Confidence Interval Calculator
A confidence interval expresses how much the estimate could move if you repeated the study. In reliability work the one-sided lower bound usually matters more than the two-sided interval — nobody is worried that equipment might be more reliable than measured.
Formula
Worked example
Twenty-five measurements give x̄ = 48.2 and s = 3.1. Construct a 95% confidence interval on the mean.
- df = 24, t(0.025, 24) = 2.0639
- Standard error = 3.1 / √25 = 3.1 / 5 = 0.620
- Margin = 2.0639 × 0.620 = 1.2796
- CI = 48.2 ± 1.28 = [46.92, 49.48]
The interval is [46.92, 49.48]. Quadrupling the sample to 100 would only halve the width — precision improves with √n, which is why the jump from a pilot study to a confirmatory study is expensive.
Common mistakes
- Reading a 95% interval as a 95% probability that the true mean lies inside it. The frequentist statement is about the procedure across repeated samples, not about this one interval.
- Using z instead of t for small samples with an estimated standard deviation. Below about n = 30 this understates the interval.
- Using a two-sided interval when the decision is one-sided. A 95% two-sided interval corresponds to a 97.5% one-sided bound.
- Applying the Wald proportion interval to zero-failure data, where it returns a nonsensical zero-width interval.
Related tools
About Reliability Pro
Confidence Interval is part of a suite of 126 reliability, maintenance and quality engineering tools covering life data analysis, accelerated testing, system reliability, FMEA and root cause, SPC, and design for reliability. It runs in the browser and as native iOS and Android apps, so the same calculation is available at a desk or in front of the asset.