Z-Score Calculator (With Steps, Percentile & TI-84 Keys)

This free z-score calculator converts a raw score to a z-score and a percentile, finds the z-score of every value in a data set, turns a z-score into a probability or p-value, and works back from a percentile to z. Every answer comes with a shaded bell curve, step-by-step working, a z-table snippet around your value, and the exact TI-84 keystrokes for your own numbers. It also compares two scores from different tests, such as the SAT and the ACT. Results update as you type.

How to Use the Z-Score Calculator

  1. Pick what you want to find. Raw score → z, z-scores of a data set, z → percentile or p-value, percentile → z, z → raw score, or compare two scores.
  2. Enter your numbers. For a single value, type x, the mean μ and the standard deviation σ. For the mean of a sample, enter the sample size n and the calculator uses the standard error σ/√n.
  3. Read the answer. You get z, the percentile, the right-tail area and the two-tailed p-value, with the area shaded on the curve.
  4. Open the extras. Step-by-step working, TI-84 keys and a z-table snippet sit under every result. Use Copy share link to send the exact calculation to a classmate or teacher.

Z-Score Formula

For one value: z = (x − μ) / σ, where x is the raw score, μ is the mean and σ is the standard deviation. For a sample mean: z = (x̄ − μ) / (σ / √n). To go back from z to a raw score, use x = μ + z·σ.

Example with steps: a test has mean 70 and standard deviation 10, and you scored 85. Subtract the mean: 85 − 70 = 15. Divide by σ: 15 / 10 = z = 1.5. The area to the left of 1.5 is 0.9332, so the score is at about the 93rd percentile. Only 6.68% of students scored higher.

Z-Score to Percentile (and Back)

A percentile is the area to the left of z under the standard normal curve, written Φ(z). Choose z → percentile / p-value and the calculator shows the left area, the right area, the area between −z and z, the area between any two z-scores, and the two-tailed p-value used in z-tests.

z-scorePercentileArea to the right
−22.28%0.9772
−115.87%0.8413
050%0.5
184.13%0.1587
1.64595%0.05
1.9697.5%0.025
297.72%0.0228
399.87%0.0013

Going the other way, Percentile → z handles a left area (percentile), a right area, a central area such as a 95% confidence level (z = ±1.96), and a two-tailed α. If you enter μ and σ it also gives the matching raw score, for example the cut-off score for the top 10% of a class.

How to Find a Z-Score on a TI-84

The TI-84 has no single z-score button, but three short methods cover every case. The calculator writes them out with your own numbers.

  • Raw score to z: on the home screen type (85−70)/10 and press ENTER.
  • z to percentile: press 2nd VARS, choose 2:normalcdf(, and enter lower −1E99, upper z, μ 0, σ 1. For the right tail use lower z and upper 1E99. Type −1E99 as (−) 1 2nd , 9 9.
  • Percentile to z: press 2nd VARS, choose 3:invNorm( and enter the area with μ 0 and σ 1. Newer TI-84 Plus CE models also offer Tail: LEFT, CENTER or RIGHT.
  • z-scores for a whole list: enter the data in L1, run 1-Var Stats, then in the header of L2 type (L1−x̄)/Sx. Find x̄ and Sx under VARS → 5:Statistics.

You can try every keystroke on our TI-84 calculator online without a physical calculator.

Z-Scores of a Data Set and Outliers

Paste a column of numbers from Excel or Google Sheets and choose sample or population standard deviation. The calculator finds the mean and SD, then lists every value with its deviation, z-score and percentile. Values with |z| above 2 are marked unusual and values with |z| above 3 are flagged as likely outliers. Need more statistics for the same data? Open the standard deviation calculator for variance, quartiles and a histogram.

Compare Two Scores From Different Tests

Raw scores on different scales can’t be compared directly, but z-scores can. Suppose the SAT has mean 1050 and SD 210, and the ACT has mean 21 and SD 5.5. A 1350 on the SAT gives z = 1.43, and a 28 on the ACT gives z = 1.27. The SAT score is relatively better, because it is further above its own mean. The Compare two scores mode does this for any pair, with both scores marked on one curve.

What Your Z-Score Means

  • |z| < 1: typical, about 68% of values fall here.
  • 1 ≤ |z| < 2: fairly common, most of the remaining 27%.
  • 2 ≤ |z| < 3: unusual, about 4.3% of values.
  • |z| ≥ 3: very rare, about 0.27%, and often treated as an outlier.

A positive z means above the mean and a negative z means below it. Z-scores assume the data is roughly normal when you turn them into percentiles. For areas between any two raw values, use the normal distribution calculator. For counts of successes, the binomial distribution calculator is the right tool, and the linear regression calculator handles relationships between two variables.

Why This Z-Score Calculator Is Different

  • Six modes in one tool, including sample-mean z-scores with the standard error.
  • Exact TI-84 keys for your numbers, including normalcdf with ±1E99, invNorm with tails, and the list method.
  • A z-table snippet that highlights your exact cell, so you can check homework against the printed table.
  • Outlier flags for a whole data set, and precise tail areas instead of rounding to 0.
  • Share links, dark mode and a layout that works on phones.

Z-Score Calculator FAQ

A z-score (standard score) tells you how many standard deviations a value is from the mean. z = 0 is exactly average, z = 1 is one standard deviation above, and z = −2 is two standard deviations below.

Find the area to the left of z under the standard normal curve, Φ(z), and multiply by 100. For example z = 1 gives 0.8413, about the 84th percentile. On a TI-84 use normalcdf(−1E99, z, 0, 1).

For a raw score, type (x − μ)/σ on the home screen. For a z from a percentile, press 2nd VARS, choose 3:invNorm( and enter the area with μ = 0 and σ = 1. For a whole list, store (L1 − x̄)/Sx into L2.

z ≈ 1.6449 for the 95th percentile (area 0.95 to the left). For a 95% confidence level, where 95% sits in the middle, the critical value is z = ±1.96.

Values with |z| greater than 2 are unusual (about 5% of a normal distribution). Values with |z| greater than 3 are commonly treated as outliers (about 0.3%). The data set mode flags both.

Use the population SD (σ, divide by n) when your data is the whole group. Use the sample SD (s, divide by n − 1) when your data is a sample from a larger group. Most class problems with raw data use the sample SD.

Convert each score to a z-score with its own test’s mean and standard deviation. The score with the higher z-score is relatively better, even when the tests use different scales, like the SAT and ACT.

Ti84CalcPro is not affiliated with or endorsed by Texas Instruments. TI-84 Plus CE is a trademark of Texas Instruments.