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Math / Z-score

Z-score Calculator

Convert an observation into standard-deviation units from a distribution’s mean. A positive z-score is above the mean, a negative score is below it, and zero is exactly at the mean. z=(x−μ)/σ, where σ is a positive population standard deviation. The result is a distance in standard-deviation units, not a probability by itself.

Your inputs

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Your result
2
standard deviations
Z-score

Example inputs shown. Adjust them to explore.

A familiar situation

Express a test result as standard deviations above or below a comparison-group mean.

How to use this calculator

  1. Enter observation x, mean μ, standard deviation σ.
  2. Select Calculate to see the answer and its breakdown.
  3. Keep a scenario, change an input and compare the results.

Use consistent units and check the assumptions below before applying the result.

How z-score is calculated

z=(x−μ)/σ, where σ is a positive population standard deviation. The result is a distance in standard-deviation units, not a probability by itself. A value of 70 in a distribution with mean 60 and SD 5 has z=(70−60)/5=2.

z=(x−μ)/σ.
A worked example

A value of 70 in a distribution with mean 60 and SD 5 has z=(70−60)/5=2.

Assumptions & useful context

The result depends on the values you enter and the method described above. Displayed numbers may be rounded; calculations use the unrounded values. Check that this model matches your situation before relying on it.

Frequently asked questions

Does z=2 always mean a 97.7th percentile?

Only under a normal-distribution assumption; the score alone does not establish a distribution shape.

Why must standard deviation be positive?

A zero spread makes division by σ undefined.

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