true value. The other 5 would miss by more, through bad luck in who was sampled.
Why do most national polls have a margin of about 3 points?
Because they interview about 1,000 people. At 95 percent confidence and a 50 percent share, 1,000 interviews give a margin of ±3.1 points, and it takes 1,068 to reach ±3. Cutting the margin to ±2 needs 2,401 interviews and ±1 needs 9,604, so 3 points is where cost and precision usually meet.
Does the margin of error depend on the size of the population?
Hardly at all once the population is large. The formula divides by the sample size, not the population size, so a poll of 1,000 people has about the same margin for a large city as for the whole country. Only when the sample is a noticeable share of the population does a correction shrink the margin; the sample size calculator covers that case.
If two candidates are 3 points apart and the margin is 3 points, is it a tie?
It is too close to call, which is not the same as a tie. Each candidate's share could be off by up to 3 points, and the gap between them can be off by about twice that, so a 3-point lead with a ±3 margin is consistent with anything from a 3-point deficit to a 9-point lead. The leader is more likely ahead than behind, but the poll alone cannot settle it.
What does the margin of error leave out?
Everything except sampling error. Pew Research Center lists the errors it does not cover: people the poll could not reach, people who declined to answer, respondents who did not answer candidly, and the model used to decide who is a likely voter. Weighting the sample also widens the real margin through the design effect. A poll with a ±3 margin can be off by more than 3 points for any of those reasons.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, 7.2.4.1 Confidence intervals for a proportion (the normal approximation p ± z × √(p(1 − p) ÷ n), its lower limit can be negative, the Wilson and exact methods for small samples)
- NIST/SEMATECH e-Handbook, 1.3.6.7.2 Critical Values of the Student's t Distribution (1.960 at 0.975 in the normal row, the z for 95 percent)
- Pew Research Center, 5 key things to know about the margin of error in election polls (±3 points for a 1,067-person sample at 95 percent; the margin covers sampling error only; the margin for the gap between two candidates is about twice that for one; subgroups have larger margins; the design effect)