Polling Margin Calculator

Work out the margin of error on a survey result and decide whether a gap between two candidates is real or within the noise. Enter the sample size, the confidence level and the headline percentage, and the page returns the error band and a significance call.

Note on Methodology and Scope

This page performs arithmetic on the figures you enter. It does not forecast any real contest and it does not model sampling error, turnout, district-level effects or polling house methodology. It is an illustrative aid for understanding how the numbers relate, not a prediction of any outcome.

An Error Band and a Yes-or-No Call

The calculator does two related jobs. First it turns a sample size, a confidence level and a percentage into a margin of error, the plus-or-minus figure you see attached to poll numbers. Then it compares two percentages and decides whether the distance between them clears that margin, which is the difference between a lead and a statistical tie.

• The core formula is the standard one for a proportion: the error is the confidence factor times the square root of p times one minus p, divided by the sample size. The result is converted to percentage points.
• The confidence level is chosen from a dropdown and supplies the factor directly: 1.645 for ninety percent, 1.96 for ninety-five, and 2.576 for ninety-nine. Ninety-five is the default and the conventional choice.
• A finite-population correction is applied when the stated population is below ten million. It shrinks the error slightly to reflect that you are sampling a meaningful fraction of a finite group rather than from an unlimited pool.
• The page also reports the interval — the result percentage minus and plus the error — to one decimal place, which is the range the true value is likely to sit in.
• A second percentage is optional. When supplied, the tool takes the absolute gap between the two figures and compares it against the margin: larger means a statistically significant difference, smaller or equal means the two are tied within the error.

Running a Margin Check

1. Enter the Sample size — the number of people actually surveyed. The default is one thousand, a common poll size and a useful starting point.
2. Leave the Confidence level at ninety-five percent unless you have a reason to move it. Switching to ninety-nine widens the band, and ninety narrows it.
3. Set the Population size. If you are unsure, leave a large number, which disables the finite-population correction and gives the standard, slightly more conservative figure.
4. Enter the Result percentage you want the margin for — typically the leader's share, or the share closest to fifty, which produces the widest band.
5. Optionally enter a Compare percentage to test a second figure against the first. The result then adds a sentence saying whether the gap is significant or a tie.
6. Read the panel top to bottom: the inputs as the tool understood them, the margin of error, the interval, and then the comparison line if you supplied a second percentage.

The Default Poll, Analysed

The page loads with a sample of one thousand, ninety-five percent confidence, a population of one million, a result of fifty percent and a comparison of forty-five percent. Here is what that produces and where each figure comes from.

• The fifty percent result gives p of one half, and half times the other half is a quarter. Dividing a quarter by one thousand and taking the square root gives about one point five eight percent, and multiplying by the 1.96 factor for ninety-five percent confidence gives roughly three point one percentage points.
• Because the population of one million is below the ten million threshold, the finite-population correction applies. Its factor is the square root of (one million minus one thousand) divided by (one million minus one), which is about zero point nine nine nine five — a very small reduction, because a thousand is only a tenth of a percent of the population.
• The margin, corrected and converted to percentage points, comes to about three point one. The page rounds it to one decimal place and reports the margin as plus or minus three point one percent.
• The interval runs from forty-six point nine to fifty-three point one percent, the result of subtracting and adding the margin to the fifty percent figure.
• The comparison figure of forty-five percent sits five points below the fifty percent result. Since five is larger than the margin of three point one, the page reports that the five-point difference is statistically significant, not a tie.
• Had the comparison been forty-eight percent instead, the gap would have been two points, which is smaller than the margin, and the tool would have called the two results statistically tied.

Different Ways to Judge Poll Certainty

Alternative	Notes
A single margin of error	The headline figure, and the one this page computes. It is simple and widely understood, but it applies to one percentage in isolation. Comparing two results each with their own error needs the difference test, which is why the tool offers the second percentage field.
The confidence interval	Reporting the range rather than a single number — forty-six point nine to fifty-three point one, say — makes the uncertainty concrete. A number near fifty produces the widest interval; results far from fifty, closer to none or all of the vote, carry narrower bands for the same sample.
Increasing the sample size	The most direct lever on the margin, but a costly one, because the error shrinks with the square root of the sample. Quadrupling the number of respondents only halves the margin, so the return on extra interviews falls away quickly.
Non-sampling error and methodology	The margin covers only random sampling variation. It says nothing about who answered, how they were selected, how responses were weighted, or how the question was worded. These effects can be larger than the stated margin, so a sound reading pairs the number with the method.

The Assumptions Behind the Number

The arithmetic is standard, but it rests on assumptions that real polls strain. The page itself notes that margin of error assumes random sampling, and it is worth taking that seriously.

• The calculation assumes a simple random sample. Real surveys use quotas, panels and weighting, and the effective error is often larger than the formula suggests once design effects are taken into account.
• The finite-population correction only engages below a population of ten million. Above that the tool uses the standard unlimited-population formula, on the basis that the sampled fraction is small enough not to matter.
• Percentage results far from fifty yield a smaller margin than results near fifty. A race polled at fifty percent carries the widest band for a given sample, so a tight two-way race is the hardest to call.
• The significance test compares the raw gap against the single margin from the first percentage. It is a straightforward comparison rather than a formal two-sample test, so it is a quick guide rather than a rigorous statistical procedure.
• The tool validates that a sample size and a result percentage are present, and shows a prompt if either is missing. Beyond that it does not police the inputs, so percentages outside zero to one hundred would produce meaningless output.

Margin of Error Questions

Why did my margin barely change when I entered a population?

Because the finite-population correction only applies below ten million and is small even then. With a sample of a thousand from a population of one million, the correction factor is about zero point nine nine nine five, which shaves a rounding-level amount off the margin. Set the population above ten million and the correction drops out entirely, leaving the standard figure.

What does statistically significant mean here?

It means the gap between the two percentages is larger than the margin of error, so the difference is unlikely to be explained by sampling noise alone. In the default case, a five-point gap against a three-point-one margin clears the bar and is called significant. If the gap had been two points, the page would have called the results a statistical tie instead.

Which confidence level should I use?

Ninety-five percent is the convention and the default, using a factor of 1.96. Choosing ninety percent uses 1.645 and gives a narrower band; choosing ninety-nine uses 2.576 and a wider one. Higher confidence means a wider interval, so the choice is a trade between certainty and precision rather than one setting being simply better.

Does the calculator need both percentages?

No. The first result percentage is required to compute the margin at all, but the comparison percentage is optional. Leave it blank and you get the margin and the interval on their own. Enter it and you also get the significance sentence. A zero in the comparison field is treated as not supplied, so the comparison block is skipped.

Last verified: October 7, 2026

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Maintained by Huang Bin · Last updated October 3, 2026