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Z-Score Calculator

Standardize a value using a supplied mean and standard deviation and interpret its distance from the mean.

Calculate

Enter a value, the reference distribution’s mean and a positive standard deviation. All three must use the same original unit.

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Formula and variables

z = (x − μ)/σ

x is the observed value, μ the reference mean and σ its positive standard deviation. The z-score itself has no units.

Worked example

A score of 85 from a reference distribution with mean 70 and SD 10 has z = (85−70)/10 = 1.5. It is 1.5 standard deviations above the mean. Substituting back gives 70 + 1.5 × 10 = 85.

When to use this formula

Use a z-score to express relative position or compare scales. Turning it into a normal percentile additionally requires a defensible normal distribution assumption.

Common mistakes

Do not divide by variance or by standard error when standardizing a single observation. A constant distribution has SD zero, so this standardization is undefined.

Is a negative z-score an error?

No. It indicates that the observation lies below the mean. A z-score of zero means the observation equals the mean.

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