Z-Score Calculator
Standardize a raw value — a test score, a measurement, one data point — into a z-score, and see its percentile and the areas under the normal curve. Switch the mode to turn a z-score back into a raw value.
The percentile and the areas assume the values follow a normal curve. With skewed data a z-score still measures distance from the mean, but the percentages do not hold.
Formula and calculation
Worked example
A score of 129.4 in a group with a mean of 100 and a standard deviation of 15 has a z-score of 1.96 — 1.96 spreads above the mean. About 97.5% of the group is expected to fall below it, so 2.5% falls above, and 95% sits within ±1.96 standard deviations of the mean in both directions.
Why standardize at all
A z-score puts everything on one scale: standard deviations from the mean. A 1300 on the SAT and a 28 on the ACT cannot be compared directly, but their z-scores can — each says how far above or below the typical score you landed, in units of that test's own spread. The same move works for a child's weight against a growth chart, a delivery time against a warehouse's normal range, or a single sensor reading against months of history. Rules of thumb follow from the curve: roughly two-thirds of values sit within one standard deviation of the mean, 95% within two, and 99.7% within three. Anything beyond three is worth a second look — usually a real outlier, occasionally a data-entry mistake.
Where this leads next
Two tools build directly on this number. The p-value calculator turns a z-score into the probability of seeing data that extreme, which is what a hypothesis test reports. The confidence interval calculator uses the same curve in the other direction, putting a margin of error around a sample mean. If you only have a list of raw numbers, find the mean and standard deviation first with the standard deviation calculator, then come back with the value you want to standardize.
What the results mean
- Z-score
- How many standard deviations your value sits from the mean. Zero is exactly average, +1 is one spread above it, and −2 is two spreads below. The sign tells you the direction; the size tells you how unusual the value is.
- Percentile
- The share of the group expected to fall below your value, assuming a normal distribution — the area under the curve to the left of z. A value at the mean is the 50th percentile.
- Raw value
- The value in its original units: the number you typed, or the value solved from a z-score in the reverse mode — that is the x in the formula.
- Area to the right
- The share expected to fall above your value — the area to the right of z. It is always 100 minus the percentile, so the two rows should add up.
- Area within ±z of the mean
- The share of a normal distribution that sits within this many standard deviations of the mean, in both directions. It is about 68% at z = 1, 95% at z = 2 and 99.7% at z = 3 — the rule of thumb behind control charts.
Frequently asked questions
How do you calculate a z-score?
Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) ÷ σ. A score of 85 in a class averaging 75 with a standard deviation of 10 gives z = (85 − 75) ÷ 10 = 1.0 — one spread above the class average.
What does a z-score of 1.96 mean?
It marks the point where 2.5% of a normal distribution sits above it and 97.5% below it. That is the value behind the familiar 95% confidence interval and the two-tailed 5% cutoff in hypothesis testing, which is why 1.96 shows up in statistics books so often.
How do I turn a z-score into a p-value?
A p-value is the tail area of the same curve: 1 − Φ(z) for an upper-tail test, or twice that for a two-tailed test. The p-value calculator on this site does that conversion and checks the result against the usual 0.05 cutoff.
Is the percentile the same as a class rank?
No. The percentile is the area under an idealized normal curve and estimates the share of the group below your value. A rank counts actual people, and real distributions are lumpier than the curve — often with a clump of high scores that pushes your percentile down.
Can I use a z-score on data that is not normally distributed?
You can always compute it, and it always means “this many standard deviations from the mean”. What you cannot do is read the percentages off the curve — incomes, waiting times and house prices have long right tails, so their real percentiles differ noticeably from the normal-curve estimate.