Z-Score & Standard Normal Calculator

Z-Score Guide

What Is a Z-Score?

A z-score (standard score) measures how many standard deviations a value is from the mean. Formula: z = (x − μ) ÷ σ. Where x = individual value, μ = population mean, σ = population standard deviation. A z-score of 0 means exactly average. Positive z: above average. Negative z: below average. Example: exam mean = 65, SD = 12. A score of 77 → z = (77−65)/12 = 1.0. A score of 41 → z = (41−65)/12 = −2.0 (two standard deviations below average).

Z-Scores and Probability

Z-scores link to probabilities via the standard normal distribution (mean = 0, SD = 1). A z-score of 1.0 corresponds to the 84th percentile — 84% of values fall below this point. Z = 2.0 → 97.7th percentile. Z = −1.0 → 15.9th percentile. Z = 1.96 → 97.5th percentile (the 5% significance level boundary in two-tailed tests). These probabilities come from tables (z-tables) or can be calculated from the cumulative normal distribution function.

Applications of Z-Scores

Standardised testing: converting raw exam scores to z-scores allows direct comparison across tests with different means and SDs. IQ: mean 100, SD 15 → IQ 130 has z = (130−100)/15 = 2.0, placing it at the 97.7th percentile. Medical: a patient's blood test result is compared to the reference range (typically mean ± 2 SD) using z-scores. Finance: z-scores compare asset returns to their historical average, adjusted for volatility. Quality control: z-scores determine whether a production batch is wit

Z-Score vs T-Score

Z-scores apply when population mean and SD are known (or sample size is large, n > 30). T-scores apply when the population SD is unknown and estimated from a sample (small n). For large samples, z and t are nearly identical. The t-distribution has heavier tails than the normal distribution — accounting for additional uncertainty from estimating SD from a small sample. At n = 30, t-critical values are only slightly larger than z-critical values. At n = 10, the difference is significant. This is w