Standardization › Mathematical Foundation

For a feature $j$ with $n$ observations, standardization transforms each value $x_{ij}$ to $z_{ij}$ using:

$$z_{ij} = \frac{x_{ij} - \bar{x}_j}{s_j}$$

where:

• $z_{ij}$ is the standardized value of feature $j$ for observation $i$
• $\bar{x}_j = \frac{1}{n}\sum_{i=1}^n x_{ij}$ is the mean of feature $j$
• $s_j = \sqrt{\frac{1}{n}\sum_{i=1}^n (x_{ij} - \bar{x}_j)^2}$ is the standard deviation of feature $j$

After standardization, each feature in your dataset is transformed so that it has a mean of zero and a standard deviation of one. This transformation ensures that all features, regardless of their original scale or units, are directly comparable and contribute equally to the analysis. In practical terms, this means that no single feature will dominate the learning process simply because it has larger numeric values. Instead, every feature is centered and scaled, allowing algorithms.especially those sensitive to feature magnitude, such as regularized regression or clustering.to perform optimally and fairly.

• Mean: $\bar{z}_j = 0$ for all features
• Standard deviation: $s_{z_j} = 1$ for all features
• Variance: $\text{Var}(z_j) = 1$ for all features

This ensures that all features contribute equally to distance calculations and regularization penalties.