Z-Score

Formula

\[ z = \frac{x-\mu}{\sigma} \]

Parameters

• \(x\): value
• \(\mu\): mean
• \(\sigma\): standard deviation

What it means

How many standard deviations \(x\) is from the mean.

What it's used for

• Standardizing data for comparison.
• Outlier detection and normalization.

Key properties

• Mean 0 and standard deviation 1 after standardization.
• Dimensionless.

Common gotchas

• If \(\sigma=0\), z-score is undefined.
• Assumptions of normality are not required for the definition, only for probabilistic interpretations.

Example

If \(x=80\), \(\mu=70\), \(\sigma=5\), then \(z=2\).

How to Compute (Pseudocode)

Input: value x, mean mu, standard deviation sigma
Output: z-score

if sigma == 0:
  return undefined
return (x - mu) / sigma

Complexity

• Time: \(O(1)\) once \(\mu\) and \(\sigma\) are known
• Space: \(O(1)\)
• Assumptions: Computing \(\mu\) and \(\sigma\) from data is a separate step (often \(O(n)\))