Z-score Calculator
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| Range from Mean | Approx. % of Data |
|---|---|
| μ ± 1σ | 68.27% |
| μ ± 2σ | 95.45% |
| μ ± 3σ | 99.73% |
What a Z-score Actually Measures
A Z-score (also called a standard score) expresses how many standard deviations a raw value sits from the mean of its distribution, calculated as z = (x − μ) / σ, where x is the raw value, μ is the population mean, and σ is the population standard deviation. A Z-score of 0 means the value equals the mean; a Z-score of +1.5 means the value sits one and a half standard deviations above it. Because the formula rescales any normally distributed variable to a common unit, Z-scores let you compare values that come from entirely different scales — a test score, a height, a reaction time — on the same footing.
From Z-score to Percentile
Once you have a Z-score, converting it to a percentile relies on the cumulative distribution function (CDF) of the standard normal distribution — the probability that a randomly drawn value from that distribution falls at or below a given Z. This calculator computes that CDF directly with a numerical approximation (Abramowitz and Stegun's rational polynomial approximation of the error function) rather than a lookup table, so it works for any Z-score, not just the ones printed in a textbook table. The well-known empirical rule — about 68% of values within ±1σ, 95% within ±2σ, and 99.7% within ±3σ — falls directly out of this same CDF, as shown in the table above.
A Common Mix-up: Population vs. Sample
This calculator uses the population mean and population standard deviation, which is the standard definition of a Z-score. If you only have a sample and are estimating the mean and spread from it, computing standard deviation correctly still matters for the result to be meaningful — see the standard deviation calculator to get σ right first. For general distribution summaries beyond a single Z-score, the statistics calculator can help.
Frequently Asked Questions
What is a Z-score?
A Z-score tells you how many standard deviations a value is from the mean of its distribution. It's calculated as z = (x - mean) / standard deviation. A Z-score of 0 means the value equals the mean, positive Z-scores are above the mean, and negative Z-scores are below it.
How do you turn a Z-score into a percentile?
The percentile is the cumulative probability of the standard normal distribution up to that Z-score, found using the standard normal CDF. This calculator computes it directly with a numerical approximation of the CDF (rather than a table), so it works for any Z-score, including ones you won't find in a printed Z-table.