Maps for Math

How many people does a poll need for a ±3% margin of error?

About 1,000. The margin of error is z times the square root of p(1 − p) divided by the number of people asked. At 95% confidence and a 50-50 split, 1,068 people gives ±3 points. Because of the square root, four times as many people only halves the margin.

0%25%50%75%100%35% ± 2.96%: 32% to 38%±3 points takes 972 people1,000 people: ±3 points
Margin of error (±)
2.96%
Low end
32%
High end
38%
People needed for ±3%
972

The margin of error only covers the luck of who got picked. Who answers, how the question is worded, and who actually votes can add more.

Challenge: Get the margin to ±2 points or less while asking no more than 2,500 people.

%
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Start with the national poll, then try the quick online survey and the big poll.

Challenge: Get the margin to ±2 points or less while asking no more than 2,500 people. The box under the picture turns green when you get it.

Stuck? Pick one of the examples from the “Try an example” menu, or press “New example.”

Understand

MOE=zp(1−p)n\text{MOE} = z\sqrt{\frac{p(1 - p)}{n}}

A poll's share wobbles depending on who happens to be asked. The typical wobble is the standard error, p(1−p)/n\sqrt{p(1 - p)/n}. Multiply by zz (1.96 for 95% confidence) to get the margin of error:

MOE=zp(1−p)n\text{MOE} = z\sqrt{\frac{p(1 - p)}{n}}

The curve shows the square root at work. The margin drops fast at first, then flattens. The bar at the top shows the range, and whether it crosses 50%.

Use

Every input has a unit menu, so you can type values in the units you already have. Results follow your units.

Show the work

  1. Standard error: the typical wobble of a sample share\text{SE} = \sqrt{\frac{p(1 - p)}{n}} = 0.01508
  2. Margin of error at 95% confidence (z = 1.96)\text{MOE} = 1.96\sqrt{\frac{0.35(1 - 0.35)}{1000}} = 2.956\ \text{points}
  3. Range35\% \pm 2.956 = 32.04\%\ \text{to}\ 37.96\%
  4. Sample size for ±3 points: solve for nn = \frac{z^2 p(1 - p)}{0.03^2} = 972

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Set how many people were asked and the share who said yes. "More settings" changes the confidence level.

  • This is for a simple random sample. Real polls weight their results, which usually widens the margin a little.

For learning and estimation. Verify with applicable codes, standards, and a qualified professional before using in design, construction, or safety-critical work.

Cheat card

MOE=zp(1−p)n\text{MOE} = z\sqrt{\frac{p(1 - p)}{n}}
n=z2p(1−p)MOE2n = \frac{z^2 p(1 - p)}{\text{MOE}^2}
SymbolMeaningUnit
nnpeople asked
ppshare answering yes
zz1.645 (90%), 1.96 (95%), 2.576 (99%)
  • A 50-50 split gives the biggest margin, so pollsters plan for it.
  • Halving the margin takes four times the people.
  • The size of the whole population barely matters once it's much bigger than the sample.

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Where it’s used

  • Computer Science
    Product teams size user surveys and experiments with this formula.
  • Medicine & Health
    Researchers plan how many patients a study needs to estimate a rate precisely.
  • Money & Shopping
    Read a poll headline and know whether a 2-point lead actually means anything.

Questions people ask

What does "margin of error ±3%" mean?

If the poll were repeated many times, about 95% of the ranges built this way would contain the true share. It measures the luck of who got picked.

Why do polls use about 1,000 people?

It gives about ±3 points, and getting to ±2 would take more than twice as many. The extra cost buys little precision.

Does a bigger country need a bigger poll?

Barely. The margin depends on how many people you ask, not on the population size, as long as the population is much larger than the sample.

Sample STA-0003 · rev 1 · model STA-M03 · Level 3 · algebra · updated 2026-09-28