Margin of Error Calculator

Measure the sampling precision of a poll or study. Compute the margin of error for a proportion or a mean, and see the interval it places around your estimate.

Picking the Right Mode

  • Proportion: your result is a percentage of people or items, such as 52% approval in a poll.
  • Mean: your result is an average measurement, such as an average delivery time or test score.
  • In both modes, higher confidence or a smaller sample widens the margin.

The Two Formulas This Calculator Uses

Margin of error is the critical value multiplied by the standard error of the estimate. What changes between modes is how that standard error is computed:

Proportion and Mean Formulas:

MOE = z × √(p(1 − p) / n)  (proportion)

MOE = z × s / √n  (mean)

Symbols:

  • • z = critical value (1.645 at 90%, 1.96 at 95%, 2.576 at 99%)
  • • p = sample proportion, as a decimal
  • • s = sample standard deviation
  • • n = sample size

For proportions, the variability comes from the proportion itself: p(1 − p) is largest at 50% and shrinks toward the extremes. For means, the variability comes from the spread of the underlying measurements, so the same sample size can yield very different margins depending on how noisy the data is.

What the Margin Actually Covers

Adding and subtracting the margin of error from your point estimate produces a confidence interval — the range of values for the population parameter that is consistent with your sample. A poll reporting 52% support with a ±3.1 point margin at 95% confidence is really reporting the interval from 48.9% to 55.1%. Because that interval straddles 50%, the poll alone cannot establish which side is ahead.

Crucially, the margin of error quantifies sampling error only: the random luck of which individuals landed in your sample. It says nothing about biased question wording, non-random sampling, or people answering untruthfully. A survey can have a tiny margin of error and still be badly wrong if the sample was not representative.

How to Shrink the Margin

Because n sits under a square root, precision improves slowly: quadrupling the sample size halves the margin of error. Moving a 1,000-person poll at 52% from ±3.1 points to ±1.55 points requires 4,000 respondents, not 2,000. The other lever is confidence: dropping from 99% to 90% narrows the margin by about 36% at the cost of weaker coverage guarantees.

If you are planning a study rather than analyzing one, it is usually easier to run the calculation in reverse — decide the margin you need and solve for n. Our sample size calculator does exactly that inversion, including the finite population correction for small groups.

Margins on Differences Are Bigger

A frequent misreading of election polls: if candidate A has 52% and candidate B has 48% with a ±3 point margin, the 4-point lead is not “outside the margin of error.” The reported margin applies to each candidate's share separately. The lead is a difference of two correlated proportions, and in a tight two-candidate race its margin of error is roughly twice the reported figure — about ±6 points here — so the race is statistically much closer than it looks.

The same caution applies to comparing subgroups (men vs. women, regions, age brackets): each subgroup has a smaller sample than the full survey, so subgroup margins are always wider than the headline number.

Worked Example: A 1,000-Person Poll

A poll of n = 1,000 voters finds 52% support for a ballot measure. At 95% confidence, the calculator finds the margin of error like this:

  1. Variability term: p(1 − p) = 0.52 × 0.48 = 0.2496.
  2. Divide by n: 0.2496 / 1000 = 0.0002496.
  3. Standard error: √0.0002496 ≈ 0.015799.
  4. Margin of error: 1.96 × 0.015799 ≈ 0.031, or 3.0966 percentage points.
  5. Interval: 52% ± 3.0966 = 48.9034% to 55.0966%.

Because the interval includes 50%, the measure's passage is not statistically settled by this poll. In mean mode the arithmetic is shorter: a sample of 64 delivery times with standard deviation 12 minutes gives MOE = 1.96 × 12 / √64 = 1.96 × 1.5 = 2.94 minutes; around a sample mean of 80 minutes, the interval runs from 77.06 to 82.94.

Frequently Asked Questions

Is margin of error the same thing as standard error?

They are related but not identical. The standard error measures the typical sampling fluctuation of the estimate, while the margin of error is the standard error multiplied by a critical value (1.96 for 95% confidence). The margin of error is therefore always larger and carries an explicit confidence level.

What does a margin of error of plus or minus 3% at 95% confidence mean?

If the same survey were repeated many times with fresh random samples, about 95% of the resulting intervals (estimate plus or minus 3 points) would contain the true population value. It does not mean there is a 95% chance the truth is within 3 points of this particular estimate, although that is a common informal reading.

Why do most national polls use about 1,000 respondents?

At n = 1,000 the margin of error for a proportion near 50% is about plus or minus 3.1 percentage points at 95% confidence, which most pollsters consider an acceptable balance of cost and precision. Because precision depends on the square root of n, cutting the margin to 1.55 points would require 4,000 interviews - four times the cost for half the error.

Does the margin of error account for biased or dishonest answers?

No. It quantifies random sampling error only. Nonsampling problems such as leading questions, low response rates, an unrepresentative sampling frame, or respondents answering inaccurately are not captured, and they can dwarf the stated margin.

When should I use the mean mode instead of the proportion mode?

Use mean mode whenever your estimate is an average of a numeric measurement - revenue per customer, minutes per task, points on a test. Use proportion mode when your estimate is the percentage of a sample falling into a category. The two modes use different standard error formulas, so picking the right one matters.