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.
How It's Calculated
Formula
z = \dfrac{x - \mu}{\sigma}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.