Confidence Interval Calculator
Calculate a two-sided Student t confidence interval for a population mean from sample statistics.
Last updated · View algorithm, constants and default inputs (JSON)
Confidence interval
About this calculator
This interval estimates a population mean from a sample mean and sample standard deviation using a two-sided Student’s t critical value. It assumes independent random observations and, especially for small samples, an approximately normal population. A 95% confidence level describes the long-run performance of the method, not a 95% probability for this one interval.
Worked example
calculated with the default values below
Sample mean 100 · Sample standard deviation 15 · Sample size 100 · Confidence level 95% Confidence interval 97.024 – 102.976 · Lower bound 97.024 · Upper bound 102.976 · Interval width 5.953 · Margin of error 2.976 · Standard error 1.5000 · Critical value 1.9843
General background references
Disclaimer
These calculators give general mathematical results. They suit study, planning and estimation — verify anything that feeds a graded submission, a published figure or a financial commitment.
Implementation source
This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.
Formula and variables
CI = x̄ ± t* × s ÷ √n
- x̄
- sample mean
- s
- standard deviation
- n
- sample size
- t*
- critical value
- CI
- confidence interval
Two-sided Student t confidence interval for a mean: mean ± t* × s/√n. Assumes independent observations and approximate normality, especially for small samples; confidence describes long-run coverage of the method.
Frequently asked questions
What does ninety-five percent confidence really mean?
It describes the method, not your particular interval. If you repeated the whole study many times, ninety-five percent of the intervals constructed this way would contain the true value. It is not a ninety-five percent probability that the true value sits in the interval you happen to have calculated — a distinction that is widely got wrong in reporting.
Why does a larger sample give a narrower interval?
Because precision improves with the square root of the sample size. Quadrupling the sample halves the width, so getting a twice-as-precise estimate costs four times the data — and beyond a point the cost stops being worth it. Note that the interval narrows around whatever your sample shows, which is no help at all if the sample itself is biased. The sample standard deviation is an estimate of population spread, so this calculator uses Student’s t critical values. For small samples, the t value is larger than the corresponding normal value and makes the interval wider; the difference shrinks as the sample grows.