Statistics
Z Score Calculator: Find Z-Scores, Percentiles And Probabilities
Last updated: October 2026
Written by Jessica Hill
A z score calculator finds how many standard deviations a value is from the mean using z = (x − μ) ÷ σ. A score of 85 on a test with a mean of 75 and a standard deviation of 5 has z = 2. In a normal distribution, that is higher than about 97.7 percent of all scores.
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A z-score turns any value into a common scale, so you can compare scores from different tests or spot unusual data. This guide shows the formula, how to read the result and how to turn it into a probability. Every value below was checked in Python.
The Z-Score Formula
A z-score measures distance from the mean in units of standard deviation:
z = (x − μ) ÷ σ
Here x is the value, μ (mu) is the mean and σ (sigma) is the standard deviation. For a score of 85 with μ = 75 and σ = 5, z = (85 − 75) ÷ 5 = 2.
The sign tells you the direction:
- Positive z: the value is above the mean.
- Negative z: the value is below the mean.
- z = 0: the value equals the mean.
If you do not have the standard deviation yet, find it first with the standard deviation calculator. For the mean, use the mean median mode calculator.
How To Use The Z Score Calculator
- Find Z: enter the value, mean and standard deviation to get the z-score and percentile.
- Z to Probability: enter a z-score to find the area to the left, the right or between two z-scores.
- Probability to Z: enter a percentile to get the z-score, and the raw value if you add the mean and SD.
From Z-Score To Percentile And Probability
For normally distributed data, each z-score matches a fixed share of the data below it. That share is the left-tail probability, and times 100, it is the percentile.
| Z-score | Area to the left | Percentile | Area to the right |
|---|---|---|---|
| −2 | 0.0228 | 2.3rd | 0.9772 |
| −1.5 | 0.0668 | 6.7th | 0.9332 |
| −1 | 0.1587 | 15.9th | 0.8413 |
| 0 | 0.5000 | 50th | 0.5000 |
| 1 | 0.8413 | 84.1st | 0.1587 |
| 1.28 | 0.8997 | 90th | 0.1003 |
| 2 | 0.9772 | 97.7th | 0.0228 |
| 3 | 0.9987 | 99.9th | 0.0013 |
To find the area above a z-score, subtract the left area from 1. To find the area between two z-scores, subtract the smaller left area from the larger one. Between z = −0.5 and z = 1.5, that is 0.9332 − 0.3085 = 0.6247, or about 62.5 percent.
These are the same areas a printed z-table lists.
From Percentile Back To A Z-Score
Working backward answers questions like "what score puts me in the top 10 percent?" Find the z-score for that percentile, then turn it into a raw value with x = μ + z × σ.
For the 90th percentile, z ≈ 1.2816. On a test with μ = 75 and σ = 5, the cutoff is 75 + 1.2816 × 5 ≈ 81.41. Scoring above about 81.4 puts you in the top 10 percent.
Common Critical Values
Statistics classes use a handful of z-scores over and over for confidence intervals and hypothesis tests. These are called critical values.
| Confidence level | Two-tailed z | One-tailed z |
|---|---|---|
| 90% | ±1.645 | 1.282 |
| 95% | ±1.960 | 1.645 |
| 99% | ±2.576 | 2.326 |
The 1.96 value comes up most. It means 95 percent of a normal distribution lies within 1.96 standard deviations of the mean, and the remaining 5 percent is split between the two tails.
Z-Score Of A Sample Mean
When you test the mean of a sample instead of one value, divide the standard deviation by the square root of the sample size. This uses the standard error: z = (x̄ − μ) ÷ (σ ÷ √n).
A machine fills bags with a mean of 58 grams and a standard deviation of 10 grams. A sample of 36 bags averages 63 grams. The standard error is 10 ÷ √36 = 10 ÷ 6 ≈ 1.67, so z = (63 − 58) ÷ 1.67 = 3. A sample mean that far from 58 would be very unusual if the machine were working normally.
Comparing Scores From Different Tests
Z-scores let you compare results measured on different scales. Convert each one to a z-score, then compare.
| Test | Your score | Mean | SD | Z-score | Percentile |
|---|---|---|---|---|---|
| Math | 82 | 70 | 8 | 1.5 | 93.3rd |
| Reading | 78 | 60 | 15 | 1.2 | 88.5th |
The reading score is further above its average in raw points (18 vs 12), but the math z-score is higher, so math is the stronger result relative to the class.
Common Z-Score Mistakes
- Swapping the order. Use value minus mean, not mean minus value. The sign matters.
- Using variance instead of standard deviation. Divide by σ, not σ².
- Forgetting √n for a sample mean. A sample average needs σ ÷ √n in the denominator.
- Reading the wrong tail. A percentile is the area to the left. "Above" questions need 1 minus that area.
- Using normal percentiles on skewed data. The z-score itself is fine, but the percentile table only fits roughly normal data.
Summary
- A z-score is (x − μ) ÷ σ: the number of standard deviations from the mean.
- In a normal distribution, a z-score maps to a percentile and a probability.
- Work backward with x = μ + zσ to find the value for a given percentile.
- Use σ ÷ √n for the z-score of a sample mean.
- Remember the critical values 1.645, 1.96 and 2.576.
Next step: enter your value, mean and standard deviation in the z score calculator at the top of this page, or explore more tools on the Calc Solver homepage.
Frequently Asked Questions
What is a good z-score?
It depends on the goal. On a test, a positive z-score means above average, and z = 1 beats about 84 percent of scores. For quality checks, a z-score near 0 is good, since it means the result is close to the target. Values beyond about ±2 are unusual, and beyond ±3 are rare, about 0.3 percent of normal data.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. A z-score of −1 is one standard deviation below average, at about the 16th percentile. Negative z-scores are just as normal as positive ones. In any normal distribution, half of all values have a negative z-score.
What is the difference between a z-score and a t-score?
Both measure distance from the mean in standard units. A z-score uses the population standard deviation, or a large sample. A t-score is used when the population standard deviation is unknown and the sample is small, so it relies on the sample standard deviation and a t-distribution, which has fatter tails than the normal curve.
What z-score is the top 5 percent?
The top 5 percent starts at about z = 1.645. That means 95 percent of values in a normal distribution fall below 1.645 standard deviations above the mean. For the top 1 percent, the cutoff is about z = 2.326, and for the top 10 percent, it is about z = 1.282.
Does a z-score have units?
No. The units cancel when you divide by the standard deviation, so a z-score is a pure number. That is what makes it useful: a z-score of 1.5 means the same thing for test points, inches or dollars. It always means one and a half standard deviations above the mean.
Try it with your own numbers
Use a free CalcSolver tool to check the examples in this guide.