Z-Score Normalization - standardisation to zero mean and unit standard deviation

Overview

z_ij ∈ (−∞,+∞); columns have mean=0 and population σ=1. Handles negative values and large-scale differences. The cost flip ensures higher z = better. Used in HELLWIG method and TAXONOMY.

Output

normalized matrix, higher is better

Data

Crisp, complete numeric matrix

Size

2+ alternatives, 3-12 criteria works best

Used for

Preprocessing

How it works

1. 1

   Compute column mean x̄_j and population standard deviation σ_j = √(Σ_i(x_ij − x̄_j)²/m). Benefit: z_ij = (x_ij − x̄_j)/σ_j. Cost: z_ij = (x̄_j − x_ij)/σ_j (direction flip so higher is better).

   $$z_{ij}=\{\begin{array}{ll}(x_{ij}-{\bar{x}}_j)/{\sigma }_j& j\in J^{+}\\ ({\bar{x}}_j-x_{ij})/{\sigma }_j& j\in J^{-}\end{array},{\sigma }_j=\sqrt{\frac{1}{m}\sum _i(x_{ij}-{\bar{x}}_j{)}^2}$$

   Hellwig 1968, p.310 (standardisation formula; pending PDF page verification)

Edge cases and pitfalls

• Sample σ (÷(m−1)) vs population σ (÷m) gives different values for small m; document which is used.

Works with

How to cite