Inverting the Chi-squared, t and F distributions on the TI83/84
The TI 83/84 series of calculators has CDF’s for the Chi-Squared, t, and F distributions. These give the probability between
a pair of scores for a given number of degrees of freedom. However, the calculator has no inverse functions that would give
critical scores as functions of a given probability.
Using Newton’s method an iteration to locate χν2,α can be implemented.
Let H ( x ) = χ 2cdf ( 0, x,ν ) − 1 + α , the only positive root of this function is x = χν2,α . From the fundamental theorem of
calculus, H ′ ( x ) =
d  2
d x 2
χ cdf ( 0, x,ν ) − 1 + α  =
χ cdf ( t ,ν ) dt = χ 2 cdf ( x,ν ) . Thus, the Newton’s iteration to find this
 dx 0
dx 
root is xn +1 = xn −
H ( xn )
χ 2cdf ( 0, xn ,ν ) − 1 + α
= xn −
.
H ′ ( xn )
χ 2 pdf ( xn ,ν )

The following program implements this iteration: Here A is α and N is ν . Prompt and Disp are I/O commands, the other
commands are CTL. This same procedure can be used to invert the t and F distributions.
: Prompt A
: Prompt N
: N→X
: For (J, 1, 9)
(
)
: X − χ 2 cdf ( 0, X, N ) + A − 1 / χ 2 pdf ( X,N ) → X
: End
: Disp X
: Stop
Al Lehnen Madison Area Technical College
10/31/2013