Math Calculators

Z-score Calculator

Compute z-scores of a normal distribution, convert between z-scores and probabilities, and find the probability between two z-scores.

Z-score Calculator

Use this calculator to compute the z-score of a normal distribution.

Z-score and Probability Converter

Provide any one value to convert between z-score and probability using the standard normal distribution.

Probability between Two Z-scores

Find the probability that a standard normal variable falls between two z-scores.

About the z-score calculator

A z-score (also called a standard score) describes how many standard deviations a raw value x lies from the mean μ of its distribution. The formula is Z = (x − μ) / σ. A positive z-score indicates a value above the mean; a negative z-score a value below the mean.

Z-scores are used to standardize scores from different distributions so they can be compared on a common scale. They also let you look up the probability associated with a value using a standard normal (z) table or the normal cumulative distribution function.

This calculator provides three tools in one: compute a z-score from raw data, convert any z-score to probabilities P(x<Z), P(x>Z), P(0 to Z), P(-Z<x<Z), and P(x<-Z or x>Z), and compute the probability between two z-scores.

All probabilities are computed using a high-precision Abramowitz-Stegun approximation of the error function, which gives accuracy to about 7 decimal places across typical z-score ranges.

Typical uses include hypothesis testing, standardizing test results, outlier detection, quality control, and statistics homework.

How to use

  • Section 1: enter your raw score x, the population mean μ, and the standard deviation σ to get Z and the related probabilities.
  • Section 2: enter any one of the six fields and press Calculate to derive the other five.
  • Section 3: enter the left and right z-scores to get the area under the curve between them.

Core formulas

Z = (x − μ) / σ. P(x<Z) = Φ(Z), P(x>Z) = 1 − Φ(Z), P(0<x<Z) = Φ(Z) − 0.5, P(-Z<x<Z) = 2Φ(Z) − 1, P(x<-Z or x>Z) = 2(1 − Φ(Z)). P(Z₁<x<Z₂) = Φ(Z₂) − Φ(Z₁).

FAQ

What is a z-score?

A z-score is the number of standard deviations a value is from the mean: Z = (x − μ) / σ.

Why do z-scores matter?

They standardize values from different distributions so you can compare them and look up their probability in a z-table.

Can a z-score be negative?

Yes. A negative z-score means the value is below the mean. The sign describes the direction from the mean.

What probability does P(x<Z) represent?

It is the area under the standard normal curve to the left of Z, equivalent to Φ(Z).