# Confidence Interval Calculator

> Two-sided Student t interval for a population mean with unknown variance. Enter confidence level explicitly; assumptions stated.

Page: https://freecalculator.tech/tools/mean-confidence-interval-calculator
Last reviewed: 2026-10-23

## Formula

Two-sided mean CI = x̄ ± t(1 − α/2, n − 1) × s/√n; α = 1 − confidence level.

- Methodology type: scientific equation.
- Methodology: NIST one-sample Student t confidence interval for a population mean with unknown variance; t quantiles computed by jStat.
- Independent approximately normal observations required for small samples. Severe skew or outliers can invalidate this interval; there is no automatic normality assurance.
- Confidence describes repeated-sampling coverage, not the probability of a fixed population mean or the fraction of individual observations in the interval.
- Two-sided mean CI = x̄ ± t(1 − α/2, n − 1) × s/√n; α = 1 − confidence level.
- Inputs used: Observations, Two-sided confidence level (enter explicitly).

## Inputs

- Observations
- Two-sided confidence level (enter explicitly)

## Worked example: Mean confidence interval

Mean = 5, SE ≈ 1.2909944, df = 3. t₀.₉₇₅,₃ ≈ 3.1824463; margin ≈ 4.1085205. 95% CI ≈ [0.8914795, 9.1085205].
1. Mean = 5, SE ≈ 1.2909944, df = 3.
2. t₀.₉₇₅,₃ ≈ 3.1824463; margin ≈ 4.1085205.
3. 95% CI ≈ [0.8914795, 9.1085205].

## Common questions

### Why use t instead of 1.96?

Population variance is unknown; the critical value depends on sample size and your confidence level. Check the formula, example, and limitations on this page before using the result for a real statistics decision.

### Is this a prediction interval?

No. It concerns the population mean, not a future individual observation.

### Does it handle proportions?

No. Proportion intervals, clustering and paired designs need different models.

### How do I use the Confidence Interval Calculator?

Enter the required values for Observations, Two-sided confidence level (enter explicitly). The calculator applies the formula on this page and shows the main result with any supporting breakdown so you can check the arithmetic.

### What formula does the Confidence Interval Calculator use?

Two-sided mean CI = x̄ ± t(1 − α/2, n − 1) × s/√n; α = 1 − confidence level. The visible formula section above lists the calculation path and the edge cases the page handles, so the result can be checked without relying on the form alone.

### Can the Confidence Interval Calculator be used for exact decisions?

Use it as a calculation aid, not as a substitute for checking the underlying rule, contract, policy, or professional advice that applies to your situation. When a result depends on local rules, personal details, prices, or dates, enter those values yourself and confirm them before acting.

## Limitations and assumptions

- Independent approximately normal sample; no robust or nonparametric method.
- No clinical inference, prediction interval or causal conclusion.
- For education and operational analysis only. No diagnosis, treatment recommendations, prescribing, dosing instructions, or reimbursement entitlement. Scientific calculations require appropriate laboratory verification; operational estimates do not replace local clinical, safety or financial policies.
- The result depends on the values you enter for this confidence interval calculator; it does not supply missing rates, rules, prices, dates, or assumptions for you.

## Sources

- NIST — https://www.itl.nist.gov/div898/handbook/prc/section2/prc221.htm

## Cite this page

- APA: Free Calculator. (2026, October 23). Confidence Interval Calculator. https://freecalculator.tech/tools/mean-confidence-interval-calculator
- MLA: "Confidence Interval Calculator." Free Calculator, 23 Oct. 2026, https://freecalculator.tech/tools/mean-confidence-interval-calculator.
- Link: Confidence Interval Calculator — Free Calculator, https://freecalculator.tech/tools/mean-confidence-interval-calculator