Z-Score Calculator
Turn a raw value into a z-score — how many standard deviations it sits above or below the mean — with its percentile under a normal curve.
How it works
z = (x − μ) ÷ σ
A z-score restates a value as its distance from the mean measured in standard deviations. Subtract the mean and divide by the standard deviation: a positive z is above the mean, a negative z below it, and zero is exactly average. Because the units cancel, z-scores let you compare values from completely different scales, and under a normal curve each z maps to a percentile.
Worked example
A test score of 85 in a class with mean 70 and standard deviation 10 gives z = (85 − 70) ÷ 10 = 1.5. The score is 1.5 standard deviations above average, which for a normal distribution is about the 93.3rd percentile.
Frequently asked questions
What counts as a high or unusual z-score?
By the empirical rule about 95% of normal data lies within z = ±2 and 99.7% within ±3, so a z beyond ±2 is uncommon and beyond ±3 is rare. There is no hard cutoff — it depends on how strict your context is.
Can a z-score be negative?
Yes. A negative z simply means the value is below the mean; −1.5 is exactly as far from the mean as +1.5, just on the low side. The sign carries the direction, the size carries the distance.
Do I use the population or sample standard deviation?
Use the standard deviation of the distribution you are comparing against. For a true z-score that is the population σ; when you only have a sample estimate, the result is an approximation and a t-score is often more appropriate for small samples.
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