Z-score Calculator

In a normal distribution: 68% of values fall within 1 standard deviation of the mean (z between -1 and +1), 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This is the empirical rule. Practical applications: if heights are normally distributed with mean 70" and SD 3", about 95% of people are between 64" and 76" (70 ± 2×3). Values beyond 3 standard deviations are extremely rare, occurring in only about 0.3% of observations — roughly 1 in 370.

Convert each raw score to a z-score using that test's mean and standard deviation. A student who scores 90 on a test with mean 80, SD 5 has z = (90-80)/5 = 2.0. Another who scores 75 on a test with mean 60, SD 10 has z = (75-60)/10 = 1.5. The first student performed relatively better — their score is 2 standard deviations above average versus 1.5. This comparison would be impossible with raw scores alone because the tests have different scales and difficulty levels.

The p-value is the probability of observing a result at least as extreme as the one obtained, assuming the null hypothesis is true. For a z-test, the p-value is the area in the tail(s) of the standard normal distribution beyond the calculated z-score. A z-score of 1.96 corresponds to a two-tailed p-value of 0.05 — the conventional threshold for statistical significance. A z-score of 2.576 gives p = 0.01. If your calculated z exceeds the critical value for your α level, the result is statistically significant.

Z-scores can be calculated for any distribution (subtract mean, divide by SD), but their probability interpretation using the standard normal table is only accurate for normally distributed data. For non-normal distributions, the z-score still tells you how many standard deviations from the mean a value is, which is useful for outlier detection. However, you cannot directly read probabilities from a normal table. For small samples from unknown distributions, use t-scores (t-distribution) instead. The Central Limit Theorem guarantees that z-scores become increasingly accurate as sample size grows, even for non-normal populations.

Standardized tests like the SAT, ACT, GRE, and IQ tests all use z-score logic to create their scaled scores. The raw score is converted to a scale designed to have a specific mean and SD (SAT: mean 500, SD 100 per section; IQ: mean 100, SD 15). A score of 700 on the SAT corresponds to z = (700-500)/100 = 2.0 — the 97.7th percentile. This standardization allows scores to be compared across different test administrations even when question difficulty varies, because all editions are calibrated to the same statistical distribution.