Z-Score Calculator – Standard Score, Percentile & Probability
Calculate z-scores and convert to percentiles using the standard normal distribution.
Z-Scores and Standard Normal Distribution
A z-score (standard score) tells you how many standard deviations a value is from the mean of its dataset. Formula: z = (x − μ) / σ where x is the value, μ is the mean, and σ is the standard deviation.
| Z-Score | Percentile | Meaning |
|---|---|---|
| −3.0 | 0.13% | Far below average (very rare) |
| −2.0 | 2.28% | Below average |
| −1.0 | 15.87% | Slightly below average |
| 0.0 | 50.00% | At the mean |
| +1.0 | 84.13% | Slightly above average |
| +2.0 | 97.72% | Above average |
| +3.0 | 99.87% | Far above average (very rare) |
The 68-95-99.7 rule states that in a normal distribution: 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ. Z-scores are used in standardized testing (SAT, IQ), quality control (Six Sigma = ±6σ), medical reference ranges, and financial risk modeling.
What does a z-score of 1.5 mean?
A z-score of 1.5 means the value is 1.5 standard deviations above the mean, placing it at approximately the 93rd percentile.
What is a good z-score?
"Good" depends on context. For test scores, higher is better. For body fat or cholesterol, the average range (z between −1 and +1) is often healthiest. In quality control, a z-score beyond ±3 flags an outlier.