Statistics - Z Score (Zero Mean) or Standard Score

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1 - About

Any raw score from any scale can be converted to Z scores (Z scale)

In statistics, the standard score is the signed number of standard deviations by which an observation or data is above the mean.


3 - Formula

<math> \text{Z Score} = \frac{\displaystyle \href{raw score}{X} – \href{mean}{\bar{X}}}{\href{standard_deviation}{\displaystyle \text{Standard Deviation}}} </math>


With this formula:

  • The mean Z-score is Z = 0
  • Positive Z scores are above average
  • Negative Z scores are below average

4 - Percentile

Z-scores can be used to find percentile rank (Raw score ~ Z-score ~ Percentile rank)

If the distribution is normal, Z=0 means a Median percentile (50th, 50 percent of the distribution falls below the mean)

5 - Documentation / Reference