Z-Score Calculator
Calculate a z-score and percentile from a value, mean, and standard deviation.
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About this calculator
How This Z-Score Calculator Works
Enter a value, the mean, and the standard deviation of its distribution, and the calculator returns the z-score and the corresponding percentile.
Worked Example
A value of 85, with a mean of 75 and a standard deviation of 10, gives a z-score of 1 — exactly one standard deviation above the mean — which falls at roughly the 84th percentile.
The Formula
z = (x − μ) ÷ σ, where x is the value, μ is the mean, and σ is the standard deviation. The z-score expresses any value as a standardized distance from the mean, measured in standard deviations — which makes it possible to compare values from completely different distributions on the same scale.
From Z-Score to Percentile
Assuming a normal (bell-curve) distribution, a z-score converts to a percentile using the standard normal cumulative distribution function — roughly, a z-score of 0 sits at the 50th percentile, +1 at about the 84th, +2 at about the 98th, and so on, with the distribution symmetric around the mean. This conversion assumes the underlying data is roughly normally distributed; for heavily skewed data, the percentile figure becomes less meaningful even though the z-score itself is still a valid standardized distance.
Common Uses
Standardized test scoring, quality control (how far a measurement is from a process average), identifying outliers in a dataset, and comparing a value from one distribution against a value from another. See the Standard Deviation Calculator to compute σ from a raw list of numbers first.
Worked example
A value of 85 with a mean of 75 and standard deviation of 10 gives a z-score of 1 — meaning it's exactly one standard deviation above the mean, at roughly the 84th percentile.
Frequently asked questions
What does a z-score tell you?
It measures how many standard deviations a value is from the mean of its distribution. A z-score of 0 is exactly average; positive values are above average, negative values are below.
How is percentile calculated from a z-score?
Using the cumulative distribution function of the standard normal distribution — it converts the z-score into the percentage of values expected to fall below it, assuming the data is normally distributed.
Does this work for data that isn't normally distributed?
The z-score itself is just a standardized distance from the mean and works regardless, but the percentile interpretation assumes a roughly normal (bell-curve) distribution — for heavily skewed data, the percentile figure may not be accurate.