The invNorm function on a TI-84 calculator finds the z-score (or x-value) that corresponds to a given area under the normal distribution curve. You’ll find it by pressing 2nd, then VARS to open the DISTR menu, and selecting invNorm(. This function is essential for statistics problems that give you a probability and ask you to work backward to find the cutoff value.
Where to Find invNorm
Press 2nd, then VARS (which opens the DISTR menu). Scroll down to invNorm( and press ENTER. On most TI-84 models, it’s option 3 in the list. If you have a newer operating system on a TI-84 Plus CE, the calculator will open an input screen with labeled fields. On older OS versions, it simply pastes “invNorm(” onto your home screen, and you type the values separated by commas.
The Syntax and What Each Input Means
The full syntax is:
invNorm(area, µ, σ, tail)
- area: The probability (between 0 and 1) you’re looking up. Enter this as a decimal, not a percentage. For 95%, type 0.95, not 95.
- µ (mean): The mean of your normal distribution. If you’re working with the standard normal distribution, this is 0.
- σ (standard deviation): The standard deviation. For the standard normal distribution, this is 1.
- tail: Tells the calculator which direction the area refers to. Options are LEFT, CENTER, or RIGHT (available on newer OS versions).
Only the area is required. If you leave out µ and σ, the calculator defaults to the standard normal distribution (mean 0, standard deviation 1). If you leave out the tail setting, it defaults to LEFT.
Finding a Z-Score Step by Step
Say your textbook asks: “Find the z-score where 90% of the area falls to the left.” Here’s exactly what to do:
Press 2nd, then VARS. Select invNorm(. If your calculator shows an input screen, type 0.90 for area, 0 for µ, 1 for σ, and select LEFT for the tail. Press ENTER or Paste, then ENTER again. You’ll get approximately 1.2816.
If your calculator uses the older command-line style, type: invNorm(0.90, 0, 1) and press ENTER. Since LEFT is the default, you don’t need to specify it. You get the same answer.
For the standard normal distribution specifically, you can skip the mean and standard deviation entirely and just type invNorm(0.90).
Using the LEFT, CENTER, and RIGHT Tail Settings
The tail setting controls how the calculator interprets the area you enter. This matters a lot, so getting it wrong will give you the wrong answer.
LEFT (the default) means the area is measured from negative infinity up to some value. If you enter 0.90 with LEFT, the calculator finds the value where 90% of the distribution sits to its left.
RIGHT means the area is measured from some value out to positive infinity. If you enter 0.05 with RIGHT, the calculator finds the value where 5% of the distribution sits to its right. This is useful for right-tailed hypothesis tests. For example, a right-tailed test at α = 0.05 gives a z-critical value of about 1.6449.
CENTER means the area is the middle portion of the distribution, symmetric around the mean. If you enter 0.95 with CENTER, the calculator finds the boundary values that capture the middle 95%. This is handy for confidence intervals and two-tailed tests.
You can find the LEFT, CENTER, and RIGHT tokens in the calculator’s catalog if they don’t appear automatically on the input screen.
What to Do Without Tail Options
Older TI-84 operating systems don’t have the tail setting at all. The function always interprets the area as left-tail. You can still solve right-tail and two-tail problems with a little arithmetic.
For a right-tail problem, subtract your area from 1 and enter that as the left-tail area. If you need the z-score where 5% is in the right tail, enter invNorm(0.95) because 1 minus 0.05 equals 0.95. This gives you approximately 1.6449.
For a two-tailed problem, use the formula 1 minus α/2. If you’re running a two-tailed test at α = 0.05, calculate 1 minus 0.025, which is 0.975. Enter invNorm(0.975) and you get approximately 1.96, the familiar z-critical value for a 95% confidence interval.
Using invNorm With Non-Standard Distributions
Not every problem uses the standard normal distribution. If a problem says exam scores are normally distributed with a mean of 500 and a standard deviation of 100, and asks for the score at the 75th percentile, you can plug those values directly into invNorm.
Enter invNorm(0.75, 500, 100) and the calculator returns approximately 567.45. This saves you from finding the z-score first and then converting it manually with the formula x = µ + zσ. The calculator handles that conversion for you.
Avoiding Input Mistakes
The most common error is entering the area as a percentage instead of a decimal. If you type 90 instead of 0.90, the calculator will return an error or a nonsensical result, because the area must be a value between 0 and 1 (exclusive). Always convert your percentage to a decimal before entering it.
Another frequent mistake is mixing up which tail you need. Read the problem carefully. Phrases like “at most,” “less than,” or “below” point to left-tail. Phrases like “at least,” “greater than,” or “above” point to right-tail. Confidence intervals and two-tailed tests use the center area or require the 1 minus α/2 conversion described above.
If you enter a negative number or a number greater than or equal to 1 for the area, the calculator will return an error. The same goes for entering 0 or a negative number for the standard deviation.