What a z-score tells you
A z-score is the number of standard deviations a value lies above (+) or below (−) the mean. It lets you compare scores from different scales: a 130 on an IQ-style scale (mean 100, SD 15) and a 650 on a test with mean 500 and SD 100 are both z = 2.0 and the 97.7th percentile.
The 68–95–99.7 rule
In a normal distribution about 68% of values fall within ±1 SD, 95% within ±1.96 (≈2) SD, and 99.7% within ±3 SD. A |z| above 3 is rare enough to be worth checking as a possible data-entry error or outlier.
| z | Percentile | z | Percentile |
|---|---|---|---|
| -3 | 0.13 | 3 | 99.87 |
| -2 | 2.28 | 2 | 97.72 |
| -1.5 | 6.68 | 1.5 | 93.32 |
| -1 | 15.87 | 1 | 84.13 |
| -0.5 | 30.85 | 0.5 | 69.15 |
Frequently asked questions
How do I calculate a z-score?
z = (x − μ) / σ: subtract the mean from the score and divide by the standard deviation. A score of 130 on a test with mean 100 and SD 15 has z = 2.0.
What percentile is a z-score of 1.96?
About the 97.5th percentile: 97.5% of a normal distribution lies below z = 1.96, and 95% lies between −1.96 and +1.96.
Can I convert a percentile to a z-score?
Yes. Choose “From percentile” above. The 90th percentile is z ≈ 1.282; the 10th is z ≈ −1.282.
Does this assume normal data?
The z-score itself is just a rescaling and works for any data. Converting it to a percentile assumes the scores are approximately normally distributed.