Student's

This is a simple description of "Studentizing" a normal curve. Once again
Mathcad can really simplify some difficult concepts in analysis. Nothing new here
but the idea is worthwhile to address in relation to HTS. The red curve is a normal
curve, the blue curve is Student's with 2 degrees of freedom and the green curve
is Student's with 5 degrees of freedom. Notice how Student's curves approach the
normal curve with progressively more degrees of freedom. Compare this with the
Access Predictor add-in.

Student's vs Normal Curve

Intuitively, what is going on here? Why do small datasets generate heavy tails
in a normal curve? Why are confidence intervals extended?... If you have many
points you can draw a circle that "tightly" fits around the distribution of points.
If you have a new set of points generated in the same fashion as the circle with
many points but you only get three points to work with... Where is the center of
the circle located? With three points the circle loosely fits the three points. The
circle has a freedom of fit. You do have a certain confidence where the center
of the circle might be.