Mistake Master
The Mean Is Steadier by Root n, and t Is the Price of Guessing Sigma
You'll learnwhere the sample mean's center, spread and shape come from, why the spread divides by the square root of n rather than n, and why replacing an unknown sigma with the sample's s moves every procedure from z onto the heavier-tailed t distribution.
A repair shop's job times are strongly right-skewed: most jobs are quick and a few drag on. Average 25 of them and the average is a different random variable from the jobs — same center, but steadier by exactly the square root of 25, and nearly symmetric even though the population never changes shape at all. This page computes that distribution exactly, every possible sample weighted by its probability, and watches the three properties arrive: the center is the population mean untouched, the spread is sigma divided by root n, and the skew dies off at rate one over root n, which is the Central Limit Theorem measured rather than promised. Then the real complication: sigma is almost never known, the sample's own s stands in, and a ratio whose denominator wobbles from sample to sample needs a wider curve than the normal. The t distribution is that accounting, and this page re-derives its critical value table by integration instead of quoting it. Twelve short steps; press Next for each one.