Z-score calculator

Standardize a value with a z-score

A z-score measures how many standard deviations a raw value lies above or below a mean. The formula is z = (x − μ)/σ. Positive results are above the mean, negative results are below it, and zero is exactly at the mean. Because the units cancel, z-scores can compare positions across differently scaled measurements.

Normal-distribution probabilities

The calculator also evaluates a numerical approximation to the standard normal cumulative distribution. The left-tail percentage is the modeled share at or below the z-score; the right tail is its complement. These probabilities describe a normal model, not the empirical rank of a value in arbitrary or skewed data.

Use the correct mean and standard deviation

The mean and standard deviation should describe the same population, reference distribution, or explicitly chosen sample summary as the raw value. The standard deviation must be positive. Mixing units, groups, time periods, or sample and population assumptions can make a mathematically correct z-score substantively misleading.

Work and verify one example

If a reference group has mean 70 and standard deviation 10, a value of 85 has z = (85 − 70) ÷ 10 = 1.5. Reversing the calculation, 70 + 1.5 × 10 returns 85. That reverse check catches a misplaced mean or standard deviation. Under a standard normal model, z = 1.5 corresponds to roughly 93.3% in the left tail and 6.7% in the right tail. It does not mean that exactly 93.3% of the observed group scored lower unless the group actually follows that model and the supplied parameters describe it.

Normal percentiles are approximations and require a defensible normal-distribution assumption. A z-score alone does not establish rarity, significance, causation, diagnosis, eligibility, or performance quality.