Primary formula
Mean: xbar +/- critical(deviation/sqrt(n)). Proportion: phat +/- z sqrt(phat(1-phat)/n).
Statistics calculator
Confidence Interval Calculator
Estimate an interval around a sample mean or proportion and inspect its standard error and margin of error.
Last reviewed: August 6, 2026Statistics calculation engine v2.0.0
Confidence Interval Calculator
Estimate an interval around a sample mean or proportion and inspect its standard error and margin of error.
Formula and assumptions
Input assumptions
Observations are suitably sampled and independent. The selected standard-error model is appropriate. Entered critical values match the intended tail area and degrees of freedom.
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 Confidence Interval Calculator does
Use this Statistics page to form common two-sided mean or proportion intervals under stated assumptions.
Confidence Interval Calculator processes the entered values locally and exposes the selected convention, calculation steps and related values.
When to use the Confidence Interval Calculator
Use this Statistics page to form common two-sided mean or proportion intervals under stated assumptions.
Choose the mode that matches the data available and the question being answered before entering values.
Formula, variables and units
Mean: xbar +/- critical(deviation/sqrt(n)). Proportion: phat +/- z sqrt(phat(1-phat)/n).
- The point estimate is xbar or phat.
- n is positive whole-number sample size.
- The critical value corresponds to confidence level and interval method.
Statistical assumptions
The calculation is deterministic, but interpretation depends on whether the selected model and entered data fit the real situation.
- Observations are suitably sampled and independent.
- The selected standard-error model is appropriate.
- Entered critical values match the intended tail area and degrees of freedom.
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.
- Mean 50, population SD 10, n=100 and z about 1.96 gives margin about 1.96.
- The resulting mean interval is about 48.04 to 51.96.
- Proportion 0.6 with n=400 gives margin about 0.048 and interval about 0.552 to 0.648.
Additional examples and interpretation
The following examples show how data, event structure, count or method changes the result.
- Mean 50, population SD 10, n=100 and z about 1.96 gives margin about 1.96.
- The resulting mean interval is about 48.04 to 51.96.
- Proportion 0.6 with n=400 gives margin about 0.048 and interval about 0.552 to 0.648.
Concept explanation
A confidence interval is generated by a procedure designed to cover the target parameter at a stated long-run rate under its assumptions.
Its half-width is the margin of error, which grows with confidence and variability and shrinks with sample size.
How to read the statistics visual
The dynamic visual is a point estimate centered between lower and upper confidence bounds. 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.
- Interpreting confidence as a posterior probability for one interval.
- Using a z critical value when a t method is required without justification.
- Ignoring sampling bias.
Limitations and method-choice notes
Statistical conventions and model assumptions can change a result or its interpretation.
- An interval does not repair nonresponse, measurement error or selection bias.
- Simple proportion intervals can perform poorly near 0 or 1 or with small samples.
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 Confidence Interval Calculator calculate?
It returns lower and upper bounds, standard error and margin of error.
What inputs does the Confidence Interval Calculator require?
a mean or proportion, sample size, confidence level and the applicable standard deviation or t critical value
Which formula does the Confidence Interval Calculator use?
point estimate plus or minus a critical value times standard error
How should I interpret the Confidence Interval Calculator result?
It returns lower and upper bounds, standard error and margin of error. Interpret it only under the assumptions stated on the page.
What assumptions are important for the Confidence Interval Calculator?
Coverage statements rely on the sampling method and model; this interval does not account for bias or guarantee the parameter lies inside.
What common mistake should I avoid in the Confidence Interval 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 Confidence Interval Calculator differ from the Margin of Error Calculator?
The tools answer related but different questions. The Confidence Interval Calculator follows point estimate plus or minus a critical value times standard error, while the Margin of Error Calculator reports its own named quantity or model.
Can I use decimal values in the Confidence Interval 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 Confidence Interval Calculator prove a statistical conclusion?
No. It performs a deterministic calculation; study design, data quality and subject-matter interpretation remain separate.
How does the Confidence Interval 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 Confidence Interval 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 Confidence Interval Calculator?
Coverage statements rely on the sampling method and model; this interval does not account for bias or guarantee the parameter lies inside.
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.