Z-Score Normalization

Z-score (zero-mean) normalization rescales an attribute's values based on its mean (μ) and standard deviation (σ):

$$v' = \frac{v - \mu}{\sigma}$$

where:

• $v$ = original value
• $\mu$ = mean of the attribute
• $\sigma$ = standard deviation of the attribute
• $v'$ = normalized value

Effect

• The transformed data has mean 0 and standard deviation 1.
• Values are typically small, ranging roughly from −3 to +3 (assuming a roughly normal distribution), though technically unbounded.

Why use it

• Useful when the min/max of an attribute are unknown or when there are outliers that would distort min-max normalization.
• Puts differently-scaled attributes (e.g., income in lakhs vs age in years) on comparable footing before applying distance-based algorithms (k-NN, k-means, clustering).

Raw values:  [200, 300, 400, 600, 1000]
                    │  subtract mean, divide by std dev
                    ▼
Z-scores:    [negative, negative, ~0, positive, large positive]

A z-score of 0 means the value equals the attribute's mean; positive/negative z-scores indicate how many standard deviations above/below the mean the value lies.