Estimate the range where a population parameter likely falls given sample data and confidence level.
Overview
A confidence interval translates sample data into a range of plausible population values. Instead of claiming the sample mean is the population mean, you report an interval plus a confidence level, such as 95%.
For large samples with known or stable standard deviation, the normal approximation uses CI = x̄ ± z × (σ/√n). Smaller samples or unknown variance often require the t distribution with degrees of freedom n − 1.
Wider intervals mean more uncertainty. They shrink when sample size grows because √n appears in the denominator. Misinterpreting confidence level as “probability the parameter is inside” is common; the correct interpretation is about the method’s long-run capture rate.
Use this calculator after checking assumptions: random sampling, approximate normality or large n, and correct standard error definition. Report units and whether σ is population or sample-based.
Steps
Input x̄ from your data.
Use sample SD or known σ as appropriate.
Provide n > 0.
Common choices: 90%, 95%, 99%.
Use t for small samples with estimated SD.
Review lower and upper limits around the mean.
Math
CI = x̄ ± z × (σ / √n) or x̄ ± t × (s / √n)
Standard error σ/√n measures uncertainty of the mean. Multiplying by critical value z or t creates a margin of error added and subtracted from x̄.
Worked cases
ME = 1.96 × (12/8) = 2.94
Interval [77.06, 82.94].
SE = 12/4 = 3; t₀.₀₂₅,15 ≈ 2.131
Wider interval than large-n case.
Higher z increases margin of error
99% CI is wider than 95% CI.
Why use this
Questions