Z-score Calculator

Enter a mean and a positive standard deviation, then either a raw value or a z-score, depending on the direction you need.

How it works

A z-score says how many standard deviations a value sits from the mean: z = (x − μ) / σ. The inverse is x = μ + zσ. σ must be greater than 0 — a zero or negative spread is rejected rather than dividing by zero.

This page does not look up a normal-table tail probability.

Formula

z = (x − μ) / σ
x = μ + zσ

Worked examples

A test score
x = 115, μ = 100, σ = 15 → z = 1
Back to the raw scale
z = 1, μ = 100, σ = 15 → x = 115

Useful notes

Whether σ is a sample s or a population σ is your choice; the algebra is the same.

FAQ

What does a z-score of 1 mean?: The value is one standard deviation above the mean. For μ = 100 and σ = 15, that is 115.
Can a z-score be negative?: Yes. Values below the mean have negative z.
Why is a zero standard deviation rejected?: You cannot divide by zero. If every observation is the same, a z-score is undefined.

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