Value Ranges of Normalization Methods

Method	Formula	Value Range
Min-Max Normalization	$v' = \dfrac{v-min}{max-min}(\text{new\_max}-\text{new\_min}) + \text{new\_min}$	Bounded exactly to [new_min, new_max] (commonly [0,1])
Z-Score Normalization	$v' = \dfrac{v-\mu}{\sigma}$	Unbounded in theory; in practice typically falls roughly in [−3, +3] for approximately normal data
Decimal Scaling	$v' = \dfrac{v}{10^j}$, where $j$ is the smallest integer such that $\max(|v'|) < 1$	Bounded to (−1, 1)

Notes

• Min-max preserves the exact relationships in the original data and gives a strictly bounded range — but it's sensitive to outliers (a single extreme value stretches/compresses everything else).
• Z-score has no hard bounds since it depends on how far a value is from the mean in standard-deviation units — extreme outliers can produce large |z| values well beyond ±3.
• Decimal scaling normalizes by moving the decimal point, controlled by the maximum absolute value in the data, guaranteeing the result always lies strictly between −1 and 1.

Min-Max:        [0 ──────────── 1]        (fixed, bounded)
Z-Score:      (-∞ ...  -3  0  +3  ... +∞)  (unbounded, ~[-3,3] typical)
Decimal Scale:   (-1 ──────── 1)           (bounded, data-dependent)