Confidence Interval Calculator
Two-sided Student t interval for a population mean with unknown variance. Enter confidence level explicitly; assumptions stated.
In short
Two-sided Student t interval for a population mean with unknown variance. Enter confidence level explicitly; assumptions stated.
Formula: Two-sided mean CI = x̄ ± t(1 − α/2, n − 1) × s/√n; α = 1 − confidence level.
What this calculator does
Two-sided Student t interval for a population mean with unknown variance. Enter confidence level explicitly; assumptions stated. Formula, worked example, sources, FAQs and limitations included.
Use it to turn Observations, Two-sided confidence level (enter explicitly) into a checked result you can compare, copy, or rerun with different assumptions.
The page shows the formula, a numeric worked example, and the assumptions that affect this statistics calculation.
Inputs and what they mean
- Observations
- — text value.
- Two-sided confidence level (enter explicitly)
- — number value.
How to use it
- Enter each value in the unit shown next to the box (use the unit converter first if your numbers are in other units).
- Check the breakdown to see every intermediate step.
- Read the limitations before relying on the result.
Formula
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).
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].
Reading the result
The headline figure is the main answer. Any breakdown underneath shows the parts that make it up, so you can check the working and see what changes when you adjust an input.
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.
Reference: NIST
Last reviewed:
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.
Sources
Method based on NIST.
Cite this page
Free Calculator. (2026, October 23). Confidence Interval Calculator. https://freecalculator.tech/tools/mean-confidence-interval-calculator
"Confidence Interval Calculator." Free Calculator, 23 Oct. 2026, https://freecalculator.tech/tools/mean-confidence-interval-calculator.
Confidence Interval Calculator — Free Calculator, https://freecalculator.tech/tools/mean-confidence-interval-calculator
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