Table of Contents
What Is a Z-Score?
A z-score (also called a standard score) tells you how many standard deviations a particular data point is from the mean. It standardizes values from any normal distribution to the standard normal distribution (mean = 0, SD = 1), enabling comparisons across different scales and distributions.
Z-scores are ubiquitous in statistics. They are used in hypothesis testing (z-tests), quality control (control charts), standardized testing (SAT, IQ scores), and any situation where you need to compare values from different distributions. A z-score of 0 means the value equals the mean; positive z-scores are above the mean, negative are below.
Formula
Common Z-Scores and Percentiles
| Z-Score | Percentile | Example |
|---|---|---|
| -3.0 | 0.13% | Extremely below average |
| -2.0 | 2.28% | Well below average |
| -1.0 | 15.87% | Below average |
| 0.0 | 50.00% | Average (at the mean) |
| 1.0 | 84.13% | Above average |
| 2.0 | 97.72% | Well above average |
| 3.0 | 99.87% | Extremely above average |
Frequently Asked Questions
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. For example, a z-score of -1.5 means the value is 1.5 standard deviations below the mean. There is nothing wrong or unusual about negative z-scores; they are simply values in the lower half of the distribution.
What z-score is considered unusual?
Values with z-scores beyond +/-2 are generally considered unusual (occurring less than 5% of the time). Values beyond +/-3 are very rare (0.3% of the time). In quality control, values beyond 3 sigma trigger investigation. However, the definition of unusual depends on context and the specific application.