A party rises from 27% to 28%, a rival falls from 25% to 24%, and the headline announces a widening lead. The arithmetic is straightforward; whether the public actually moved is a separate question.
A poll estimates opinion from a sample, so small differences need more evidence than a single pair of rounded figures. The familiar margin of error describes only one part of that uncertainty.
Where the familiar three points come from
Under a simple random-sampling model, an approximate 95% margin for a percentage is 1.96 × √[p(1−p)/n], with p expressed as a fraction and n the sample size. At 50% in a sample of 1,000, that gives about 3.1 percentage points either side of the estimate.
The calculation is narrower for a smaller percentage: at 20% in the same sample, it is about 2.5 points. These are mathematical illustrations under the stated model, not confidence intervals calculated for a particular published poll.
The British Polling Council’s explanation of sampling uncertainty discusses the conventional approximation and the further errors which affect real polling. Quota samples and modelled estimates do not automatically acquire a valid probability-sampling interval because someone prints “±3%” underneath them.
What does 95% mean?
In the idealised repeated-sampling exercise, an interval procedure with 95% coverage captures the true population proportion in about 95 out of 100 repetitions. Once an interval has been calculated, that frequentist statement is about the procedure; it does not assign a 95% probability to a fixed true value being inside this particular interval.
That distinction matters less in everyday reporting than another one: an interval around a party share does not cover every reason the estimate might be wrong. Weighting, non-response, question wording and turnout assumptions can introduce uncertainty or bias beyond the sampling calculation.
A lead has its own uncertainty
Party shares from the same poll are related because respondents usually choose one party. The margin around Party A’s percentage is therefore not the margin around A’s lead over B.
For an illustrative simple random multinomial sample, the variance of the difference is [pA + pB − (pA − pB)²]/n. At 30% versus 28%, with 1,000 respondents, the approximate 95% margin around the two-point lead is about 4.7 points. That example shows why simply attaching three points to the lead gives the wrong calculation.
How much does a bigger sample help?
Sampling uncertainty falls with the square root of sample size. Moving from 1,000 to 4,000 interviews roughly halves the idealised margin, rather than making the estimate four times as precise. A subgroup of 100 people has far less information than the full poll of 1,000 containing it.
Large samples can still be systematically unrepresentative. Research on the sensitivity of survey weights examines the consequences of important characteristics being absent from the adjustment. More interviews using the same biased recruitment process do not necessarily remove that bias.
What should a reader do with a small movement?
Look for repetition across fieldwork periods and independent polling organisations, checking that methods stayed comparable. Avoid treating a rounded point as a precisely measured change, and be especially cautious with the crossbreaks in polling tables.
An average of polls can smooth some random variation, but shared methodological error can survive the averaging. Our guide to polling accuracy explains how performance can be assessed once there is an election result to compare with the estimates.
Sources and further reading
- British Polling Council: uncertainty and sampling
- Hartman and Huang: sensitivity analysis for survey weights
Reviewed 7 October 2026. Worked examples are illustrative unless an election or poll is identified.