How z-score calculator works
A z-score (standard score) answers "how unusual is this value?" in units everyone can compare: standard deviations. The formula is z = (x − μ) / σ — the value minus the mean, divided by the standard deviation. A z-score of +2 says the value sits two standard deviations above the mean, whatever the original units were, so exam scores, birth weights and sensor readings all become directly comparable once standardised.
The sign carries the direction and the size carries the surprise. Negative z-scores fall below the mean, positive above, and the magnitude is what you check against the normal distribution: about 68% of a normal distribution lies within ±1 standard deviation, 95% within ±2, and 99.7% within ±3 — so |z| beyond 3 is already rarer than 1 event in 370 on average.
The calculator converts the z-score into probabilities with the normal curve's cumulative distribution, computed through the error function (a standard rational approximation accurate to about six decimal places). You get the percentile — the share of the population at or below your value — the share above it, and, if you enter a second value, the area between the two, which is the probability a random draw lands in that range.