Standardization vs Min-Max Scaling

Many models — anything that computes distances, dot products, or gradients across features with different natural scales — are sensitive to the raw units a feature happens to be measured in. A feature ranging from 0 to 1,000,000 (say, market cap in dollars) will numerically dominate a feature ranging from -1 to 1 (say, a momentum z-score) in any distance-based or gradient-based calculation, even if the momentum feature is far more predictive. Rescaling features onto a common footing before training is standard practice, and the two most common ways to do it — standardization and min-max scaling — make very different assumptions about what "common footing" should mean.

The two transformations, in words

Standardization (also called z-scoring) subtracts the feature's mean and divides by its standard deviation:

$$z = \frac{x - \bar{x}}{\sigma} .$$

In plain English: after standardization, a feature has mean 0 and standard deviation 1, and a value of $z=2$ means "two standard deviations above the average," regardless of the feature's original units. The transformed values are not bounded to any fixed range — a genuine extreme value stays extreme, just measured in standard-deviation units instead of raw units.

Min-max scaling instead maps the feature linearly onto a fixed range, usually $[0, 1]$:

$$x' = \frac{x - x_{\min}}{x_{\max} - x_{\min}} .$$

In plain English: the smallest value in the data becomes exactly 0, the largest becomes exactly 1, and everything else is placed proportionally in between. Every rescaled value is guaranteed to land inside $[0, 1]$, by construction.

Why the outlier behavior differs so much

Min-max scaling anchors its entire range to just two numbers: the minimum and the maximum. A single extreme outlier — one $50 million trade where every other trade is under $500,000 — becomes the new maximum, and every other value gets compressed into a tiny sliver near 0. Standardization is less extreme (an outlier inflates the standard deviation, pulling other z-scores toward 0), but not immune — a heavy enough outlier still distorts the scale for everyone.

Worked example: one extreme value, two treatments

Suppose a feature (daily trade notional, in $000s) has 99 values clustered between 10 and 200, plus one outlier at 5,000. Min-max scaling maps the outlier to exactly 1.0 and squeezes the 99 normal values into the tiny range roughly $[0.0, 0.038]$ — they've lost almost all their relative spread. Standardization, with a mean pulled up to about 59 and a standard deviation inflated to around 495, maps the 99 normal values to z-scores mostly between $-0.1$ and $0.28$ — still compressed, but nowhere near as severely collapsed as the min-max version, and the outlier itself sits at a z-score around 10, clearly flagged as extreme rather than silently becoming the new "1.0."

Standardization is built around exactly the mean-and-standard-deviation description shown in the explorer above — a feature is described by how many standard deviations a value sits from the center, which is a more outlier-aware framing than "where does this fall between the smallest and largest value seen."

What this means in practice

Min-max scaling is a reasonable default only when you're confident the feature has no meaningful outliers and its true range is well known and stable (for example, a feature that's already a bounded ratio or percentage). Standardization is the safer general-purpose choice for financial features, which are routinely fat-tailed, but when outliers are severe enough to distort even the mean and standard deviation, neither is ideal — robust scaling using the median and interquartile range is usually the better choice.