Z-score
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Z-Score & Normal Distribution Calculator
Z-score (x-mean)/sd and an approximate left-tail percentile via an erf expansion.
Page updated 2026-09-04.
Calculator
Calculated Results
Approx percentile
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Approx above
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How many standard deviations away, and what percentile that means
A value of 82, against a mean of 70 and standard deviation of 8, has a z-score of 1.50 -- 1.5 standard deviations above the mean. Under a normal distribution, that corresponds to roughly the 93.32nd percentile, meaning only about 6.68% of values would be expected to fall above 82.
Z-score itself is a simple standardization: z = (x - mean) / standard deviation = (82 - 70) / 8 = 12 / 8 = 1.5. It converts any raw value into a universal 'how many standard deviations from average' scale, which is what makes it possible to compare values from completely different distributions (different means, different scales) on the same footing.
The jump from z-score (1.5) to percentile (93.32%) relies on the well-established shape of the normal distribution -- roughly 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3 (the empirical rule), with the precise 93.32% figure coming from the exact normal distribution formula rather than these rounded approximations.
The math behind the percentile, and its real assumption
Percentile uses Abramowitz and Stegun erf 7.1.26. Not a stats package. Converting a z-score into a precise percentile requires evaluating the normal distribution's cumulative function, which has no simple closed-form formula -- this calculator uses a well-known polynomial approximation (Abramowitz and Stegun's formula 7.1.26) to compute it accurately without needing a full statistical software package.
This entire calculation assumes the underlying data is actually normally distributed (the classic bell curve) -- for data that's skewed, has heavy tails, or otherwise doesn't follow a normal distribution, the percentile figure this produces wouldn't accurately reflect the real-world percentile in that specific dataset.
A negative z-score would indicate a value below the mean, and its corresponding percentile would be below 50% -- the same formula and logic apply symmetrically in both directions from the mean.
Related statistical tools
This calculation needs a mean and standard deviation as inputs -- if you're calculating those from a raw dataset first, the Standard Deviation & Variance Calculator produces exactly those two figures.
For a related but different statistical question -- how large a sample needs to be for a given confidence level -- the Margin of Error & Sample Size Calculator covers that.
Frequently Asked Questions (FAQ)
How is the z-score of 1.50 calculated?
Z-score = (value - mean) / standard deviation = (82 - 70) / 8 = 1.5. It expresses how many standard deviations the value 82 sits above (positive) or below (negative) the mean of 70.
How does a z-score of 1.5 translate to a 93.32 percentile?
Percentile uses Abramowitz and Stegun erf 7.1.26. Not a stats package. It uses the cumulative normal distribution function, which has no simple closed-form formula -- this calculator applies a well-established polynomial approximation to compute the precise percentile that corresponds to any given z-score under a normal distribution.
Does this calculation assume the data is normally distributed?
Yes, entirely. The percentile figure is only meaningful if the underlying data actually follows a normal (bell curve) distribution. For skewed or otherwise non-normal data, this calculated percentile wouldn't accurately reflect the real percentile within that specific dataset.
What would a negative z-score mean?
It would indicate the value sits below the mean, with a corresponding percentile below 50% -- the same formula and standardization logic apply symmetrically for values above or below the mean.
Where do I get the mean and standard deviation this calculator needs?
From your own dataset's summary statistics, or calculate them directly from raw data first using a standard deviation calculator -- this z-score tool requires those two figures as inputs rather than computing them from a raw list itself.
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