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1. Why Scales Differ

Features like monthly income (tens of thousands) and monthly bill (hundreds to thousands) live on different ranges. Without scaling, large-magnitude features dominate distances and gradients.

2. Who Needs Scaling

• Scale-sensitive: KNN, SVM, PCA, K-Means (distance / projection math)
• Scale-insensitive: Decision trees, random forests (per-feature splits)

3. Two Core Methods

• Z-score standardization: mean 0, std 1; unbounded
• Min-max normalization: map values into [0, 1]

4. Telecom Churn Context

import pandas as pd

df = pd.read_csv("telecom_churn.csv")
print(df[["Monthly Income", "Monthly Bill"]].describe())

Income spans roughly $5k–$200k+; bills span about $710–$36k—clear scale imbalance.

5. Z-Score Standardization

Formula: \(Z = (x - \mu) / \sigma\)

income_mean = df["Monthly Income"].mean()
income_std = df["Monthly Income"].std()
df["Income_Zscore"] = (df["Monthly Income"] - income_mean) / income_std
print(df[["Monthly Income", "Income_Zscore"]].head(3))

After transform, mean ≈ 0 and std ≈ 1.

6. Min-Max Scaling

Formula: \(X_{\mathrm{scaled}} = (x - x_{\min}) / (x_{\max} - x_{\min})\)

bill_min = df["Monthly Bill"].min()
bill_max = df["Monthly Bill"].max()
df["Bill_MinMax"] = (df["Monthly Bill"] - bill_min) / (bill_max - bill_min)
print(df[["Monthly Bill", "Bill_MinMax"]].head(3))

Min maps to 0.0, max to 1.0; all values sit in [0, 1].

7. Avoid Data Leakage

Split train/test first. Fit mean, std, min, max on training only; transform test with those same training statistics—never scale the full dataset before the split.

8. Quick Comparison

Aspect	Z-Score	Min-Max
Range	Unbounded	[0, 1]
Target	Mean 0, std 1	Min→0, max→1
Outliers	Often more robust	Can squeeze inliers