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Margin of Error Calculator

Find how far a poll or survey percentage could be off from the sample size and confidence level, with an optional small-population correction.

Level: Grade 12 to college

  • Common Core standard HSS.IC.B.4Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
Printable worksheet

Margin of Error Calculator

Know the answer you need? Pick an input to work backward.

Inputs

The share of the sample with the answer you care about. Use 50% if unknown; it gives the largest margin.

Leave blank for a large or unknown population.

Confidence level

Result

Margin of error

3.099%

At 95% confidence, a result of 50% from 1000 responses has a margin of error of ±3.1 percentage points.

Interval lower bound
46.901%
Interval upper bound
53.099%
Standard error
0.0158
Critical value (z*)
1.96

See the step-by-step solution ↓

Step-by-step solution

  1. Look up the z* value for 95% confidence. It sets how wide the range must be to be that sure

    z*=1.96

  2. Find the standard error, how much a poll result typically wobbles from sample to sample. Use p as a decimal and n as the sample size

    SE=0.5×(1−0.5)1,000=0.015811

  3. Multiply z* by the standard error to get the margin of error E

    E=1.96×0.015811=0.03099

  4. Multiply by 100 to read the margin in percentage points

    0.03099×100=3.098975%

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out how much a poll or survey result could be off, just because you asked some people instead of everyone. Say a poll reports that 50% of people agree, "plus or minus 3 points". That plus-or-minus figure is the margin of error. It tells you how far the sample result could reasonably be from the answer you'd get by asking the whole population. The gap comes purely from which people happened to be picked.

Enter the sample percentage, the number of responses and a confidence level. You'll get the margin in percentage points, the range it creates and the standard error behind it. Did you survey a large share of a small, known group? Then add the population size. The calculator will apply the finite population correction, which shrinks the margin.

Where it's used

Why this math matters outside the classroom:

  • Political polls. A poll of 1,000 people finding 50% support has a 95% margin of error of 1.96 × √(0.25 ÷ 1000), about 3.1 points.
  • Market research. A company surveying 400 customers on a yes-or-no question has a margin of up to 4.9 points at 95% confidence.
  • School surveys. A student surveying 100 of 600 classmates uses the finite population correction to cut a 9.8-point margin to about 9.0.
  • Reading the news. Readers see that a 48% to 46% lead with a 3-point margin of error is too close to call.
  • Invoice audits. An auditor who finds errors in 10% of 200 sampled invoices reports 10% ± 4.2 points at 95% confidence.

Grade level and standards: Grade 12 to college; Common Core State Standards for Mathematics HSS.IC.B.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Enter the sample proportion as a percentage. Planning a survey and don't know the result yet? Leave it at 50%. That gives the widest, most cautious margin.
  2. Enter the number of completed responses as the sample size.
  3. If your sample is more than a few percent of the whole group, you can enter the total population size.
  4. Choose the confidence level and read the margin of error in percentage points.

What the inputs mean

Sample proportion
The share of people who gave a particular answer. The margin is largest at 50% and shrinks toward 0% or 100%. At exactly 0% or 100% the formula breaks down, so the calculator suggests a Wilson interval instead (a range method that works at the extremes).
Population size (optional)
This only matters when the sample is a noticeable part of the population, such as surveying 200 of a company's 800 employees. For a national poll, leave it blank.
Custom confidence level
Between 50% and 99.99%.

Formula and method

E = z* × √(p(1 − p) / n)
With a finite population:  E = z* × √(p(1 − p) / n) × √((N − n) / (N − 1))
z* = 1.645 (90%), 1.960 (95%), 2.576 (99%)

Here's what the letters mean. E is the margin of error. p is the sample proportion as a decimal, so 50% is 0.5. n is the number of responses, and N is the size of the whole population. z* (the critical value) is a fixed number for each confidence level.

The standard error, √(p(1 − p)/n), measures how much a sample result typically changes from one sample to the next. Multiplying it by z* stretches it wide enough to match the confidence level you chose.

Because n sits under a square root, precision improves slowly. Four times the sample size only cuts the margin of error in half.

The finite population correction shrinks the margin when your sample uses up a large part of the population. If you ask everyone (n = N), the correction is zero, and so is the sampling error.

Worked example

A poll of 1,000 adults finds 50% support for a proposal. What is the margin of error at 95% confidence?

  1. Critical value at 95%: z* ≈ 1.96.
  2. Standard error: √(0.5 × 0.5 ÷ 1,000) ≈ 0.0158114.
  3. Margin: 1.96 × 0.0158114 ≈ 0.0309898, or about 3.099 percentage points.
  4. Range: 50% ± 3.099 points, from about 46.901% to 53.099%.
  5. If the 1,000 people came from a town of 10,000 adults, the correction factor √(9,000 ÷ 9,999) ≈ 0.9487 cuts the margin to about 2.94 points.

Result: The margin of error is about ±3.1 percentage points (±2.94 with a population of 10,000).

Understanding the result

Picture repeating the poll many times. A margin of ±3 points at 95% confidence means about 95% of those ranges would contain the true population percentage.

The margin applies to each percentage on its own. The gap between two candidates in the same poll is less certain than either number alone. When the race is close, it can be up to about twice the margin.

Smaller groups within the poll have their own, larger margins. If only 200 of the 1,000 people are under 30, results for that group carry a margin of about ±6.9 points.

Assumptions and limitations

  • The sample is a simple random sample, so everyone in the population had an equal chance of being picked.
  • Each person's answer doesn't affect anyone else's.
  • The sample is large enough for the normal approximation (the bell-curve shortcut behind the formula) to hold. As a rule of thumb, you need at least about 10 responses on each side of the question.
  • The margin of error covers random sampling error only. It says nothing about bias from who chose to answer, how the questions were worded or people not telling the truth. Those errors can be bigger than the margin itself.
  • Weighted samples, and cluster or stratified surveys (ones that sample by groups or by subgroups), have a design effect. That usually makes the true margin larger than this formula shows.
  • For very small samples, or percentages near 0% or 100%, a Wilson interval gives a more reliable range.

Frequently asked questions

Why do so many polls use about 1,000 people?

At 95% confidence and 50% support, 1,000 responses give a margin of about ±3.1 points. Going much further costs a lot for little gain. You'd need about 4,000 responses just to halve it to ±1.5 points.

Does the population size matter?

Surprisingly little, unless the sample is a sizable part of it. A random sample of 1,000 has almost the same margin whether it comes from a city or a whole country. Enter the population size only when you're sampling a noticeable share of it.

Why use 50% when I don't know the result?

The product p(1 − p) is largest when p = 0.5. So 50% gives the biggest possible margin for a given sample size. If you plan around it, your real margin will be no larger than the one you report.

Is the margin of error the same as the standard error?

No. The standard error is one unit of how much samples vary. The margin of error is the standard error times the critical value. At 95% confidence, that makes it about 1.96 times larger.

How this calculator is checked

Method: Normal-approximation (Wald) margin of error for a single proportion, z* × √(p(1 − p)/n), with the standard finite population correction √((N − n)/(N − 1)) from sampling theory. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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