Z-Score Calculator
- 2.0000
- z-score
- 97.72%
- Percentile
- 2.28%
- Area beyond
- 2.00 above
- Standard deviations
The z-score calculator expresses how far a value sits from the mean in standard deviations, and converts that to a percentile. A z-score of 2 means two standard deviations above the mean, which is the 97.7th percentile, but only if the underlying data is roughly normally distributed.
How it works
z = (x - mean) / standard deviation
- x
- the raw value you are placing
- standard deviation
- must be above zero: a distribution with no spread has no meaningful z-scores
The percentile comes from the cumulative normal distribution, computed with the Abramowitz and Stegun error-function approximation, accurate to about 1.5 × 10⁻⁷, far beyond the precision of any real dataset.
- z = 0 is exactly the mean, the 50th percentile.
- z = ±1 covers about 68 percent of values between them.
- z = ±2 covers about 95 percent; z = ±3 about 99.7.
Examples
An IQ score
Value
130
Mean
100
SD
15
Result
z 2.0000 · 97.72nd percentile
(130 − 100) / 15. Two standard deviations above the mean puts the score above about 98 percent of the population.
A value below the mean
Value
85
Mean
100
SD
15
Result
z −1.0000 · 15.87th percentile
One standard deviation below. The 68 percent within ±1 leaves 16 percent in each tail.
Frequently asked questions
What does a negative z-score mean?
That the value is below the mean. The sign carries the direction and the magnitude carries the distance. Z = −1.5 is one and a half standard deviations below, at roughly the 6.7th percentile.
Does the percentile assume a normal distribution?
Yes, and this is the main caveat. The z-score itself is valid for any distribution, but converting it to a percentile requires the data to be roughly normal. For strongly skewed data (incomes, for instance) the percentile will be badly wrong even though the z-score is arithmetically correct.
Should I use the sample or population standard deviation?
Whichever matches your situation. If your mean and SD describe an entire population, use the population figure. If they are estimates from a sample, use the sample figure, and be aware the resulting z-score is itself an estimate.
What is the difference between a z-score and a t-score?
They express the same distance on different scales. A t-score is 50 + 10z, which avoids negative numbers and decimals. A z of −1.5 becomes a t of 35. Psychological and educational testing prefer t-scores for exactly that reason.