Student-t Mean Confidence Interval
Confidence interval for a population mean from a small sample — uses the t-distribution, not the normal.
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The engineering
When you estimate a population mean from a small sample and don't know the true variance, the sampling distribution has fatter tails than the normal — that's the Student-t correction. This card returns the two-sided interval x̄ ± t·s/√n along with the standard error and the t critical value for your chosen confidence level and n−1 degrees of freedom.
The t-inflation matters most for small n: at n=5 the 95% t factor is 2.78 versus the normal 1.96, so ignoring it understates your interval by ~40%. By n≈30 the two nearly coincide. If your data are wildly non-normal or you have outliers, the interval is only as good as the mean itself — check a histogram first.
This implementation gets the t-quantile from an inverse-normal plus a Cornish-Fisher series; it's accurate to a few parts in 10⁴ for df ≥ 3, well inside bench tolerance.
Where this math comes from
William Sealy Gosset derived the distribution in 1908 while working as a brewer at Guinness in Dublin, wrestling with tiny sample sizes from batches of barley and yeast. Guinness barred employees from publishing trade research, so he wrote under the pseudonym 'Student' — and the name stuck to the distribution forever.
Gosset's paper 'The Probable Error of a Mean' gave engineers a way to reason honestly about uncertainty when only a handful of measurements are available. Ronald Fisher later put it on rigorous footing and connected it to degrees of freedom, and the two corresponded for decades over the practical statistics that QA and reliability work still rely on.
- 1908W. S. Gosset ('Student')Publishes the t-distribution for small-sample inference, writing under a pseudonym for Guinness.
- 1925Ronald A. FisherFormalizes degrees of freedom and popularizes the t-test in Statistical Methods for Research Workers.
- 1988P. J. Acklam / numerical communityCompact rational approximations make the inverse-normal (and thus t) fast to evaluate on any machine.
- 2012NIST/SEMATECHe-Handbook of Statistical Methods codifies the t-based confidence interval as standard QA practice.
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