Z-Score Calculator

z = (x − μ) / σ. Subtracting μ centers the distribution at zero. Dividing by σ rescales it so 1 unit of z means 1 standard deviation in the original units.

A z of +1.5 means 'this value is 1.5 standard deviations above the mean.' Because the normal distribution is symmetric and fully determined by its mean and SD, that single number is all you need to find the percentile, the probability of seeing a value at least that extreme, or the gap between two distributions on a common scale.

Reverse the formula to go back: x = μ + z · σ. Useful for finding the raw score that corresponds to a target percentile (e.g. 'what SAT score is the 95th percentile?').

Suppose SAT scores have μ = 1050 and σ = 210, and a student scores 1300.

z = (1300 − 1050) / 210 = 250 / 210 ≈ 1.190.

Φ(1.190) ≈ 0.8830 → the student beat about 88.3% of test-takers. About 11.7% scored higher.

To find what score is the 90th percentile, invert: x = 1050 + 1.2816 · 210 ≈ 1319 (since Φ⁻¹(0.90) ≈ 1.2816).

For any normal distribution, roughly 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD. These come straight from the z-table: Φ(1) − Φ(−1) ≈ 0.6827.

Practical use: if a manufacturing process has μ = 100 mm and σ = 2 mm, parts outside 94–106 mm are 'three-sigma' outliers — you should see them roughly 3 in 1000 if the process is in control.

If the underlying distribution is strongly skewed (income, response times, body-fat percentage at the extremes), normal-based percentiles will mis-rank the tails. Check a histogram or QQ-plot first.

If your sample size is small (n < 30) and the population SD is unknown, prefer Student-t intervals instead of normal-based ones. The t adjusts for the extra uncertainty in estimating σ from a small sample.

Categorical data has no meaningful z-score. Standardize only continuous variables.

Using sample SD without noting it: z = (x − x̄) / s is conceptually correct but technically a t-score for inference. Report which one you used.

Confusing percentile with probability of equal scores. Φ(z) is the probability of a randomly drawn value being LESS than x, not equal to x — continuous distributions assign probability 0 to any single point.

Comparing z-scores across different distributions without checking that both are roughly normal. Two z = +2 values can mean very different things if one population is skewed.