Z-Score Reference
| Z-Score | Percentile | Data within |
|---|---|---|
| ±1.0 | 16th / 84th | 68.3% |
| ±1.645 | 5th / 95th | 90% |
| ±1.96 | 2.5th / 97.5th | 95% |
| ±2.0 | 2.3rd / 97.7th | 95.4% |
| ±2.576 | 0.5th / 99.5th | 99% |
| ±3.0 | 0.13th / 99.87th | 99.7% |
Z-Score Calculator
Calculate the z-score (standard score) for any data point. Find the percentile rank and probability from a z-score, or calculate a z-score from a raw value, population mean, and standard deviation.
Standard Score Calculator
Frequently Asked Questions
What does a z-score tell you?
A z-score tells you how many standard deviations a data point is from the mean. A z-score of 0 means exactly average. A z-score of +2 means 2 standard deviations above average (about the 97.7th percentile). A z-score of −1 means 1 standard deviation below average (about the 16th percentile).
What is a good z-score for a test?
It depends on context. In a normal distribution, z-scores above +1 are in the top 16%, above +2 in the top 2.3%. For standardized tests, schools often look at z-scores above +1.5 to +2 as strong performance. For quality control, z-scores beyond ±3 are considered outliers (less than 0.3% of data).
What a z-score expresses
z = (x − μ) ÷ σ
A z-score states how many standard deviations a value sits from the mean. A z of 1.5 means one and a half standard deviations above average; −2 means two below.
Why standardise
It makes different scales comparable. A score of 85 on a test averaging 70 with a standard deviation of 10 gives z = 1.5. A score of 42 on a different test averaging 30 with a standard deviation of 8 gives exactly the same z — equally strong performances despite unrelated raw numbers.
The empirical rule
For roughly normal distributions:
| Within | Contains about |
|---|---|
| ±1 SD | 68% |
| ±2 SD | 95% |
| ±3 SD | 99.7% |
So a z-score beyond ±2 is already unusual, occurring in about 5% of cases, and beyond ±3 is rare.
Z-scores and percentiles
Each z-score maps to a percentile — the proportion of the distribution falling below it. z = 0 is the 50th percentile, z = 1 is roughly the 84th, z = 2 roughly the 97.7th.
The assumption that matters
Converting z-scores to percentiles assumes an approximately normal distribution. For strongly skewed data — income being the classic case — that mapping is wrong, sometimes badly so.
The z-score itself is still computable for any distribution, since it is just a rescaling. It is the probability interpretation that depends on normality. Check the shape of your data before reading percentages off a z-table.