T-Distributions (Degrees of Freedom) – AP Statistics

Learning Objectives

Small Sample Uncertainty: When $n < 30$ and we don't know the true population standard deviation $\sigma$, we must use our sample's standard error $s / \sqrt{n}$. This adds massive uncertainty to our model.

Fat Tails: The t-distribution accounts for this uncertainty by physically squashing the top of the normal bell curve and pushing that probability into the edges ("fat tails"), meaning extreme outliers are much more common.

Degrees of Freedom ($n-1$): As sample size grows ($df \to \infty$), our estimate of $s$ becomes perfect, and the fat t-curve morphs exactly into the tight Normal $Z$-curve!

Tags

t-TestDegrees of FreedomFat TailsNormal

       STUDENT'S T TEST STATISTIC

       $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

t-Curve ($df$)

Normal Z-Curve (Reference)

Sample Size Control

Sample Size ($n$) = 2

CURVE PROPERTIES

Degrees of Freedom: df = 1

CRITICAL VALUE ($95\%$ CONFIDENCE)

Normal $Z^*$: 1.960

Student's $t^*$: 12.706

Extreme Caution ($df=1$): Look how massive the $t^*$ value is (12.7)! Because $n=2$ is such a tiny sample, we are completely blind to the true standard deviation. The curve is smashed down and the tails stretch brutally wide.