Z-Score Calculator

What It Does

This calculator restates a value as a distance from the mean measured in standard deviations: z = (x − μ) / σ. Enter the value, the mean it is read against and that distribution's standard deviation, and it returns the z-score together with what the standard normal curve says about that position — the percentile Φ(z), the probability above it and the two-tailed probability. The model is the normal distribution and the figures are the curve's, not your data's, so the answer is a count of spreads and carries no unit.

Real-World Usage

A z-score is how different scales are put on one line: a child's height against a growth chart, a blood measurement against a reference range, a test score against a cohort, a delivery time against a service promise. In quality control a z of 3 or beyond is the usual flag for an out-of-control point; in finance the same figure, read against a normal model, is what a daily loss estimate means. Because the score has no unit, it can be compared with a z from a completely different measurement — the mean and the spread have already been divided out.

Local Context

The z-score is unit-free and country-independent: the value, the mean and the spread must be in the same unit, whatever that unit is, and the result is read against a distribution rather than against any local standard. Nothing on the page is converted or regionalised, and the reference curve is the standard normal one used in published tables everywhere. Only the display follows the page, where a Portuguese catalogue writes 1,5 and 93,32 % for a z that an English catalogue prints as 1.5 and 93.32 %.

What is a z-score actually for?

It restates a value in units of the spread, which is what makes figures on different scales comparable. Two readings — 85 in a test whose mean is 70 with a spread of 10, and 1,300 in one whose mean is 1,000 with a spread of 200 — are both exactly 1.5 spreads above their own middle, so they sit at the same place in their own distributions. Raw marks and percentages cannot say that; z-scores can, and the same trick underlies standardised test scores, growth charts and control limits.

Does the percentile tell me what share of my own data is below this value?

No — it is the share the normal model predicts, not a count from your list. Φ(1.5) = 0.9332 says that under a normal distribution with this mean and this spread about 93.32 % of values would fall below 85. Whether your actual data follows that curve is a separate question: a skewed or bunched set can share the same mean and spread and put a very different share below the same value. Read the percentile as the model's answer, and read the sample when you have the sample.

What does the two-tailed probability mean?

It is the chance the model puts on a value landing at least this far from the mean in either direction — twice whichever tail is smaller. At z = 1.5 the lower tail is 0.9332, so the upper tail is 0.0668 and the two-tailed figure is 0.1336: about 13.4 % of values would sit 1.5 or more spreads away from the middle, high or low. Use the one-tailed figure when only one direction matters (above a limit, later than promised) and the two-tailed figure when either side counts, which is the usual reading of a test statistic.

Why is a standard deviation of zero — or a negative one — refused?

z = (x − μ) / σ divides by the spread, so a σ of zero has no answer at all: every value would sit exactly on the mean and the step that turns a distance into a z-score breaks down. A negative σ is not a spread — a spread cannot be smaller than zero — so it is refused rather than squared into something that looks valid. A data set whose values are all equal is a different question, and the standard-deviation page reports that honestly as 0.