What is a margin of error?

The margin of error tells you how far your sample statistic is likely to be from the true population value at a given confidence level.

Does population size affect margin of error?

Yes, if your sample is a large fraction of the population, you can apply the finite population correction to slightly reduce the margin of error.

How can I reduce the margin of error?

Increase the sample size, use a lower confidence level, or use a proportion closer to 0.5 only when necessary; 0.5 is the most conservative.

Margin of Error Calculator

Quickly compute the margin of error for a survey or experiment. Choose between proportion (percent/yes-no) and mean (numeric values), set your confidence level, and optionally apply the finite population correction (FPC). You can also estimate the sample size needed to achieve a target margin of error.

Confidence level

CL used to find z-score

Sample size (n)

# of respondents

Observed proportion (p)

Use 0.5 for worst case

Population size (N)

for FPC

Margin of error

—

in same units; for proportions this is in proportion terms

Lower bound

—

Upper bound

—

Needed n

—

from sample size tab

Margin of error formulas

For a proportion (worst case when p=0.5):

\( ME = z \sqrt{ \frac{p(1-p)}{n} } \)

If population size N is known and not huge, apply finite population correction:

\( ME_{FPC} = ME \times \sqrt{ \frac{N - 1}{N - n} } \)

For a mean: \( ME = z \frac{s}{\sqrt{n}} \)

Common z-scores

Confidence level	z-score
90%	1.645
95%	1.96
99%	2.576

What to report

For a survey you'll often write: “With a sample of n=400, at the 95% confidence level, the margin of error is ±4.9 percentage points.”

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

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Formula (extracted text)

For a proportion (worst case when p=0.5): \( ME = z \sqrt{ \frac{p(1-p)}{n} } \) If population size N is known and not huge, apply finite population correction: \( ME_{FPC} = ME \times \sqrt{ \frac{N - 1}{N - n} } \) For a mean: \( ME = z \frac{s}{\sqrt{n}} \)

Variables and units

No variables provided in audit spec.

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
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