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How to use
A z-score expresses how far a value is from the mean in standard-deviation units. The tool supports raw-to-z, z-to-raw and percentile-to-z modes. Percentile calculations use the standard normal distribution.
- Choose a conversion mode.
- For raw-to-z enter value, mean and standard deviation; reverse mode uses z instead of the raw value.
- For percentile-to-z enter a percentile strictly between 0 and 100.
- Review z, lower/upper percentile areas and two-tail probability; raw value appears when mean and SD are supplied.
How it is calculated
Example
If x=85, mean=75 and SD=10, then z=(85−75)/10=1. Under the standard normal distribution, about 84.13% of the area lies below z=1.
Important notes
The z-score itself is a standardization. Turning it into a normal percentile or tail probability is appropriate only when a normal-model interpretation is reasonable; strongly skewed data may not support that assumption.
Worked examples and interpreting results
A value of 130 with mean 100 and standard deviation 15 has z-score 2: two standard deviations above the mean.
| Case | Calculation | Result |
|---|---|---|
| Difference from mean | 130 − 100 | 30 |
| z-score | (130 − 100) ÷ 15 | 2 |
How to check the result
Choose raw value to z-score. Standardization itself does not assume normality, but this calculator’s percentile conversion uses a normal model.
Frequently asked questions
What does z=0 mean?
The raw value is exactly at the mean.
Does z=2 mean the top 2%?
Not exactly; under a normal model about 97.72% lies below z=2 and about 2.28% above.
Can I recover the raw score?
Yes, with z, mean and standard deviation: x=μ+zσ.
What does the sign of a z-score mean?
A positive z-score is above the mean, a negative one is below it, and zero equals the mean. The magnitude |z| is the number of standard deviations separating the observation from the mean.