Normal and NiMlInRH Distributions restart; with(plots); with(stats); with(statevalf); with(pdf); We first wish to compare the normal distribution (red) with the NiMlInRH distribution with 9 degrees of freedom (blue). p0:=plot(normald(x),x=-5..5,color=red): p1:=plot(studentst(x),x=-5..5,color=blue): display(p0,p1); We see the normal curve is higher in the middle but lower on the extremes. Let's compute the difference of the curves. plot(normald(x)-studentst(x),x=-4..4); Next let's compare the normal distribution (red) with the NiMlInRH distribution with 24 degrees of freedom (blue), which we would often use when NiMvJSJuRyIjRA==. p0:=plot(normald(x),x=-5..5,color=red): p2:=plot(studentst(x),x=-5..5,color=blue): display(p0,p2); We have the same relationship as before, but the curves are much closer. Let's again compute the difference of the curves. plot(normald(x)-studentst(x),x=-4..4); With this small difference, you can see why we often, for convenience sake, use NiMlInpH instead of NiMlInRH for large populations (when NiMlIm5H is greater than or equal to 25). What is the total accumulated absolute error in using NiMlInRH instead of NiMlInpH when NiMvJSJuRyIjRA==? dif:=abs(normald(x)-studentst(x)); int(dif,x=-infinity..infinity); evalf(%); This is pretty small. Let's see what happens when NiMvJSJuRyIjSQ==. dif:=abs(normald(x)-studentst(x)); int(dif,x=-infinity..infinity); evalf(%); This total accumulated absolute error will keep getting smaller as NiMlIm5H increases.