Sampling Distributions for Sample Means

For a simple random sample, $\mu_{\bar{x}} = \mu$ and $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$, and the shape is normal when the population is normal or approximately normal when $n$ is large enough for the Central Limit Theorem. Because $\sigma$ is essentially never known, $s$ replaces it, giving the standard error $\frac{s}{\sqrt{n}}$, and $\frac{\bar{x} - \mu}{s/\sqrt{n}}$ follows a t distribution with $df = n - 1$: symmetric, centered at 0, heavier-tailed than the normal, and approaching it as df grows without reaching it.

The spread gets rebuilt wrong: $\frac{s}{n}$ instead of $\frac{s}{\sqrt{n}}$, or the data's own spread of 9.2 reported as the mean's spread of 1.84, or the sample's $s$ expected to shrink toward zero as $n$ grows when it is estimating a fixed $\sigma$. And the t distribution gets treated as cosmetic: a z critical value used where t belongs, $n$ used for df instead of $n - 1$, or t and z called interchangeable at $n = 12$, where $t^{*} = 2.201$ exceeds $z^{*} = 1.960$ by 12%.