The TI-84 has no z-score button, so you type the formula z = (x − μ) ÷ σ on the home screen. For x = 85, μ = 70, and σ = 10, enter (85−70)÷10 to get z = 1.5. To turn that into a probability, press 2nd VARS, 2 for normalcdf(, and use −1E99 to 1.5.
The z-score formula
A z-score tells you how many standard deviations a value sits above or below the mean. The formula is z = (x − μ) ÷ σ, where x is your value, μ is the mean, and σ is the standard deviation.
A z-score of 1.5 means the value is one and a half standard deviations above the mean. A z-score of −2 means it is two standard deviations below. A z-score of 0 is exactly average. It is one of the core TI-84 statistics calculators because almost every normal distribution problem starts here.
If you only have raw data, find the mean and standard deviation first with 1-Var Stats. The standard deviation calculator walks through those steps.
How to calculate a z-score on a TI-84
- Press ( to open a parenthesis.
- Type the value, press −, and type the mean.
- Press ) to close the parenthesis.
- Press ÷ and type the standard deviation.
- Press ENTER.
The parentheses matter. Without them, the TI-84 divides the mean by σ first and then subtracts, which gives a wrong answer. Typing 85−70÷10 returns 78, while (85−70)÷10 returns the correct 1.5.
If the mean or the value is negative, use the (−) key for the negative sign and the − key only for subtraction. Mixing them up is the top cause of ERR:SYNTAX in the TI-84 error codes guide.
Converting z to a probability with normalcdf
normalcdf( finds the area under the normal curve between a lower and an upper bound. That area is the probability of landing in that range.
- Press 2nd, then VARS to open DISTR.
- Press 2 for normalcdf(.
- For lower, type −1E99. Press (−), 1, 2nd , for E, then 9 9.
- For upper, type your z-score, such as 1.5.
- Leave μ as 0 and σ as 1, then select Paste and press ENTER.
The answer, .9331927987, is the area to the left of z = 1.5. For the area to the right, use 1.5 as the lower bound and 1E99 as the upper bound, which gives .0668072013.
You can also skip the z-score. Enter normalcdf(−1E99, 85, 70, 10) with the raw value, mean, and standard deviation, and you get the same .9331927987. Practice both forms on the free TI-84 calculator online, which has the full DISTR menu.
Reading a z-table vs using the calculator
A printed z-table and normalcdf give the same left tail area, but the calculator is faster and more exact. Most tables round z to two decimals and the area to four, so 1.5 reads as .9332.
| Method | Input | Area to the left |
|---|---|---|
| Z-table | Row 1.5, column .00 | .9332 |
| TI-84 normalcdf | normalcdf(−1E99, 1.5) | .9331927987 |
Tables only list left tail areas. For a right tail you subtract from 1, and for an area between two z-scores you subtract one table value from another. normalcdf handles all three directly through its bounds.
Worked example
A class's test scores have a mean of 70 and a standard deviation of 10. You scored 85. How unusual is that?
Your z-score is (85 − 70) ÷ 10 = 1.5. normalcdf(−1E99, 1.5) gives .9331927987, so you scored higher than about 93.3 percent of the class. That puts you near the 93rd percentile.
The area above you is 1 − .9331927987 = .0668072013. About 6.7 percent of students would score higher than 85 if the scores follow a normal curve.
Now flip the question. What score puts you in the top 10 percent? Press 2nd VARS, 3 for invNorm(, and enter an area of .9 with μ = 70 and σ = 10. The TI-84 returns 82.81551566, so you need about 83 points. invNorm(.9) alone gives the matching z-score, 1.281551566.
Common mistakes
- Forgetting the parentheses around x − μ, which makes the TI-84 divide before it subtracts.
- Using −99 instead of −1E99 for the lower bound. −99 still works for z-scores, but it fails with raw values like test scores.
- Reading the left tail area as the answer when the question asks "more than". Use the right tail instead.
- Mixing up Sx and σ. Use the standard deviation the problem gives you for the population.
FAQ
How do you calculate a z-score on a TI-84?
Type the formula on the home screen. Enter (x − μ) ÷ σ with your numbers, keeping the parentheses around the subtraction, then press ENTER. For a value of 85, a mean of 70, and a standard deviation of 10, type (85−70)÷10 and the screen shows 1.5.
Does the TI-84 have a z-score function?
No. There is no menu item that returns a z-score. You type the formula yourself. The closest built in tools are normalcdf(, which turns a z-score into an area, and invNorm(, which turns an area back into a z-score. Both live in the DISTR menu under 2nd VARS.
What is a good z-score?
It depends on what you measure. For test scores, a positive z-score beats the average, and a z of 1 beats about 84 percent of people. In statistics class, a z-score beyond 2 or below −2 usually counts as unusual, since only about 5 percent of values land there.
How do you find probability from a z-score on a TI-84?
Press 2nd VARS, then 2 for normalcdf(. For the probability below z, use −1E99 as the lower bound and z as the upper bound. For above z, use z and 1E99. Leave μ at 0 and σ at 1, then press ENTER to see the area.
What is the difference between z-score and normalcdf?
A z-score is a position, the number of standard deviations from the mean. normalcdf gives an area, the probability of landing between two points. You often find the z-score first, then feed it into normalcdf, though normalcdf can also take the raw value, mean, and σ directly.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. A score of 60 in a class with a mean of 70 and a standard deviation of 10 has z = −1. The sign shows the direction, and the size shows how far from the mean the value sits.