Z‑Score Calculator

Mastering Z‑Scores: Complete Guide

A z‑score (standard score) quantifies how many standard deviations an observation is from the mean. It is a cornerstone of statistical analysis, enabling comparisons across different distributions and forming the basis of hypothesis testing and confidence intervals. This Professional Z‑Score Calculator offers three essential functions: converting raw values to z‑scores, computing probabilities from z‑scores, and finding z‑scores corresponding to given percentiles. Each result is accompanied by a visual curve and a step‑by‑step explanation.

Calculating a Z‑Score from Raw Data

The formula is z = (x − μ) / σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. For example, if a student scored 85 on a test with a class mean of 75 and standard deviation of 10, their z‑score is (85−75)/10 = 1.0—they are one standard deviation above average.

Probabilities from Z‑Scores

The standard normal cumulative distribution function (CDF), often denoted Φ(z), gives the probability that a random variable from a standard normal distribution is less than or equal to z. The calculator provides left‑tail, right‑tail, and two‑tail probabilities, which are directly used in significance testing (p‑values).

Finding Z‑Scores from Percentiles

The inverse CDF (quantile function) maps a cumulative probability back to a z‑score. For instance, to find the z‑score that cuts off the top 5% of the distribution, you would input the 95th percentile, yielding z ≈ 1.645.

The Empirical Rule (68–95–99.7)

For any normal distribution, approximately 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean. This rule is a quick way to gauge the rarity of a z‑score.