Z-Score

Z-scores standardize a value relative to a distribution by measuring how many standard deviations it is from the mean. This is useful for Outliers and Normalisation.

Definition:
The Z-score of a value $x$ is given by: $Z=\frac{x-\bar{x}}{s}$

where $\bar{x}$ is the sample mean and $s$ is the sample standard deviation.

Interpretation:

• $Z=0$: The value equals the mean.
• $|Z|>2$: Indicates a possible outlier (if normality is assumed).
• Z-scores allow comparisons across different distributions.

Assumptions:

• Data is approximately normally distributed.
• Useful primarily when comparing existing values to a distribution.

Use Cases:

• Standardizing data for machine learning algorithms.
• Detecting anomalies.
• Ranking or scoring values.

Related terms:

• Z-Test
• Z-Normalisation
• Z-Score

2. Modified Z-Score

• Formula:
  $M=\frac{0.6745\cdot (X-\text{median})}{\text{MAD}}$
  • $MAD$: Median Absolute Deviation
• Procedure:
  • Use this method for datasets with extreme outliers.
  • Points with $M>3.5$ are typically anomalies.