Z-Score Calculator
Calculate a z-score with the formula z = (x − μ) / σ. Enter a value, the distribution mean, and standard deviation to find how many standard deviations that value is from the mean.
Calculator Inputs
The individual data point you want the z-score for.
The average of the distribution the value belongs to.
How spread out the distribution's values are. Must be greater than zero.
- Z-Score
- 1.50
About these results
CalcSpectrum calculators produce automated estimates using established formulas and the specific values you enter. Because real-world outcomes depend on input accuracy, mathematical assumptions, and rounding conventions, please verify critical figures with an appropriately qualified financial, tax, medical, legal, or other professional, as relevant.
How It's Calculated
Formula
A z-score tells you how many standard deviations a value sits from its distribution's mean. A z-score of 0 means the value equals the mean exactly. A positive z-score means the value is above the mean; a negative z-score means it's below. The magnitude matters as much as the sign — a z-score of 2 is twice as many standard deviations from the mean as a z-score of 1. This calculator applies the formula directly: enter a value, the distribution's mean, and its standard deviation, and it returns the z-score. This version reports the z-score itself only — it does not convert that score into a percentile or probability (that requires the normal distribution's cumulative distribution function, which is out of scope here).
Worked Examples
Standard case: x = 85, μ = 70, σ = 10
- Subtract the mean from the value: x − μ = 85 − 70 = 15
- Divide by the standard deviation: z = 15 / 10 = 1.5
- Result: z = 1.5 — the value is 1.5 standard deviations above the mean.
Below the mean: x = 55, μ = 70, σ = 10
- Subtract the mean from the value: x − μ = 55 − 70 = −15
- Divide by the standard deviation: z = −15 / 10 = −1.5
- Result: z = −1.5 — the value is 1.5 standard deviations below the mean.
Frequently Asked Questions
What does a negative z-score mean?
A negative z-score means the value is below the distribution's mean. The sign only indicates direction — below (negative) or above (positive) the mean — while the absolute value tells you how far away it is, in standard deviations. A z-score of −2 is just as far from the mean as a z-score of +2, only on the other side.
Why must the standard deviation be positive?
The standard deviation measures spread, and spread can't be negative — a distribution's values are either identical (standard deviation of exactly 0) or vary by some positive amount. Dividing by zero is undefined, and a negative standard deviation isn't a valid measurement, so this calculator requires σ to be strictly greater than zero.
Does a z-score tell me a probability?
Not by itself. The z-score only tells you how many standard deviations a value is from the mean. Converting that into a percentile or probability requires looking the z-score up against the normal distribution's cumulative distribution function (CDF), which this calculator does not currently compute.
Can x and the mean be equal?
Yes. If the value equals the mean, the z-score is exactly 0, meaning the value sits right at the center of the distribution — neither above nor below average.