Z-Score Calculator

A Z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a dataset. It standardizes values from different scales so they can be compared on a common basis.

Z-score formula: Z = (X − μ) / σ

What each variable means:

• X: the individual data point (observed value)
• μ (mu), the population mean (or sample mean x̄ if using a sample)
• σ (sigma), the population standard deviation (or s for sample standard deviation)
• Z: the resulting score; positive means above average, negative means below average

Standard Normal Distribution reference:

• Z = 0 → exactly at the mean (50th percentile)
• Z = +1 → 84.13th percentile (1 SD above mean)
• Z = −1 → 15.87th percentile (1 SD below mean)
• Z = +2 → 97.72nd percentile
• Z = −2 → 2.28th percentile
• Z = +3 → 99.87th percentile (used in “six sigma” quality control)

Worked example 1 — Exam scores: Class mean: 74 points. Standard deviation: 8 points. Student scored 90. Z = (90 − 74) / 8 = 16 / 8 = +2.0 This student scored in the top 2.28% of the class.

Worked example 2 — Height: Average male height in the US: 69.1 inches (5'9"), SD = 2.9 inches. A man is 6'3" (75 inches). Z = (75 − 69.1) / 2.9 = 5.9 / 2.9 = +2.034 He is taller than approximately 97.9% of men.

Worked example 3 — Finding a raw score: If the mean test score is 500 and SD is 100 (like the SAT), what score corresponds to the 90th percentile? 90th percentile Z ≈ +1.282 X = μ + Z × σ = 500 + 1.282 × 100 = 628

Uses: Quality control (detecting defects), finance (risk assessment — 2-sigma and 3-sigma events), academic grading (curved grades), medical research (identifying outlier lab values), and standardized testing (SAT, GRE scores).