Z-Score Calculator

A z-score (standard score) tells you how many standard deviations a data value lies from the mean. The formula z = (x − μ) / σ converts any normal distribution to the standard normal distribution with mean 0 and standard deviation 1. Positive z means above average; negative z means below. A z-score of 2 indicates the value sits exactly two standard deviations above the mean — a relatively uncommon outcome in a bell curve.

Example: exam score x = 85, class mean μ = 75, standard deviation σ = 5. Then z = (85 − 75) / 5 = 10 / 5 = 2. The student scored two standard deviations above average. If another student scored 70, z = (70 − 75) / 5 = −1 — one standard deviation below. Z-scores make comparisons fair across different scales: a 680 SAT section score and a 85% class exam can both be expressed as z-values relative to their respective distributions.

The standard normal cumulative distribution function Φ(z) gives P(Z ≤ z) — the area under the bell curve to the left of z. By symmetry, Φ(0) = 0.50 exactly: half the distribution falls below the mean. The famous value z = 1.96 corresponds to Φ(1.96) ≈ 0.975 = 97.5%, which is why 95% confidence intervals use ±1.96 standard errors. z = −1.96 leaves 2.5% in each tail for a two-tailed 5% significance test.

Reverse calculation finds the raw score from a z-value: x = μ + zσ. If you need the 90th percentile of a distribution with μ = 100 and σ = 15, find z ≈ 1.28 from tables, then x = 100 + 1.28 × 15 ≈ 119.2. This calculator supports three modes: compute z from x, find probability from z, and solve for x given z.

Z-scores connect directly to the standard deviation calculator and average calculator — you need μ and σ before standardizing. They underpin hypothesis tests via the p-value calculator and margin-of-error formulas in the confidence interval calculator. Always confirm data are approximately normal (or n is large for means) before interpreting z-probabilities strictly.

Empirical rule quick reference for normal data: about 68% of values fall within z = ±1, 95% within z = ±2, and 99.7% within z = ±3. Use this tool to verify homework, identify outliers, standardize features for comparison, or read statistical tables without flipping textbook pages.

When comparing two values from the same distribution, subtract their z-scores to see how many standard deviations apart they are. A shift from z = 1 to z = 2 represents one additional standard deviation of separation — a useful frame for quality control limits and standardized test reporting.