Updated June 17, 2026
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A z-score is the number of standard deviations a data point sits above or below the mean. Formula: z = (x - mu) / sigma. A z-score of 0 is exactly at the mean; +2 is two SDs above; -1 is one SD below.
The formula
z = (x - mu) / sigma
- x is the observed value
- mu is the population mean
- sigma is the population standard deviation
For sample data, substitute the sample mean (x-bar) and sample SD (s) in place of mu and sigma.
Worked example
A student scores 78 on a test where the class mean is 70 and the standard deviation is 8.
| Step | Calculation | Result |
|---|---|---|
| Subtract mean | 78 - 70 | 8 |
| Divide by SD | 8 / 8 | z = 1.0 |
A z-score of 1.0 means the student scored one standard deviation above the mean, placing them at approximately the 84th percentile in a normal distribution.
What different z-scores mean
| Z-score | Percentile (normal dist.) | Meaning |
|---|---|---|
| -2.0 | ~2nd | Well below average |
| -1.0 | ~16th | Below average |
| 0.0 | 50th | Exactly the mean |
| +1.0 | ~84th | Above average |
| +2.0 | ~98th | Well above average |
| +2.5 | ~99th | Highly unusual |
Z-score vs standard deviation: what is the difference?
Standard deviation is a property of the whole data set, measuring its overall spread. A z-score is a property of a single value within that data set, measuring how far that one point sits from the mean in units of SD. You need the standard deviation to compute any z-score, but they answer different questions: SD describes the distribution; z-score describes one observation within it. Use the standard deviation calculator to find your SD first, then divide to get z.
What does a z-score of 2.5 mean?
A z-score of 2.5 means the value sits 2.5 standard deviations above the mean. In a normal distribution, only about 0.6% of values lie above z = 2.5, making this an unusually high observation. Whether that is meaningful depends on context: in quality control it might flag a defect; in research it approaches statistical significance.
What is the 95% z-score?
The z-score that cuts off the top 5% of a normal distribution (leaving 95% below) is approximately 1.645. For a two-tailed 95% confidence interval, the critical value is 1.96, because 2.5% sits in each tail. The confidence interval article covers how this is applied in practice.
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