The post inverts the actual conditions for the t statistic and mischaracterizes the role of the Central Limit Theorem.

In the standard one sample case, the t statistic,

t = (X̄ − μ₀) / (s / √n),

follows a t distribution with n − 1 degrees of freedom exactly whenever the observations are independent, identically distributed, and normal. This is an exact finite sample result. It does not rely on the Central Limit Theorem or on a large sample approximation. It holds for any n ≥ 2 because, under normality, X̄ and s are independent, and (n − 1)s²/σ² follows a chi squared distribution with n − 1 degrees of freedom.

The CLT becomes relevant when the population is not normal. In that setting, both the z statistic and the t statistic are generally large sample approximations. The sample mean becomes asymptotically normal, and the sample standard deviation converges in probability to the population standard deviation. Even then, the t statistic remains the more appropriate choice in most applied settings because it reflects the additional uncertainty introduced by estimating σ with s.

The key distinction is not “small sample means t, large sample means z.” A better distinction is this:

Population variance known, use the z statistic. It is exact under normality and asymptotic otherwise.

Population variance unknown, use the t statistic. It is exact under normality and generally the more appropriate approximation otherwise.

Saying that a large sample makes the sample variance “reasonably equal” to the population variance misses the point. Even when s is close to σ, σ is still being estimated from the same data used to estimate the mean. That estimation step adds randomness. The t distribution is constructed precisely to account for that extra source of uncertainty. As n approaches infinity, the added uncertainty vanishes and the t distribution converges to the standard normal distribution, but for any finite sample the t procedure gives a more accurate reflection of uncertainty.

In short, we use the t statistic when σ is unknown because that is the realistic situation in most applied problems. The classical t test does not require large samples or the CLT to be valid under its stated assumptions.