Z Score Calculator

The z-score calculator computes z = (x - mu) / sigma, where mu is the mean and sigma is the standard deviation, and any one of the four variables, z, x, mu, or sigma, can be solved for from the other three. A fifth mode inverts the standard normal cumulative distribution to recover a z-score from a probability or percentile. Standard normal probabilities are computed with the Abramowitz and Stegun 7.1.26 erf approximation, which has a worst-case absolute error near 1.5e-7 across the real line, so the reported tail values stay accurate well into the extreme tails.

1. Any of the four variables, z, x, mu, or sigma, can be set as the one being solved for, with the remaining three supplied as knowns.
2. Accepted formats include decimals, fractions, mixed numbers, scientific notation, and comma-grouped numbers, such as 1.96, 1 3/4, 1-3/4, 1.5e2, and 1,250.
3. Results report the solved unknown, the z-score itself, and four standard normal probabilities, left-tail, right-tail, two-tailed, and inner, plus the corresponding percentile.
4. Decimal places, displaying probabilities as percentages, and a step-by-step derivation panel are all available as advanced display options.
5. A batch mode scores many observations at once against the shared mean and sigma entered above, either typed in row by row or imported from a single-column CSV, with the results exportable back to CSV.

Formula reference. Standard score: z = (x - mu) / sigma. Standard normal CDF: Phi(z) = 0.5 * (1 + erf(z / sqrt(2))). Right-tail: 1 - Phi(z). Two-tailed: 2 * (1 - Phi(|z|)). Inner: 1 - 2 * (1 - Phi(|z|)). Inverse for the z from probability mode uses Acklam's rational approximation, accurate to roughly 1.15e-9 in the central region.

A z-score expresses how unusual an observation is relative to its distribution by counting standard deviations from the mean. The same calculation powers hypothesis tests, percentile reporting, outlier flags, and process control charts, so the tool fits any context where raw numbers need to be made comparable across scales.

• Statistics coursework: Verify textbook problems where x = 78, mu = 70, sigma = 4 yields z = 2 and a percentile of 97.72.
• Standardized testing: Convert a raw SAT or GRE score to a z-score against the published mean and standard deviation, then read the percentile to position the result inside the cohort.
• Quality control: Score a production measurement against the target mean and process sigma to flag whether it sits inside the natural plus or minus three sigma control band.
• Finance and risk: Standardize a daily return against trailing sample statistics to estimate how heavy a single-day move is, recognizing that real return distributions have fatter tails than the normal model assumes.
• Clinical reference ranges: Compare a lab value to a population mean and sigma, then read the two-tailed probability to gauge how far from typical the observation falls.
• Survey research: Standardize Likert composite scores or attitude indices across instruments with different scales so they can be compared on a common metric.
• Sports analytics: Score a single-game stat line against a player's season mean and sigma to identify breakout or slump performances by deviation count rather than raw totals.
• Inverse lookup: Use the z from probability mode to recover the critical value for a 95 percent confidence interval (z = 1.96) or a one-tailed 99 percent threshold (z = 2.326).