Computing Quantiles Using the TI-83+/84
Let’s compute quantiles for the normal, chi-square, and t distributions. We will use the invNorm, c2 cdf
and tcdf commands (found under DISTR).
What is the a'th quantile of random variable X, where 0 § a § 1? As seen below, it’s the value q for which
the (shaded) area under pdf HX L equals a, i.e., PHX § qL = a. Another way to denote the a’th quantile is Qa .
ü Normal Distribution
invNorm returns Qa for X ~nIm, s2 M. Just enter invNorm@a, m, sD.
Example. For X ~nH5, 234L, Q0.23 = invNormI0.23, 5,
234 M º -6.3022.
ü Solver
No built-in function returns chi-square and t quantiles (although the TI-84’s invT command returns t quantiles). And so use the Solver command to find roots.
From background math, r is a root of function f HxL when f HrL = 0. For instance, the roots of f HxL = x2 - 9
are ≤3 because x2 - 9 = 0 yields x2 = 9 and thus x = ≤3. To compute this using Solver:
è Punch Math ê Solver, hit Æ, punch CLEAR, and next to “0=” enter f HxL (here x2 - 9)
è Hit ENTER and next to "X =", input an estimated root (say 2.3) but do not punch ENTER!
è Leaving the cursor on the same line as the variable for which you're solving, punch SOLVE (hit
ALPHA then ENTER). Solver solves for the variable on whose line the cursor rests (this is how you
identify the variable for which to solve when your equation has other variables).
è When the "crawling ants" vanish from the upper right corner, the answer appears next to X (this value is
automatically stored in the variable for later use)
è Although here we used variable X, use whatever variable you wish (since you may want to store several
roots for later use). To more quickly find a root, enter a better estimate that is near the root sought.
ü t Distribution
On the TI-84, invT(a, n) returns Qa for X ~tHnL. Otherwise, use Solver on the TI-83+, as follows:
è We seek the value Q that satisfies PHX § QL = a, i.e., tcdf H-¶, Q, nL = a
è To use Solver, we need the function whose root we seek. Since PHX § QL - a = 0, the left side of this
equation is that function. And so next to "0 =" enter tcdf H- E99, Q, nL - a and proceed as before.
Example. For X ~tH7L, compute Q0.23 . Thus, we seek Q that satisfies PHX § QL = 0.23 and so in Solver,
enter the equation tcdfH- E99, Q, 7L - 0.23. Then enter an estimate (for the t-distribution, zero suffices).
The quantile is approximately -0.7817.
ü Chi-square Distribution
To find Qa for X ~ c2 HnL, proceed as before except use c2 cdf H0, Q, nL - a. For an estimate of the quantile,
simply enter the mean of X (here it’s n, as discussed earlier in the semester).
Example. For X ~ c2 H7L, compute Q0.61 . Enter the equation c2 cdf H0, Q, 7L - 0.61 and then use 7 as an
estimate to get Q0.61 º 7.3842.