How the Z-score is calculated
A Z-score restates a raw value in units of standard deviation from the mean, which makes values from different scales comparable. The percentile then reads that Z-score against the standard normal distribution's cumulative probability.
- x — the value
- μ — the mean
- σ — the standard deviation
- Φ(z) — standard normal cumulative distribution function
More detail
Why Z-scores make different scales comparable
A test score of 85 out of 100 and a score of 850 out of 1000 can't be compared directly, but if you know each distribution's mean and standard deviation, converting both to Z-scores puts them on the same footing: 'how many standard deviations above or below average.' A Z-score of +1.5 means the same relative standing no matter what the original units were.
Reading the percentile. A percentile of 93 means the value sits above roughly 93% of a normally distributed population — not that it beats 93% of any specific dataset, unless that dataset is actually normal. For skewed real-world data, treat the percentile as an approximation.
Frequently asked questions
What's the Z-score for a value of 85 with mean 70 and standard deviation 10?
z = (85 − 70) / 10 = 1.5, which sits at about the 93.32nd percentile of a standard normal distribution — enter these exact numbers above to see it.
What does a negative Z-score mean?
A negative Z-score means the value is below the mean. For example, a value of 50 with mean 70 and standard deviation 10 gives z = −2, roughly the 2.28th percentile — well below average.
Why does this calculator show "—" when standard deviation is 0?
Dividing by a standard deviation of 0 is undefined — a dataset with zero spread has every value equal to the mean, so "how many standard deviations away" has no meaning.
How accurate is the percentile shown here?
It uses the Abramowitz & Stegun 7.1.26 erf approximation, with a maximum absolute error of about 1.5×10⁻⁷ — accurate enough for any practical statistics use, though not an exact closed-form value (none exists for the normal CDF).