Z-score Calculator
Calculate z-scores, percentiles, and probabilities using the standard normal distribution.
| Z-score | Percentile | Area Left |
|---|---|---|
| -3.00 | 0.13% | 0.0013 |
| -2.00 | 2.28% | 0.0228 |
| -1.00 | 15.87% | 0.1587 |
| 0.00 | 50.00% | 0.5000 |
| 1.00 | 84.13% | 0.8413 |
| 1.645 | 95.00% | 0.9500 |
| 1.960 | 97.50% | 0.9750 |
| 2.00 | 97.72% | 0.9772 |
| 2.576 | 99.50% | 0.9950 |
| 3.00 | 99.87% | 0.9987 |
Convert a raw score to a z-score (standard deviations from the mean), work backward to find a raw score from a given z-score, or find the probability area between two z-scores.
How Z-score Calculator Works
The z-score formula, z = (X − μ) / σ, measures how many standard deviations a value X sits away from the mean μ of its distribution. A positive z-score means the value is above average; a negative one means it's below average.
Once a z-score is known, the calculator uses a normal cumulative distribution function (CDF) approximation to look up the percentile — the percentage of a standard normal distribution that falls at or below that z-score — along with the area to the right and the two-tailed probability (the combined area in both extreme tails beyond ±z).
"Find Raw Score" mode reverses the same formula (X = μ + z×σ) to answer questions like "what raw score corresponds to a z-score of 1.5," while "Area Between Z" integrates the area under the standard normal curve between two given z-scores, showing what fraction of the distribution falls in that range.
See It In Action
Who Uses Z-score Calculator and Why
- Comparing a test score against a class's mean and standard deviation to judge relative performance.
- Checking whether a manufacturing measurement is unusually far from a process's expected mean.
- Finding what raw score corresponds to a target percentile using the Find Raw Score mode.
- Determining what proportion of a distribution falls between two thresholds using Area Between Z.
Mistakes to Avoid
- Applying a z-score calculation to data that clearly doesn't follow a normal (bell-curve) distribution — the percentile and probability results assume normality.
- Confusing one-tailed probability (area beyond z in one direction) with two-tailed probability (combined area in both extreme tails), which matters a lot for hypothesis testing.
- Mixing up which value is the raw score X and which is the mean μ when working the formula by hand alongside the calculator.
Tips for Best Results
- A z-score of exactly 0 always corresponds to the 50th percentile — use that as a quick sanity check that the formula was applied correctly.
- Two-tailed probability is always double the one-tailed probability for the same z-score on a symmetric normal distribution.
Fixing Common Problems
My percentile doesn't match what I expected. — Confirm whether you need the one-tailed or two-tailed probability for your situation — they measure different things, and mixing them up is a common source of mismatched results.
Terms Explained
Standard normal distribution: A bell-curve distribution with mean 0 and standard deviation 1, which z-scores are measured against.
Percentile: The percentage of a distribution that falls at or below a given z-score.