Worked example
Known-sigma mean
- Input
- x̄ = 100; σ = 15; n = 100; 95%
- Output
- ≈ [97.0601, 102.9399]
The z critical value is about 1.96.
Confidence Interval Calculator
Choose an interval method that matches the information available instead of applying one formula to every sample. Known population standard deviation uses a z critical value, an estimated mean uses Student’s t, and an integer success count with sample size uses the Wilson interval.
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How to use this calculator
Enter interval method, confidence level, sample size, and either the mean with standard deviation or the whole-number success count required by that method.
Review sampling independence, population shape, how the standard deviation was obtained, sample design, and whether the chosen level was prespecified.
Read lower confidence bound, upper confidence bound, margin of error, standard error, point estimate, and critical value as applicable and compare the displayed assumptions before reusing the answer.
Formula
Mean interval = sample mean ± critical value × standard deviation ÷ √n. The t method uses n − 1 degrees of freedom. The Wilson proportion interval adjusts both center and half-width by z² ÷ n instead of using the fragile Wald interval.
Worked example
For mean 100, known standard deviation 15, n = 100, and 95% confidence, the z interval is about 97.06 to 102.94 with a margin near 2.94.
Reference guide
Formula details and result guidance
Follow the variables, derivation, precision rule, and worked cases below to reproduce the calculation independently.
Variables
| Symbol | Meaning | Unit | Role |
|---|---|---|---|
| θ̂ | Sample estimate | measured quantity or proportion | Input |
| x, n | Successful observations and sample size | whole observations | Input |
| SE | Standard error | same as estimate | Intermediate |
| q* | Critical quantile | none | Intermediate |
| L, U | Lower and upper bounds | same as estimate | Output |
Derivation
- A z mean interval with known population standard deviation uses x̄ ± z*σ/√n.
- A t mean interval with sample standard deviation uses x̄ ± t*s/√n with n − 1 degrees of freedom.
- For a proportion, the Wilson interval adjusts both center and half-width and remains within [0, 1].
Rounding and precision
Critical values and standard errors are retained without intermediate display rounding; bounds show up to 6 decimal places. The selected confidence level is a repeated-sampling coverage target, not a probability assigned to fixed bounds after calculation.
Worked and boundary examples
Worked example
Small-sample t interval
- Input
- x̄ = 10; s = 2; n = 16; 95%
- Output
- ≈ [8.934, 11.066]
Using t accounts for estimating variability with 15 degrees of freedom.
Boundary case
No observed successes
- Input
- x = 0; n = 20; Wilson 95%
- Output
- Lower bound 0; upper bound above 0
Wilson avoids a misleading zero-width interval at the boundary.
Common mistakes
- Using a z interval with a sample standard deviation as if population variability were known.
- Interpreting 95% confidence as a 95% posterior probability for the already computed interval.
- Ignoring sampling design, dependence, bias, or nonrepresentative data because the arithmetic produces bounds.
How to interpret the result
Under the selected model and repeated use of the procedure, the stated share of intervals is designed to cover the parameter. The single computed interval still depends on the sample and assumptions.
Scope
Appropriate for
- Reproducing a z, t, or Wilson interval from summary statistics.
- Comparing how sample size and confidence level change interval width.
Do not use for
- Repairing biased or dependent samples through arithmetic alone.
- Making clinical, policy, or safety decisions without subject-matter review.
Tools for the same task
Next tools in the workflow
Evidence and sources
- Mean confidence intervals use an estimate plus or minus a critical value times a standard error.National Institute of Standards and Technology: Confidence Limits for the MeanSupports: Student t confidence limits for a mean when population variability is estimated. Reviewed 2026-08-22.
- Standard-normal quantiles supply critical values for z intervals.National Institute of Standards and Technology: Confidence IntervalSupports: Normal critical-value intervals for a mean with known population standard deviation. Reviewed 2026-08-22.
- The Wilson score interval uses the observed success count and sample size to estimate a binomial proportion.National Institute of Standards and Technology: Proportions Control Charts — Wilson IntervalSupports: The Wilson score confidence bounds for a binomial proportion. Reviewed 2026-08-22.
- Student t critical values depend on probability and degrees of freedom.National Institute of Standards and Technology: Critical Values of the Student t DistributionSupports: Reference critical values for selected Student t probabilities and degrees of freedom. Reviewed 2026-08-22.
Assumptions and limits
- The interval quantifies repeated-sampling uncertainty under the selected model. It does not mean the fixed parameter has the displayed probability of lying inside this one computed interval.
- Confidence Interval Calculator uses only the values entered in the form and does not infer an institution, policy, contract, population, or project condition.
- Working calculations retain full numeric precision; display rounding is separate and should not be copied into a later calculation when the unrounded value is available.
Sources
- National Institute of Standards and Technology: Confidence Limits for the MeanStudent t confidence limits for a mean with estimated variability. Reviewed 2026-08-22.
- National Institute of Standards and Technology: Confidence IntervalNormal critical-value intervals for a mean with known population standard deviation. Reviewed 2026-08-22.
- National Institute of Standards and Technology: Proportions Control Charts — Wilson IntervalWilson score interval bounds for a binomial success count and sample size. Reviewed 2026-08-22.
- National Institute of Standards and Technology: Critical Values of the Student t DistributionReference t critical values by probability and degrees of freedom. Reviewed 2026-08-22.
- National Institute of Standards and Technology: Normal DistributionOfficial formula or method source used by this calculator. Reviewed 2026-08-22.
Related calculators
Verification
How this calculator is checked
Trust comes from reproducible evidence, not model confidence. You can inspect the formula, assumptions, worked example, and cited sources on this page.
- Versioned calculationFormula logic is kept separate from the interface and covered by automated registry and behavior checks.
- Cited evidenceSources linked for independent checking: 5.
- Transparent AI useAI may assist drafting or adversarial review. Agreement between models is not proof, and no human expert review is claimed unless a named reviewer is shown.
Frequently asked questions
How does the Confidence Interval Calculator work?
Mean interval = sample mean ± critical value × standard deviation ÷ √n. The t method uses n − 1 degrees of freedom. The Wilson proportion interval adjusts both center and half-width by z² ÷ n instead of using the fragile Wald interval.
What should I enter in the Confidence Interval Calculator?
Enter interval method, confidence level, sample size, and either the mean with standard deviation or the whole-number success count required by that method, then verify sampling independence, population shape, how the standard deviation was obtained, sample design, and whether the chosen level was prespecified.
When should I avoid relying only on the Confidence Interval Calculator?
The interval quantifies repeated-sampling uncertainty under the selected model. It does not mean the fixed parameter has the displayed probability of lying inside this one computed interval.
Last updated 2026-08-25 · mean-z-mean-t-wilson-counts-v2 · content-2026-08-22 · Published by YunFanLabs