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Transformations: Z-Scores

Learning outcomes

On watching this video, students should be able to:

  1. Explain what a linear transformation is and identify a z-score as a type of linear transformation.
  2. Calculate a z-score and interpret it as a measure of the distance of a data point from the mean in units of standard deviation.
  3. Describe the properties of a variable transformed to z-scores, in particular that they have a mean of zero and standard deviation of 1, and explain why this makes them useful for comparing groups or individuals across variables that have different measurement units.
  4. Link z-scores to percentile points and explain their use for reference (normative) ranges.
  5. State the 68, 95, 99.7 rule or 3 sigma rule which refers to the percentage of observations within ±1z,  ±2z and  ±3z units from the mean in the standard normal distribution.
  6. Use the 3 sigma rule to make approximate probability statements about an observation.