What Is a Z-Score?
A z-score (also called a standard score) tells you how far a value is from the mean, measured in standard deviations. If a student scores 85 on a test where the class average is 80 and the standard deviation is 10, their z-score is 0.5 — meaning they scored half a standard deviation above average.
Z-scores let you compare values from different distributions. A z-score of 2.0 on one test means the same thing as a z-score of 2.0 on a completely different test — both are two standard deviations above their respective means.
The Formula
z = (x - μ) / σ
- x = the raw value you want to standardize
- μ = the mean (average) of the distribution
- σ = the standard deviation of the distribution
Z-Scores and Percentiles
Every z-score maps to a percentile via the standard normal distribution. Some commonly used values:
| Z-Score | Percentile | Meaning |
|---|---|---|
| -2.0 | 2.28% | Far below average |
| -1.0 | 15.87% | Below average |
| 0.0 | 50.00% | Exactly average |
| 1.0 | 84.13% | Above average |
| 1.96 | 97.50% | Statistically significant (p < 0.05) |
| 2.0 | 97.72% | Far above average |
Frequently Asked Questions
What is a z-score?
A z-score tells you how many standard deviations a value is from the mean. Positive means above, negative means below.
What does a negative z-score mean?
A negative z-score means the value is below the mean. A z-score of -1.5 means 1.5 standard deviations below average.
How do I find the z-score in Excel?
Use =STANDARDIZE(x, mean, standard_dev). For example: =STANDARDIZE(85, 80, 10) returns 0.5.
What z-score is statistically significant?
For a two-tailed test at the 0.05 significance level, z-scores beyond ±1.96 are statistically significant. For 0.01 level, the threshold is ±2.576.
Can z-scores be greater than 3?
Yes, but values beyond ±3 are rare in normally distributed data (only 0.3% of values). They may indicate outliers or non-normal distributions.