Statistics Tools

Comprehensive Z-Score Calculator

Compute z-scores, percentiles, p-values, and visualize the normal distribution

Inputs

Enter parameters or paste a dataset

Choose how to provide μ and σ

The observation you want to standardize

Average of the distribution

Must be > 0

Probability & Critical Values

Optional: compute ranges, percentiles, and critical z

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Quick Scenarios

Summary

Inputs, derived statistics, and interpretation

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Visualizations

Normal curve, CDF, and tail areas

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Z Table & Critical Values

Reference probabilities and common critical z-scores

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Reference

Formulas, interpretation, and usage notes

Core Formula
z = (x − μ) / σ
Standardize a value by expressing it as the number of standard deviations from the mean.
Interpreting z

Rules of thumb for normal-ish data.

  • |z| < 1: very typical
  • |z| ≈ 2: unusual
  • |z| ≥ 3: extreme / potential outlier
Percentiles

Convert z into a percentile using the standard normal CDF Φ(z).

  • Percentile = 100 × Φ(z)
  • Right tail = 1 − Φ(z)
  • Two-tailed p = 2 × min(tails)
When to use

Z-scores are most interpretable when data is approximately normal.

  • Standardizing scores (tests, ratings, KPIs)
  • Outlier checks
  • Probability under a normal model

Understanding Z-Scores

What is a z-score?

A z-score tells you how many standard deviations a value is above or below the mean.

  • Positive z: above the mean
  • Negative z: below the mean
  • z = 0: exactly at the mean
  • Magnitude indicates unusualness
Inputs matter

Your mean and standard deviation determine the scaling.

  • Use population σ when you know it
  • Use sample s (n−1) when estimating from data
  • σ must be > 0
  • Check units of x, μ, and σ
Assumptions

Probabilities rely on a normal model for X.

  • Normal approximation may be poor for skewed data
  • Outlier rules of thumb are heuristic
  • For small samples, consider t-scores
  • Always sanity-check with plots