A Z-score standardizes a value relative to the mean and standard deviation of a distribution: z = (x − μ) / σ. It expresses how many standard deviations a value is from the mean.

How do I convert a Z-score back to a raw value?

Use x = μ + zσ. Provide the mean and standard deviation to reconstruct the original value from a Z-score.

What probabilities does the calculator report?

It provides left-tail P(Z ≤ z), right-tail P(Z ≥ z), two-tailed P(|Z| ≥ |z|), and the percentile (100 × P(Z ≤ z)) for the standard normal distribution.

Can I compute Z-scores for an entire dataset?

Yes. Paste your values and choose sample (n − 1) or population (n) standard deviation. The tool returns mean, SD, and a list of Z-scores.

Which standard deviation should I use?

Use the population SD when you know the full population’s variability. Use the sample SD (n − 1) when estimating variability from a sample.

Are extreme Z-scores outliers?

Values beyond ±2 are unusual under normality; beyond ±3 are rare. Outlier status depends on context, assumptions, and diagnostics beyond the Z-score alone.

How accurate are the probabilities?

Probabilities are computed via a numerically stable approximation to the normal CDF based on the error function (Abramowitz and Stegun, 1964), providing high accuracy for typical use cases.

Z-Score Calculator

A professional-grade Z-score calculator for students, data analysts, and researchers. Compute the Z-score of a value, convert a Z-score back to a raw value, or standardize an entire dataset. Instantly get left/right/two-tailed probabilities and percentiles under the standard normal distribution.

Calculator

Z from value Value from Z Dataset Z-scores

Value (x)

Mean (μ)

Standard Deviation (σ)

Tail for p-value

Results

Z-score —

Left-tail P(Z ≤ z) —

Right-tail P(Z ≥ z) —

Two-tailed p-value —

Percentile —

Classification —

Dataset summary

—

Note: If all values are identical (SD = 0), all Z-scores are reported as 0.

Data Source and Methodology

Primary references:

NIST/SEMATECH e-Handbook of Statistical Methods (2012), Normal Distribution and Summary Statistics. Available at: https://www.itl.nist.gov/div898/handbook/
Abramowitz, M., & Stegun, I. A. (1964). Handbook of Mathematical Functions, National Bureau of Standards, Chapter 7 (Error Function). Reference

Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.

The Formula Explained

Core definitions:

Inline: z = (x − μ) / σ

Display:

$$ z = \frac{x - \mu}{\sigma} $$

$$ x = \mu + z\,\sigma $$

$$ \Phi(z) = \Pr(Z \le z) = \tfrac{1}{2}\left[1 + \operatorname{erf}\!\left(\frac{z}{\sqrt{2}}\right)\right] $$

$$ p_{\text{right}} = 1 - \Phi(z), \quad p_{\text{two}} = 2\min\{\Phi(z), 1 - \Phi(z)\} $$

For a dataset {x₁,…,xₙ}:

$$ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i, \quad s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2}, \quad \sigma_{\text{pop}} = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2} $$

Each standardized score: $$ z_i = \frac{x_i - \bar{x}}{s} \quad \text{or} \quad \frac{x_i - \bar{x}}{\sigma_{\text{pop}}} $$

Glossary of Variables

x — raw value (observation) to standardize.
μ — mean of the distribution (population or assumed mean).
σ — standard deviation of the distribution (population SD, must be > 0).
z — Z-score, measured in standard deviations from the mean.
Φ(z) — standard normal CDF, the left-tail probability.
Percentile — 100 × Φ(z).
Left/Right/Two-tailed p-value — probabilities under the standard normal curve.
s — sample standard deviation (n − 1), used to standardize sample data.

How It Works: A Step-by-Step Example

Worked Example

Suppose a test has μ = 100 and σ = 15. A student scores x = 110.

Compute the Z-score using z = (x − μ) / σ: z = (110 − 100) / 15 ≈ 0.6667.
Compute the percentile: Φ(0.6667) ≈ 0.7475 → 74.75th percentile.
Right-tail probability: 1 − Φ(0.6667) ≈ 0.2525.
Two-tailed p-value: 2 × min(0.7475, 0.2525) = 0.5050.
Interpretation: About 0.67 SD above average, which is typical.

Frequently Asked Questions (FAQ)

What is a Z-score?

A Z-score standardizes a value relative to the mean and standard deviation: z = (x − μ) / σ. It indicates how many standard deviations a value is above or below the mean.

When should I use sample vs population standard deviation?

Use the population SD (σ) when the variability of the entire population is known. Use the sample SD (s, with n − 1 in the denominator) when you estimate variability from a sample.

Does a Z-score assume normality?

Z-scores are most interpretable under approximate normality. For highly skewed or heavy-tailed data, probabilities derived from the standard normal may be misleading.

How precise are the probabilities?

They are calculated via an accurate approximation to the normal CDF using the error function (erf). For most practical purposes, the precision matches standard statistical tables.

Are Z-scores the same as standard scores?

Yes. “Standard score” is another name for Z-score, representing units of standard deviation from the mean.

What percentile corresponds to z = 0, 1, and 2?

z = 0 → 50th percentile; z = 1 → ~84.13th percentile; z = 2 → ~97.72nd percentile (under the standard normal model).

Can I export dataset Z-scores?

Yes. After computing, use “Copy Z-scores (CSV)” to put a CSV list on your clipboard.

Authorship & Review

Tool developed by Ugo Candido.
Content verified by CalcDomain Expert Team.
Last reviewed for accuracy on: September 14, 2025.