StratumZ-Score (Standard Score)
How many standard deviations a value sits from the mean — and how to read it. The z-score strips away the units of a measurement and leaves a single, comparable number.
What it is
A z-score — also called a standard score — answers one question about a value: how far is it from the mean, measured in standard deviations? A z of 0 means the value is exactly average. A z of +1 means it sits one standard deviation above the mean; a z of −1.5 means it's a standard deviation and a half below. The sign tells you the direction, the magnitude tells you the distance.
That makes the z-score a unit-free yardstick. A height in centimetres and an income in dollars can't be compared directly, but their z-scores can — both are now just "so many standard deviations from typical." Standardizing a whole variable means replacing every value with its z-score, which re-centers the data on 0 and rescales it to a standard deviation of 1.
The formula
Subtract the mean, then divide by the standard deviation:
z = (x − μ) / σ using the population mean μ and SD σ
z = (x − x̄) / s using the sample mean x̄ and SD sThe subtraction shifts the value so the mean becomes 0; the division rescales it so one standard deviation becomes one unit. Whatever the original units, the output is the same kind of number — a count of standard deviations.
How to read it
| Z-score | Reading |
|---|---|
z = 0 | Exactly the mean — perfectly average. |
z = +2 | Two standard deviations above the mean. |
z = −1 | One standard deviation below the mean. |
|z| > 2 | Roughly the outer 5% under a normal distribution — unusual. |
|z| > 3 | About the outer 0.3% — genuinely rare. |
Those last two rows come straight from the 68–95–99.7 rule: if a variable is normally distributed, about 95% of values land within 2 standard deviations of the mean, so a value beyond |z| = 2 is in the leftover 5%. The z-score is just that rule applied to a single value.
Why it matters
Because it removes units, the z-score is the basis of standardization — a routine step before methods that compare variables on a common footing. PCA and clustering standardize first so that a variable measured in large numbers doesn't dominate one measured in small numbers; otherwise the analysis would just track whichever column happened to have the bigger range.
A z-score also connects directly to probability. Under a normal distribution, a z maps to a tail probability: the further from 0, the less of the curve lies beyond it. That tail area is exactly what a p-value reports for a z-test — a z of 1.96 corresponds to a two-sided p of about 0.05. So the z-score is the bridge between "how unusual is this value?" and "how surprising would it be by chance?"
(x − mean(x)) / sd(x) and the standardized values recompute live as the data change. The exact mean and standard-deviation conventions are on the conventions page.
See also
- Try it free
- P-value calculator — a z maps to a p-value; enter a z statistic and read the tail probability
- Related terms
- Normal Distribution · Standard Deviation · P-Value
- In the app
- See standardization at work in PCA and clustering
The z-score is the simplest tool for asking "how unusual is this?" — and the unit-free scale that lets you compare anything to anything. Pair it with the normal distribution to turn a distance into a probability.