Z-Score Calculator
Turn a raw score into a z-score and percentile — with the exact areas to the left, right, and between the mean and your value — or run it in reverse: give a percentile and get the z-score and matching raw score back.
Steps
What a z-score tells you
A z-score restates any value as “how many standard deviations from the mean” — z = (x − μ) ÷ σ. That single number places the value on the standard normal curve, where areas are known: about 68% of values fall within ±1σ, 95% within ±2σ, and 99.7% within ±3σ. Because z strips the units away, it lets you compare across scales — a 75 on a test with mean 60, σ 12 (z = 1.25) beats an 82 where the mean is 75 and σ is 8 (z = 0.875).
How it’s calculated
Forward: z = (x − μ) ÷ σ; percentile = Φ(z), the standard normal CDF, computed here via the error function: Φ(z) = ½(1 + erf(z/√2)). Area right = 1 − Φ(z); mean-to-x = |Φ(z) − 0.5|. Reverse: z = Φ⁻¹(p) using the Acklam rational approximation (accurate to ~10⁻⁹), and x = μ + zσ.
Results update as you type and are for education, not professional advice — double-check any number that matters.
Worked example
A score of 75 on a test with mean 60 and σ 12 gives z = (75 − 60) ÷ 12 = 1.25. The area left is Φ(1.25) = 0.8944, so the score sits at the 89.4th percentile; 0.1056 of scores lie above it and 0.3944 lie between the mean and 75. Reverse mode: the 90th percentile of the same test is z = 1.2816, a raw score of 60 + 1.2816 × 12 = 75.38.
Common mistakes
- Dividing by the variance instead of the standard deviation — z uses σ, not σ².
- Reading a negative z as an error — it just means the value is below the mean.
- Applying normal-curve percentiles to clearly skewed data — z-scores describe position, but the area-to-percentile step assumes normality.
Where it is used
- Standardized test reports (SAT, IQ) and growth charts.
- Quality control limits — flagging output beyond ±3σ.
- Comparing values measured on different scales.
Frequently asked questions
What is a good z-score?
It depends on direction. z = 0 is exactly average; z = +1.25 beats about 89% of the population; z = −2 sits at roughly the 2nd percentile. Beyond ±3, values are rare — about 1 in 370 on each side for a normal distribution.
How do I convert a z-score to a percentile?
Take the area under the standard normal curve to the left of z: percentile = Φ(z) × 100. This calculator computes Φ with the error function, so z = 1.25 → 89.44 without a lookup table.
What if my z is negative?
Nothing changes mechanically — Φ(−1.25) = 1 − Φ(1.25) = 0.1056, the 10.6th percentile. Negative simply means below the mean.
Can a percentile of exactly 0 or 100 be converted back?
No — the normal curve never quite reaches 0 or 1, so Φ⁻¹ is undefined there. Enter values strictly between 0 and 100 (99.99 works; 100 does not).
Does this assume my data is normal?
The z-computation itself does not, but translating z into a percentile does. For heavily skewed data, the percentile line will be off even though the z-score is still a valid distance-from-mean measure.