Z-Score Calculator — Standard Score & Normal Percentile
Convert a value to a z-score using the mean and a positive standard deviation.
Last updated · View algorithm, constants and default inputs (JSON)
About this calculator
Convert a value to a z-score using the mean and a positive standard deviation. The percentile and tail probability use a standard normal approximation; they are meaningful only when that model fits your data.
Worked example
calculated with the default values below
Value 85 · Mean 70 · Standard deviation 10 Z-score 1.5000 · Percentile rank 93.32 % · Deviation from mean 15.0000 · Probability above this value 6.68 %
General background references
Disclaimer
These calculators give general mathematical results. They suit study, planning and estimation — verify anything that feeds a graded submission, a published figure or a financial commitment.
Implementation source
This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.
Formula and variables
z = (x − μ) ÷ σ; P = Φ(z) × 100
- x
- Value
- μ
- Mean
- σ
- Standard deviation
- z
- Z-score
- P
- Percentile (%)
The standard-normal CDF Φ is approximated by: t = 1 ÷ (1 + 0.2316419|z|), φ = 0.3989423 × e^(−z² ÷ 2), Q = φ × t × (0.3193815 + t(−0.3565638 + t(1.781478 + t(−1.821256 + t × 1.330274)))); Φ(z) = 1 − Q for z > 0, otherwise Q. The percentile and upper-tail probability are approximations and require a suitable normal model.
Frequently asked questions
Why does a z-score of 1.5 mean different things across datasets?
A z-score is unitless: it is only (value minus mean) divided by standard deviation, so 1.5 always means one and a half standard deviations above the mean regardless of scale. The calculator turns that into a percentile using the normal CDF, about 93.3% for z = 1.5, but the underlying spread in raw units depends entirely on your entered standard deviation.