Z-Score Calculator
Calculate the z-score from a value, or find the value from a z-score. Includes percentile and probability. See also Standard Deviation Calculator and Percentile Calculator.
How to Calculate Z-Score
A z-score (standard score) tells you how many standard deviations a value is from the mean. A positive z-score means the value is above the mean; a negative z-score means it is below. Z-scores are used to compare values from different distributions and to find percentiles in a normal distribution.
Z-Score Formulas
Find Z-Score: z = (x − μ) / σ
Find Value: x = μ + z × σ
Where: x = value, μ = mean, σ = standard deviation
Example Calculation
A student scores 85 on a test with mean 75 and std dev 10.
z = (85 − 75) / 10 = 10 / 10 = 1.0
Percentile ≈ 84.13% (the student scored better than ~84% of students)
Z-Score Reference Table
| Z-Score | Area (Left) | Percentile |
|---|---|---|
| -3.0 | 0.0013 | 0.13% |
| -2.5 | 0.0062 | 0.62% |
| -2.0 | 0.0228 | 2.28% |
| -1.5 | 0.0668 | 6.68% |
| -1.0 | 0.1587 | 15.87% |
| -0.5 | 0.3085 | 30.85% |
| 0.0 | 0.5000 | 50.00% |
| 0.5 | 0.6915 | 69.15% |
| 1.0 | 0.8413 | 84.13% |
| 1.5 | 0.9332 | 93.32% |
| 2.0 | 0.9772 | 97.72% |
| 2.5 | 0.9938 | 99.38% |
| 3.0 | 0.9987 | 99.87% |
Frequently Asked Questions
What does a z-score of 0 mean?
A z-score of 0 means the value is exactly equal to the mean. It is at the 50th percentile.
What is a "good" z-score?
It depends on context. In testing, a z-score of 1.0 or higher means above average. In quality control, values within ±2 standard deviations (z between −2 and 2) are typically considered normal.
Can z-scores be negative?
Yes. A negative z-score means the value is below the mean. For example, z = −1.5 means the value is 1.5 standard deviations below the mean.
What is the empirical rule (68-95-99.7)?
In a normal distribution, about 68% of values fall within ±1 standard deviation, 95% within ±2, and 99.7% within ±3 standard deviations of the mean.