Finding a critical value on your TI-84 means using the inverse of a probability distribution function to locate the boundary point for your hypothesis test or confidence interval

A critical value is the number that marks where your test statistic needs to cross to reject the null hypothesis. On a TI-84, you do not calculate it by hand — you use the built-in distribution functions to find it when ready. The method depends on which distribution you are working with: t-distribution (most common for small samples), z-distribution (normal distribution), chi-square, or F-distribution.

The core idea is the same for all of them: you tell the calculator what probability you want (your significance level, usually 0.05 or 0.01), and it returns the value that cuts off that probability in the tail of the distribution. For a two-tailed test, you split your significance level in half.

Key Takeaways

  • The TI-84 uses inverse distribution functions (invT, invNorm, invChi2, invF) to find critical values, not the forward distribution functions.
  • For a t-distribution critical value, press 2nd, VARS, scroll to invT, enter your significance level (or half of it for two-tailed), then your degrees of freedom.
  • For a z-distribution critical value, use invNorm the same way, but you do not need degrees of freedom.
  • Two-tailed tests require you to divide your significance level by 2 before entering it into the calculator.
  • The calculator returns only the positive critical value; for two-tailed tests, your critical region includes both the positive and negative versions.

Finding a t-distribution critical value

The t-distribution is what you use when your sample size is small (usually under 30) or when you do not know the population standard deviation. Press 2nd, then VARS to open the DISTR menu. Scroll down until you see invT and press ENTER.

The calculator will ask for two pieces of information. First is the area (probability) to the left of the critical value. If your significance level is 0.05 and you are doing a one-tailed test, enter 0.95 (because 1 − 0.05 = 0.95). If you are doing a two-tailed test with significance level 0.05, enter 0.975 (because you split 0.05 in half to get 0.025 in each tail, and 1 − 0.025 = 0.975).

Second is your degrees of freedom, which for a t-test is usually your sample size minus 1. Press ENTER and the calculator returns your critical value. Write it down — this is the boundary your test statistic must cross.

Finding a z-distribution critical value

The z-distribution (also called the standard normal distribution) applies when your sample is large or when you already know the population standard deviation. Open the DISTR menu the same way: 2nd, VARS. Scroll to invNorm and press ENTER.

Enter the area to the left of your critical value. For a one-tailed test with significance level 0.05, enter 0.95. For a two-tailed test with significance level 0.05, enter 0.975. The z-distribution does not require degrees of freedom, so press ENTER and you get your answer.

The z-critical value is usually smaller in absolute value than the t-critical value for the same significance level, because the z-distribution has thinner tails. This is why t-tests are more conservative when sample sizes are small.

Finding a chi-square critical value

Chi-square tests are used for categorical data and goodness-of-fit tests. Open the DISTR menu and scroll to invChi2 (or χ²cdf on some versions — look for the inverse function). Press ENTER.

Enter the area to the left of your critical value. For a significance level of 0.05 in a right-tailed test, enter 0.95. Then enter your degrees of freedom, which depends on your test (for a goodness-of-fit test, it is the number of categories minus 1). Press ENTER to get your critical value.

Chi-square distributions are always right-tailed, so you do not split your significance level in half for a two-tailed test — chi-square tests are inherently one-tailed.

Finding an F-distribution critical value

F-tests compare variances across groups and are used in ANOVA. Open the DISTR menu and find invF. Press ENTER.

The F-distribution requires two sets of degrees of freedom: the numerator degrees of freedom (usually the number of groups minus 1) and the denominator degrees of freedom (usually the total sample size minus the number of groups). Enter the area to the left of your critical value first — for a significance level of 0.05, enter 0.95. Then enter the numerator df, then the denominator df. Press ENTER to get your critical value.

Understanding one-tailed versus two-tailed tests

A one-tailed test looks for a difference in one direction only. Your critical region is in one tail of the distribution. If your significance level is 0.05, you enter the area as 0.95 (or 1 − 0.05).

A two-tailed test looks for a difference in either direction. Your critical region is split between both tails. If your significance level is 0.05, you divide it by 2 to get 0.025 in each tail, so you enter the area as 0.975 (or 1 − 0.025). The calculator returns the positive critical value; the negative critical value is its opposite. Your test statistic must be more extreme than either one to reject the null hypothesis.

The choice between one-tailed and two-tailed depends on your research question, not on your data. If your hypothesis predicts a direction (greater than or less than), use one-tailed. If it just predicts a difference (not equal to), use two-tailed.

Common mistakes and how to avoid them

The most frequent error is entering your significance level directly instead of subtracting it from 1. Remember: the calculator wants the area to the left of the critical value, not the area in the tail. If you want 0.05 in the right tail, you enter 0.95.

Another common mistake is forgetting to divide by 2 for two-tailed tests. If you are doing a two-tailed test with α = 0.05, you must enter 0.975, not 0.95. This will give you a larger critical value, which is correct — two-tailed tests are more conservative.

Make sure you are using the inverse function (invT, invNorm, invChi2, invF), not the forward distribution function (tcdf, normalcdf, etc.). The forward functions tell you the probability given a test statistic; the inverse functions tell you the test statistic given a probability. They do opposite things.

Frequently Asked Questions

What is the difference between invT and tcdf?

invT takes a probability and returns a test statistic (the critical value). tcdf takes a test statistic and returns a probability (the p-value). You use invT to find the critical value before you run your test; you use tcdf to find the p-value after you calculate your test statistic.

Do I need to memorize the degrees of freedom formula?

For a one-sample t-test, degrees of freedom is n − 1 (sample size minus 1). For a two-sample t-test, it is n₁ + n₂ − 2. For chi-square goodness-of-fit, it is the number of categories minus 1. Your textbook or assignment will tell you which formula to use for your specific test.

What if my calculator shows an error when I enter invT?

Check that your area is between 0 and 1 (not a percentage like 95). Also make sure your degrees of freedom is a positive whole number. If you entered 0.05 instead of 0.95, the calculator may reject it because 0.05 is too far into the tail for some distributions.

Can I use the critical value to find the p-value?

Not directly. The critical value is a boundary; the p-value is the probability of observing your test statistic if the null hypothesis is true. Use tcdf, normalcdf, or the appropriate forward distribution function to find the p-value from your test statistic.

Why is my critical value different from the table in my textbook?

Tables round to fewer decimal places and may use slightly different degrees of freedom rounding. The TI-84 is more precise. If the values are close (within 0.01 or so), you are doing it correctly. If they differ by more than that, check that you entered the right significance level and degrees of freedom.