Z-Score Calculator

Convert any value to a z-score, find its percentile rank, and calculate tail probabilities.

A z-score tells you how many standard deviations a value sits above or below the mean: z = (x − μ) / σ. A z of 0 is exactly average, +1.5 is one-and-a-half SDs above the mean, −2 is two below. Z-scores let you compare values measured on different scales and read probabilities from the standard normal table. A |z| beyond about 1.96 marks the outer 5% of a normal distribution — the usual two-tailed significance cutoff. Results are validated against SciPy to full precision.

Calculate Z-Score

When to use a z-score

Standardize a value when you need to compare it across different distributions (e.g. an SAT score vs an ACT score), find its percentile, or decide whether it is unusually extreme. Z-scores require the population mean and standard deviation; if you only have a sample and must estimate σ, a t-test (and the t-distribution) is the honest choice.

How to interpret it

Under a normal distribution, about 68% of values fall between z = ±1, 95% between ±2, and 99.7% between ±3 (the empirical rule). Convert a z to a percentile with the z-table or the normal distribution calculator.

Worked example

A test has μ = 500 and σ = 100. A score of 680 gives z = (680−500)/100 = 1.8, which falls at about the 96th percentile — higher than roughly 96% of test-takers. Enter x, μ, and σ above to get z and its percentile.

Percentile to z-score (and back)

For a normal distribution, a person at the 20th percentile has a z-score of about −0.84: they sit 0.84 standard deviations below the mean. Common conversions (computed with SciPy):

Percentile1st5th10th20th25th50th75th80th90th95th99th
z-score−2.326−1.645−1.282−0.842−0.67400.6740.8421.2821.6452.326

Going the other way: z = −1 is the 15.9th percentile, z = 1 the 84.1th, z = 1.5 the 93.3th and z = 2 the 97.7th. For any other value use the z-table or the normal distribution calculator.

Common mistakes

Guide: Understanding Z-Scores

A z-score (or standard score) tells you how many standard deviations a value is above or below the mean of a distribution. The formula is: z = (x − μ) / σ

A z-score of +1.5 means the value is 1.5 standard deviations above the mean. A z-score of −2 means it is 2 standard deviations below the mean. Z-scores allow you to compare values from different distributions on a common scale.

The percentile tells you what percentage of the population scores at or below a given value. A z-score of 0 is the 50th percentile — exactly half the distribution is below it. A z-score of +1 is approximately the 84th percentile.

Example: if a student scores at the 92nd percentile on a standardised test, 92% of students scored at or below them.

Known Mean & SD: Use this when you know the population parameters — for example, IQ tests have a defined mean of 100 and SD of 15, or when your textbook provides the distribution parameters.

From raw data: Use this when you have a dataset and want to evaluate where a specific value sits relative to that data. The calculator will estimate the mean and SD from your data first.

Left tail P(X ≤ x): The probability that a randomly chosen value falls at or below your value. This is the same as the percentile expressed as a proportion.

Right tail P(X ≥ x): The probability that a randomly chosen value falls at or above your value.

Two-tailed P(|X| ≥ |z|): The probability of being this extreme in either direction. Used in hypothesis testing.

In a standard normal distribution:

  • |z| < 1.0: Within one SD of the mean — 68% of observations
  • |z| < 2.0: Within two SDs — 95% of observations
  • |z| < 3.0: Within three SDs — 99.7% of observations
  • |z| > 3.0: Very rare — only ~0.3% of a normal distribution

Values with |z| > 2.5–3 are commonly flagged as potential outliers.

Related Tools

Normal Distribution Confidence Interval Descriptive Statistics T-Test Calculator