How to Find Critical Values on a TI-84

You need a critical value for your hypothesis test and you have a TI-84 in your hands during an exam or a homework session. Use invNorm for Z critical values when the population standard deviation is known, and invT for t critical values when sigma is estimated from the sample. The most common mistake is treating “n ≥ 30 means use Z” as a rule: the choice between Z and t depends on whether the population standard deviation is known (Z) or estimated from the sample (t), not on sample size alone.

The TI-84 Plus CE guidebook documents invNorm, accessed via 2ND DISTR 3. For a standard normal distribution (mean 0, standard deviation 1), the basic syntax is invNorm(area), where area is the left-tail probability. For a test at α = 0.05 with two tails, you need the critical value that leaves α/2 = 0.025 in each tail. Enter invNorm(0.025) and the calculator returns -1.95996. The positive critical value for the right tail is 1.95996, conventionally rounded to 1.96.

Using the Tail Option on Newer OS Versions

On OS version 5.0 and later, invNorm accepts a third argument for the tail: invNorm(area, μ, σ, tail). The tail parameter accepts -1 for left-tail, 0 for center (two-tailed), and 1 for right-tail. For a one-tailed test at α = 0.05, enter invNorm(0.05, 0, 1, 1) to get 1.64485 directly, avoiding the manual halving of alpha. The TI-84 guidebook confirms this syntax. For a two-tailed test at the same α, use invNorm(0.05, 0, 1, 0) and the calculator returns 1.95996. This feature removes the most common failure mode: using a one-tailed critical value for a two-tailed test, or vice versa.

invT exists on TI-84 Plus CE calculators with OS version 5.0 or later, released in 2017. Access it via 2ND DISTR 4. The syntax is invT(area, df), where area is the left-tail probability and df is the degrees of freedom. For a test at α = 0.05 with two tails and 15 degrees of freedom (df = n - 1 = 15), enter invT(0.025, 15). The calculator returns -2.1314. The positive critical value is 2.1314, which matches the printed table value for df = 15 at α = 0.05 (2.131, often rounded to 2.13 in older tables).

Why Degrees of Freedom Matter for t Critical Values

The critical value for the t-distribution increases as degrees of freedom decrease. At df = 5, the two-tailed critical value at α = 0.05 is 2.5706; at df = 30, it is 2.042; and as df approaches infinity, it converges to the Z critical value of 1.96. If you use an incorrect df, the critical value will be wrong. The TI-84 guidebook requires the df parameter for invT, so check your sample size: for a one-sample t-test, df = n - 1; for a two-sample t-test with equal variances, df = n₁ + n₂ - 2. For a one-tailed test, halve the alpha: enter invT(0.05, 15) to get 1.753, the one-tailed critical value at α = 0.05 with 15 df.

The TI-84 Plus CE guidebook states that no chi-square inverse function (invχ²) and no F inverse function (invF) exist as built-in commands. The DISTR menu provides χ²cdf (for area left of a value) and χ²pdf (for density), as well as Fcdf and Fpdf, but no direct inverse. This is a hard limitation: you cannot get a chi-square or F critical value with a single keystroke sequence. To obtain a chi-square critical value for a goodness-of-fit test with df = 5 and α = 0.05 (right-tail test), you must use the χ²cdf command and a trial-and-error approach. Start with a guess, such as 11.07 (the known critical value from tables), and enter χ²cdf(11.07, 1E99, 5). If the result is 0.05, your guess is correct. If not, adjust the guess upward or downward. For the F-distribution, use Fcdf(guess, 1E99, numerator df, denominator df) and solve for the value that yields the target right-tail probability.

This manual process is slow and error-prone during an exam. A faster workaround is to pre-compute critical values for common alpha levels (0.10, 0.05, 0.01) and degrees of freedom using a different tool, such as Excel or Google Sheets with the CHISQ.INV.RT or F.INV.RT commands, and then store them in a note on your TI-84. The TI-84 guidebook does not document any chi-square or F inverse commands, so rely on external tables or software for these distributions. If you must use the TI-84, the χ²cdf trial-and-error method is your only option. The NIST/SEMATECH e-Handbook of Statistical Methods provides exact tables for chi-square and F critical values that you can memorize for the most common df values.

To get a chi-square critical value for a right-tail test (e.g., goodness-of-fit) at α = 0.05 with df = 5, use the χ²cdf command from the DISTR menu. The syntax is χ²cdf(lower, upper, df). For a right-tail probability, set lower to your candidate critical value, upper to a large number like 1E99, and df to 5. The command returns the right-tail area. If the returned area is not exactly 0.05, adjust the candidate value. For df = 5 and α = 0.05, the correct critical value is 11.07. Enter χ²cdf(11.07, 1E99, 5) to confirm a return of approximately 0.05. For other df values, use the same method. For df = 10, the critical value is 18.307; for df = 1, it is 3.841. This approach works for any df but requires iteration.

For the F-distribution, the TI-84 offers Fcdf from the DISTR menu, with syntax Fcdf(lower, upper, numerator df, denominator df). For a right-tail test at α = 0.05 with numerator df = 3 and denominator df = 20, the critical value is 3.10 (from tables) or exactly 3.098 (from software). Enter Fcdf(3.10, 1E99, 3, 20) to confirm a return of about 0.05. Again, trial and error is required. Pre-compute critical values for your specific df combinations using a tool like Excel with the F.INV.RT command to avoid this hassle during an exam.

After computing a critical value on your TI-84, verify the result with a second method. Use the normalcdf command (for Z), tcdf (for t), χ²cdf (for chi-square), or Fcdf (for F) to compute the area beyond your critical value. For a Z critical value of 1.96 from invNorm, enter normalcdf(1.96, 1E99) and confirm a return of 0.025. For a t critical value of 2.131 with df = 15, enter tcdf(2.131, 1E99, 15) to get 0.025. This cross-check catches entry errors, such as using the wrong df or the wrong tail. The TI-84 guidebook recommends this approach to avoid misreading the output.

The most frequent error is using a one-tailed critical value for a two-tailed test. If you enter invNorm(0.05) for a two-tailed test at α = 0.05, you get -1.64485, which corresponds to a one-tailed critical value, not the two-tailed threshold of 1.96. For a two-tailed test, halve alpha and use invNorm(0.025) or use the tail option: invNorm(0.05, 0, 1, 0) on OS 5.0+. The second failure mode is using the wrong degrees of freedom for t. A common mistake is using n instead of n - 1. For a sample of size 20, df = 19, not 20. Enter invT(0.025, 19) to get the correct critical value of 2.093. For chi-square and F, using the left-tail instead of the right-tail critical value is a third failure mode. Hypothesis tests for chi-square (goodness-of-fit, independence) and F (ANOVA) use the right-tail critical value. The left-tail value is near zero and is not the threshold for rejection. The TI-84’s χ²cdf and Fcdf commands default to left-tail area unless you set the lower bound to your candidate value and the upper bound to a large number.

The single deciding factor is whether you know the population standard deviation (σ) or estimate it from the sample standard deviation (s). If σ is known, use invNorm. If σ is estimated from s, use invT. The sample size does not determine the choice: at n = 100 with unknown σ, use t; at n = 10 with known σ, use Z. The TI-84 guidebook supports this distinction. For a confidence interval for a population mean, use t unless σ is given. For a hypothesis test for a proportion, use Z because the standard error is derived from the null proportion (known). For a test of a single variance, use chi-square. For ANOVA, use F. Do not fall back on the “n ≥ 30” shortcut; it is not a mathematical truth and it will produce the wrong critical value when σ is unknown.