CalcOS calculator
Z-Score Calculator
Calculate the z-score (standard score) of a raw value, given the population mean and standard deviation. Includes probability percentile lookup.
Enter your details
How this calculator works
Standardize your data points with our free z-score calculator. A z-score (or standard score) indicates how many standard deviations a raw data point is above or below the population mean. Simply enter your raw value, mean, and standard deviation to calculate the z-score and find its corresponding standard normal cumulative probability.
- Method
- Finite-number validation, z = (x − μ) / σ standardization, and numerical standard-normal cumulative probability approximation
- Source
- NIST/SEMATECH e-Handbook of Statistical Methods — Normal Data and Standardization
- Last reviewed
- 2026-08-26
Method limitations
- Raw value and mean must be finite values on the same measurement scale.
- Standard deviation must be finite and strictly greater than zero.
- Percentile and tail-percentage outputs assume an appropriate normal-distribution model.
- The normal CDF is a numerical approximation and extreme-tail display values may round to 0% or 100%.
- A z-score alone does not establish statistical significance, causation, diagnosis, or practical importance.
Engine-backed example
Standardize a raw value: worked example
What is the z-score of 85 when the population mean is 70 and the standard deviation is 10?
- Raw value
- 85
- Population mean
- 70
- Population standard deviation
- 10
- Percentile
- 93.32%
- Percentage Above Raw Value
- 6.68%
- Standard Deviation Units
- +1.50 σ
Subtracting 70 from 85 and dividing by 10 gives a z-score of 1.5. Under a normal-distribution model, approximately 93.32% of values lie at or below this standardized value.
Trust and review
Who created and reviewed this calculator?
- Author
- CalcOS Editorial TeamResponsible for statistical terminology, examples, assumptions, and normal-model interpretation boundaries.
- Reviewed by
- CalcOS Scientific ReviewTechnical review of standardization, signed inputs, positive scale requirements, cumulative normal probability approximation, rounding, engine parity, and interpretation limitations.
- Substantive review date
- 2026-08-26Z-score standardization, signed raw values and means, strictly positive standard deviation, standard normal CDF outputs, numerical approximation, display precision, and worked-example parity.
Sources
- NIST/SEMATECH e-Handbook — Normal Data and Standardization
- NIST/SEMATECH e-Handbook — Standard Normal CDF Table
- NIST/SEMATECH e-Handbook — Location and Scale Parameters
Assumptions used by this calculator
- The raw value and population mean may be any finite signed values measured on the same scale.
- The supplied standard deviation is finite and strictly greater than zero.
- The z-score is computed as (raw value − mean) divided by standard deviation.
- Percentile and cumulative-probability outputs assume the standardized variable follows a normal distribution.
- The standard normal cumulative probability is numerically approximated and displayed to governed precision.
- A z-score describes relative position; it does not by itself establish rarity, causation, diagnosis, or statistical significance.