Primary formula
Proportion E=z sqrt(p(1-p)/n). Mean E=z sigma/sqrt(n). Optional FPC=sqrt((N-n)/(N-1)).
Statistics calculator
Margin of Error Calculator
Calculate the modeled half-width around a mean or proportion estimate and preview the resulting interval.
Last reviewed: August 6, 2026Statistics calculation engine v2.0.0
Margin of Error Calculator
Calculate the modeled half-width around a mean or proportion estimate and preview the resulting interval.
Formula and assumptions
Input assumptions
The selected standard-error model and critical value are appropriate. Sampling is sufficiently independent and representative. Finite correction assumes sampling without replacement from known N.
Precision note
Statistics calculations use Decimal.js for deterministic arithmetic; displayed values are rounded only after the selected method is evaluated.
How NexaCalc evaluates the inputs
Parse
Finite numeric observations and method-specific inputs are validated locally.
Calculate
The selected Statistics method runs in a pure TypeScript engine.
Explain
The result includes substitutions, ordered data, supporting measures and method warnings.
Visualize
A tool-specific dataset visual appears only after a valid calculation.
What the Margin of Error Calculator does
Use this Statistics page to connect confidence, variability and sample size to sampling precision.
Margin of Error Calculator processes the entered values locally and exposes the selected convention, calculation steps and related values.
When to use the Margin of Error Calculator
Use this Statistics page to connect confidence, variability and sample size to sampling precision.
Choose the mode that matches the data available and the question being answered before entering values.
Formula, variables and units
Proportion E=z sqrt(p(1-p)/n). Mean E=z sigma/sqrt(n). Optional FPC=sqrt((N-n)/(N-1)).
- E is margin of error.
- z is the confidence critical value and n is sample size.
- p or sigma describes variability; N is optional finite population size.
Statistical assumptions
The calculation is deterministic, but interpretation depends on whether the selected model and entered data fit the real situation.
- The selected standard-error model and critical value are appropriate.
- Sampling is sufficiently independent and representative.
- Finite correction assumes sampling without replacement from known N.
Step-by-step worked example
Use the default example inputs to reproduce the primary worked example. The result breakdown is produced by the same pure TypeScript engine used by the page.
- For p=0.5, n=400 and 95% confidence, E is about 0.049 or 4.9%.
- For sigma=10 and n=100, E is about 1.96.
- With N=1000 and n=400, FPC is about 0.775 and adjusted proportion E is about 0.038.
Additional examples and interpretation
The following examples show how data, event structure, count or method changes the result.
- For p=0.5, n=400 and 95% confidence, E is about 0.049 or 4.9%.
- For sigma=10 and n=100, E is about 1.96.
- With N=1000 and n=400, FPC is about 0.775 and adjusted proportion E is about 0.038.
Concept explanation
Margin of error is half the width of the corresponding symmetric confidence interval under the selected model.
Larger samples reduce random sampling error at a square-root rate, while higher confidence widens the margin.
How to read the statistics visual
The dynamic visual is a point estimate with symmetric error arrows and interval endpoints. It appears only after a valid calculation and uses the current result.
Visual proportions are normalized for readability and are explanatory rather than a substitute for a full statistical plot.
Common mistakes
Check dataset parsing, method choice and denominator before using a result.
- Confusing margin of error with total interval width.
- Ignoring units for mean estimates.
- Assuming margin covers bias and nonresponse.
Limitations and method-choice notes
Statistical conventions and model assumptions can change a result or its interpretation.
- Margin of error describes modeled random sampling variability only.
- Coverage, measurement, response and design errors are not included.
Educational and privacy note
This calculator provides educational statistics calculations and does not replace statistical consulting, instructor guidance or professional analysis.
Input values are processed in the browser calculator session and are not sent to an external statistics service.
Frequently asked questions
What does the Margin of Error Calculator calculate?
It returns the modeled half-width and previews lower and upper interval endpoints.
What inputs does the Margin of Error Calculator require?
confidence level, sample size and either a proportion or mean variability inputs
Which formula does the Margin of Error Calculator use?
E=z sqrt(p(1-p)/n) or E=z sigma/sqrt(n), with optional finite-population correction
How should I interpret the Margin of Error Calculator result?
It returns the modeled half-width and previews lower and upper interval endpoints. Interpret it only under the assumptions stated on the page.
What assumptions are important for the Margin of Error Calculator?
Margin of error excludes bias, nonresponse, coverage error and unsuitable design; it is not total survey error.
What common mistake should I avoid in the Margin of Error Calculator?
Select the mode that matches the question, keep units or probabilities consistent and do not treat a model output as stronger evidence than the inputs support.
How does the Margin of Error Calculator differ from the Confidence Interval Calculator?
The tools answer related but different questions. The Margin of Error Calculator follows E=z sqrt(p(1-p)/n) or E=z sigma/sqrt(n), with optional finite-population correction, while the Confidence Interval Calculator reports its own named quantity or model.
Can I use decimal values in the Margin of Error Calculator?
Yes where the selected model permits them. Whole-number count fields reject decimals, while measurements, probabilities and dataset values accept finite decimals.
Does the Margin of Error Calculator prove a statistical conclusion?
No. It performs a deterministic calculation; study design, data quality and subject-matter interpretation remain separate.
How does the Margin of Error Calculator visual help?
The visual uses the current inputs and result to show the model structure, selected region or interval. It appears only after a valid calculation.
Why might another Margin of Error Calculator return a different value?
Different conventions, critical values, rounding rules or event assumptions can change results. NexaCalc labels the selected method and rounds only for display.
What are the main limitations of the Margin of Error Calculator?
Margin of error excludes bias, nonresponse, coverage error and unsuitable design; it is not total survey error.
References
- OpenStax Introductory Statistics 2e, Measures of the Center of the Data. Relevance: mean, median and mode. Last verified: August 5, 2026. Source.
- OpenStax Introductory Statistics 2e, Measures of the Spread of the Data. Relevance: range, variance and standard deviation. Last verified: August 5, 2026. Source.
- NIST/SEMATECH e-Handbook, Measures of Scale. Relevance: statistical spread and scale measures. Last verified: August 5, 2026. Source.
- NIST/SEMATECH e-Handbook, Percentiles. Relevance: ordered data, ranks and percentile-method differences. Last verified: August 5, 2026. Source.
- OpenStax Introductory Statistics 2e, Two Basic Rules of Probability. Relevance: conditional, multiplication and addition rules. Last verified: August 6, 2026. Source.
- OpenStax Introductory Statistics 2e, Poisson Distribution. Relevance: discrete event-count probability models. Last verified: August 6, 2026. Source.
- OpenStax Introductory Statistics 2e, Confidence Intervals chapter review. Relevance: interval estimation, margin of error and sample size. Last verified: August 6, 2026. Source.
Statistics methods and references reviewed on August 6, 2026.
Educational disclaimer
This calculator provides mathematical results from the values, conventions and methods you enter. Verify important academic, engineering or professional work independently.