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Z-Score Calculator

Standardize any value and instantly see where it falls on the normal distribution

A z-score (also called a standard score) measures how many standard deviations a value is from the mean of its distribution. It answers the question: is this value typical, unusually high, or unusually low?

The formula is simple: z = (x − μ) / σ, where x is your value, μ is the mean, and σ is the standard deviation. A z-score of 0 means the value equals the mean. A z-score of +2 means the value is 2 standard deviations above the mean — in the top ~2.3% of a normal distribution.

Z-scores are essential for comparing values across different scales — for example, comparing a SAT score to an IQ score — and for finding percentiles from the standard normal table. Enter a value, mean, and standard deviation below to compute your z-score and the corresponding percentile.

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FAQ

What is a z-score?
A z-score (standard score) measures how many standard deviations a data point lies above or below the mean. The formula is z = (x − μ) / σ. Z-scores let you compare values from different distributions on a common scale and look up percentiles using the standard normal table.
How do you calculate a z-score?
Subtract the mean (μ) from your value (x), then divide by the standard deviation (σ): z = (x − μ) / σ. For example, if a test score is 85, the mean is 75, and the standard deviation is 10, then z = (85 − 75) / 10 = 1.0. The AI will show this calculation step by step and plot the result on the normal curve.
What does a z-score of 2 mean?
A z-score of +2 means the value is 2 standard deviations above the mean. In a normal distribution, about 95% of values fall between z = −2 and z = +2, so a z-score of 2 is in approximately the top 2.3% — a fairly unusual result. A z-score of −2 is symmetrically in the bottom 2.3%.
How do you convert a z-score to a percentile?
Use the standard normal cumulative distribution function (CDF), also called the Φ function. It gives the probability P(Z ≤ z) — the fraction of the distribution below your z-score. For example, z = 1.0 corresponds to the 84th percentile (84% of values fall below). The AI calculates and visualizes this shaded area automatically.
When should you use z-scores?
Use z-scores when you need to compare values from different distributions (e.g., SAT vs ACT scores), identify outliers (|z| > 2 or 3 is often flagged), or convert raw scores to percentiles. Z-scores require the data to be approximately normally distributed for percentile interpretations to be accurate.
What is the difference between a z-score and a percentile?
A z-score is a raw standardized distance from the mean (can be any number, positive or negative). A percentile is the percentage of values that fall below a given point (always 0–100). They are related: every z-score maps to exactly one percentile via the normal CDF. For instance, z = 0 is always the 50th percentile.

Related Topics

Normal Distribution CalculatorP-Value CalculatorStatistics Calculator