Z-Score Calculator & Percentile

Understanding the z-score calculator — concept and principle

A z-score calculator tells you how far a value lies from the mean, measured in standard deviations, and converts that distance into a percentile and p-values. It is used to compare results from different scales (a test score against a height, a lab value against its reference), to flag outliers and to run quick z-tests.

130 sits two SD steps above 100: z = 2, the 97.72nd percentile.

How it is calculated

z = (x − μ) ÷ σ, where x is your value, μ the mean and σ the standard deviation, all in the same unit.
Percentile = Φ(z) × 100, the share of a normal population below x.
One-tailed p = 1 − Φ(|z|), the area beyond |z| in one tail.
Two-tailed p = 2 × (1 − Φ(|z|)), the area beyond |z| in both tails.

Φ is the standard normal cumulative distribution function. The z-score itself has no unit, because the units cancel in the division.

The principle

Subtracting the mean centres the data at zero; dividing by the standard deviation removes the scale. The result is a pure number that means the same thing in every data set: z = 2 is two standard deviations above average, whether the data are exam marks or blood pressure. If the data are normal, z also follows the standard normal distribution, so its tail areas give probabilities. The absolute value |z| is used for p-values so that a value equally far below the mean gives the same p.

Worked example

Inputs: x = 130, mean = 100, SD = 15.
z = (130 − 100) ÷ 15 = 30 ÷ 15 = 2.
Φ(2) ≈ 0.97725, so the percentile is 97.72%.
One-tailed p = 1 − 0.97725 ≈ 0.02275.
Two-tailed p = 2 × 0.02275 ≈ 0.0455.

Each point is the calculator re-run with only “Value x” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Value x	z
32.5	-4.5
65	-2.333333333
97.5	-0.1666666667
130	2
162	4.166666667
195	6.333333333
260	10.66666667

How to read the result

z = 0 is exactly average. Roughly 68% of normal values have |z| < 1, 95% have |z| < 1.96 and 99.7% have |z| < 3. Values with |z| above 2 are uncommon and above 3 are often investigated as possible outliers. In a z-test, a two-tailed p below 0.05 corresponds to |z| > 1.96. Many intelligence tests are scaled to mean 100 and SD 15, so the example corresponds to a score about 2 SD above average.

Common mistakes

Using a sample SD from a tiny sample as if it were the true σ.
Reading the percentile as a probability that the value is "correct".
Reporting a one-tailed p when the hypothesis was two-sided.

Keep in mind: Percentiles and p-values from z assume a normal distribution. For skewed data the z-score still measures distance, but its percentile can be misleading. A p-value is a statistical measure, not proof of an effect.

Frequently asked questions

Can a z-score be negative?

Yes. A negative z means the value is below the mean; z = −1.5 is 1.5 SD below average.

What is a good z-score?

There is no universal good value; it depends on whether higher is better. It only tells you position relative to the mean.

How do I turn a z-score back into a raw value?

Use x = μ + z × σ. For z = 2, μ = 100, σ = 15, x = 130.

Sources: NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.6.6.1 Normal Distribution · Moore, D. S., McCabe, G. P. and Craig, B. A., Introduction to the Practice of Statistics (W. H. Freeman)

Checked 8 Nov 2026. Standard formulas, computed in your browser with double-precision numbers.

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