Margin of Error Calculator

About the Margin of Error Calculator

Every poll and survey result comes with uncertainty. If 52 percent of 1,000 people surveyed say they support a policy, the true figure across the whole population is probably not exactly 52 percent — a different 1,000 people would have given a slightly different answer. The margin of error puts a number on that uncertainty: a result of 52 percent with a margin of ±3.1 points means the true figure is likely somewhere between about 48.9 and 55.1 percent.

This margin of error calculator works for a survey percentage or a sample mean. Enter the sample size and, for a percentage, the result (or 50 percent if you want the survey's overall margin); for a mean, the standard deviation. Choose a confidence level and, if your sample is a sizeable share of a small population, enter the population size to apply the finite population correction. The calculator shows the margin, how it was calculated, the likely range of the true value, and how much a larger sample would help.

How to Use the Margin of Error Calculator

Choose Percentage for polls and surveys or Mean for averages.

For a percentage, enter the sample percentage. Use 50% for the overall margin of a survey.

For a mean, enter the standard deviation.

Enter the sample size and, optionally, the population size.

Choose the confidence level — 95% is standard.

The Formulas

  percentage:  E = z × √( p(1 − p) ÷ n )
  mean:        E = z × σ ÷ √n

  finite population correction:  E × √( (N − n) ÷ (N − 1) )

  z = 1.645 (90%),  1.960 (95%),  2.576 (99%)

Here p is the sample proportion as a decimal, n the sample size, σ the standard deviation and N the population size.

Step-by-Step Example: A Poll

A poll of 1,000 people, result 50%, 95% confidence.

  √(0.5 × 0.5 ÷ 1,000) = √0.00025 = 0.015811
  E = 1.96 × 0.015811 = 0.03099

The margin of error is ±3.1 percentage points. At 99% confidence it widens to ±4.07 points; quadrupling the sample to 4,000 would narrow it to ±1.55 points.

Step-by-Step Example: A Small Population

500 responses from a company of 2,000 employees.

  Without correction:  1.96 × √(0.25 ÷ 500) = ±4.38 points
  Correction factor:   √((2,000 − 500) ÷ 1,999) = 0.8662
  With correction:     4.38 × 0.8662 = ±3.80 points

Because a quarter of all employees responded, the result is more precise than the same sample from an unlimited population.

Step-by-Step Example: A Mean

A sample of 100 with standard deviation 10.

  E = 1.96 × 10 ÷ √100 = 1.96 × 1 = ±1.96

The sample mean is likely within about 2 units of the population mean.

Why 50 Percent Gives the Largest Margin

The expression p(1 − p) is largest at p = 0.5, where it equals 0.25. At 20 or 80 percent it is 0.16, and at 10 or 90 percent 0.09. So a result near 50 percent has the widest margin, and extreme results have narrower ones. Pollsters report a single margin for the whole survey, calculated at 50 percent, because it is the upper limit for every question in it.

The Square Root Law

The margin of error falls in proportion to 1 ÷ √n. Going from 250 to 1,000 respondents halves the margin from about 6.2 to 3.1 points; going to 4,000 halves it again to 1.55. Each halving costs four times as many interviews, which is why most national polls settle on 1,000 to 2,000 people — beyond that, the gain in precision rarely justifies the cost.

Margin of Error for Differences

When comparing two groups within a poll — men and women, or two regions — each subgroup is smaller than the whole sample, so its margin is larger. A subgroup of 250 people has a margin of about ±6.2 points on its own. The margin for the difference between two results is larger still, roughly 1.4 times the individual margins when they are similar. A two-point lead in a poll with a ±3 point margin is therefore well within the noise.

What the Margin of Error Leaves Out

The margin of error covers only random sampling error — the luck of who happened to be chosen. It does not include non-response bias (the people who answer may differ from those who do not), coverage bias (some groups may be hard to reach), question wording, or people giving answers they think are expected. In practice these can be larger than the sampling margin, which is why polls sometimes miss by more than their stated margin of error.

Choosing a Confidence Level

The confidence level sets how often the range should contain the true value. At 95 percent, about one survey in twenty will miss by more than the margin through bad luck alone. Moving to 99 percent makes that one in a hundred, at the cost of a margin about 31 percent wider.

Understanding Your Result

The headline is the margin of error.

The how it was found line gives the z value, standard error and any population correction.

The what it means line gives the likely range of the true value.

The larger sample line shows the effect of quadrupling the sample size.

When Should You Use This Calculator?

Use it to report the precision of a survey or poll.

Use it to judge whether a difference between two results is meaningful.

Use it to check a margin quoted in a news report.

Use it for coursework on sampling and estimation.

Common Mistakes

Applying the whole-sample margin to subgroups. Subgroups have larger margins.

Treating the margin as covering all error. It excludes bias.

Reading a lead within the margin as meaningful. Differences need their own margin.

Forgetting the confidence level. A 99% margin is wider than a 95% one.

Using the finite population correction when the population is large. It makes almost no difference then.

Using the margin for a percentage with a mean. Means need the standard deviation, not the p(1 − p) formula.