The t-Distribution

Every confidence interval and test for a mean needs to know how spread out the population is. Usually you don't know sigma, so you estimate it with the sample standard deviation s. But s is itself a number computed from your one sample, and it jiggles from sample to sample. That extra layer of guessing makes your estimate of the mean's spread less reliable than the normal curve assumes. The t-distribution is built to absorb that extra wobble: it looks like the normal curve but with fatter tails, so a wider range of values counts as plausible.

The exact shape of the t-curve depends on degrees of freedom, df = n - 1, which tracks how much information the sample carries about the spread. With few degrees of freedom, s is a shaky estimate of sigma, so the tails are noticeably fatter than normal, and critical values are larger. As df climbs, s settles down and becomes a trustworthy stand-in for sigma, so the t-curve's tails thin out and the whole curve slides back toward the standard normal. By the time df hits a couple hundred, t and z are nearly identical.

This sets up every mean-based procedure that follows: one-sample t-intervals, t-tests, paired t-tests, and eventually two-sample t-procedures all lean on this same curve and the same df logic. The question is never just 'is my sample small,' it's 'am I estimating sigma.' Small samples from a population where sigma is somehow known still use z; any sample, large or small, where sigma is estimated from s uses t.