Calculate Z score

Express an observation as standard deviations from a mean and estimate its normal percentile.

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ToolZ Score Calculator
Input
Output
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How it works

The score is z = (value − mean) ÷ standard deviation. An error-function approximation converts z into the percentage below the observation under a normal model.

  • Mean zero and deviation one represent the standard normal starting point.
  • A non-positive deviation is rejected because it cannot scale distance.

Worked example

A test score below average
A score of 85 against a mean of 100 and standard deviation of 15
Input
											Value: 85
Mean: 100
Standard deviation: 15
Output
												Z score: -1
Percentile below: 15.87
Percentile above: 84.13

When to use this

Test scores are standardized across cohorts, quality measurements compare distance from target, and outlier lessons convert observations to normal percentiles.

Edge cases

  • A score below the mean has a negative z and a percentile below 50% under the normal approximation.
  • Small samples can make z-score outlier rules misleading, as NIST notes.
  • A zero standard deviation is rejected because the denominator would be zero.

References