The formula
How to calculate margin of error
The margin of error states how far a survey result might sit from the true value purely because a sample was used rather than the whole population. It is the "plus or minus three points" quoted alongside opinion polls.
The margin is widest at 50% and narrows towards the extremes, because a proportion near zero or one has less room to vary. Pollsters often quote the 50% figure as a conservative headline.
What to enter:
- Observed proportion (%)
- Sample size
- z-score for the confidence level — 1.645 for 90%, 1.96 for 95%, 2.576 for 99%
Everything recalculates as you type, and the numbers in the address bar update with it, so a link to this page carries your figures with it.
Units matter more here than the arithmetic itself: the formula assumes a specific set of units for each input, stated next to the field, and converting into those units first is usually the difference between a correct result and one that is wrong by a clean power of ten.
Why margin of error matters
A margin of error calculation gets used both to check work already done by hand and to explore how a formula behaves without redoing the algebra every time an input changes — this page exists for both, since the underlying arithmetic is the same either way.
It is useful for checking a manual calculation before submitting or acting on it, and equally useful for building intuition about a formula by adjusting one input at a time and watching how the result moves in response — a much faster way to understand a relationship than working through several versions of the algebra by hand.
This calculation sits in a long tradition of being done first by hand with tables and slide rules, then with a scientific calculator, and now with a page like this one — the underlying mathematics has not changed at any point in that history, only the speed and convenience of getting from the inputs to the answer. Understanding the formula itself, shown above, is still worth doing even when a tool computes it instantly, since it is what makes the result trustworthy rather than just fast.
It is worth keeping a note of which inputs were used to produce a given result, particularly where the figure is going into a report or a further calculation — reproducing a result later, or explaining how it was reached, is far easier with the original inputs to hand than by trying to reverse-engineer them from the output alone.
Worked example
Take the figures the calculator starts with:
- Observed proportion: 52 %
- Sample size: 1,000
- z-score for the confidence level: 1.96
That gives:
- Margin of error: 3.097 percentage points
- Lower bound: 48.903 %
- Upper bound: 55.097 %
The figures above are the calculator's own default values, shown purely so the working is visible rather than hidden — the same steps apply exactly to your own numbers, entered in the fields at the top of this page.
Reading the result
A 52% result with a 3.1-point margin means the true value plausibly lies between 48.9% and 55.1%. A poll showing 52% against 48% is therefore not evidence of a lead.
Where this goes wrong. The margin covers sampling error only. Non-response bias, question wording and coverage error are usually larger and are not captured by any formula.
A result that is wrong by an exact factor of ten, a hundred or a similar round number is almost always a units error rather than a mistake in the formula itself — checking each input against the unit stated beside it is the fastest way to track it down.
Because 1,000 respondents is the industry standard sample size, and at 50% that gives roughly ±3.1 points. Going to ±1.5 points would need around 4,000 respondents — four times the cost.
Barely, unless the sample is a large fraction of the population. Polling 1,000 people gives similar precision whether the population is 100,000 or 60 million.
It returns margin of error. With 52 % observed proportion, 1,000 sample size and 1.96 z-score for the confidence level, that comes to 3.097 percentage points. Change any field and the figure moves with it.
Generally, no more than the least precise input justifies — a result reported to six decimal places from inputs measured to two significant figures is implying a precision the calculation does not actually have. The calculator shows full precision so you can round appropriately for your own use.
Yes — the equation shown in the formula section above is the standard form used in textbooks and reference material for this calculation, not a simplified or approximate version.
Yes, in the sense that it applies the correct standard formula and returns an accurate result for the inputs given — but check your own course or publication's requirements for how results should be rounded, presented and referenced, since those conventions vary and are not something a calculator can know on your behalf.