Project 03: Normalization, Distance & Sampling¶

Chapter 4 — Data Quality and Preprocessing

What This Project Does¶

This project demonstrates how to prepare numeric data for analysis and how to sample datasets correctly:

Min-Max Normalization — rescale values to [0, 1] or any target range
Z-Score Standardization — center at zero, scale by standard deviation
Euclidean Distance — compute straight-line distance in n-dimensional space
Distance Matrix — compute all pairwise distances in a dataset
Simple Random Sampling — with and without replacement
Stratified Sampling — sample proportionally from each class

The demo explicitly solves Exercise Q4 and Q5 from the Chapter 4 exercise sheet and prints Expected vs. Computed comparisons.

Concepts Covered¶

Technique	Use Case	Section in Notes
Min-max normalization	Scale to [0, 1]; bounded input (neural nets)	6.1
Z-score standardization	Center + scale; Gaussian-assumption algorithms	6.2
Euclidean distance	Measure similarity between two data points	7
Distance matrix	All pairwise distances in a dataset	7
Simple random sampling	Select a representative subset randomly	3.1-3.3
Stratified sampling	Ensure all classes are represented in sample	3.4

Files¶

File	Language	Description
`normalization_distance.py`	Python 3	Full implementation, pure stdlib
`normalization_distance.R`	R	Same operations using base R

Usage¶

Python¶

python normalization_distance.py

R¶

Rscript normalization_distance.R

Exercise Q&A Solved in This Project¶

Q4: Min-Max Normalize [31,38,42,29,46,23,83,43,51,55,27,35] to [0,1]¶

Formula: v' = (v - 23) / (83 - 23) = (v - 23) / 60

v	v' (Expected)	v' (Computed)
31	0.133	0.133
38	0.250	0.250
42	0.317	0.317
29	0.100	0.100
46	0.383	0.383
23	0.000	0.000
83	1.000	1.000
43	0.333	0.333
51	0.467	0.467
55	0.533	0.533
27	0.067	0.067
35	0.200	0.200

Q5: Euclidean Distance between x=(1,3,-2,5) and y=(2,4,1,6)¶

differences: -1, -1, -3, -1
squares:      1,  1,  9,  1
sum:         12
distance:    sqrt(12) ~= 3.464

Expected: 3.464 Computed: 3.464

The Scale Problem Demonstration¶

When attributes have different units, the one with the larger numeric range dominates distance calculations.

Example (from lecture — Bernhard, Gwyneth, James):

Person	Age (years)	Age (decades)
Bernhard	43	4.3
Gwyneth	38	3.8
James	42	4.2

Computing Euclidean distance in years vs. decades gives different relative rankings — even though the data is identical. The solution is to normalize first.

Excel How-To¶

Min-Max Normalization¶

Suppose your data is in A2:A13. To normalize to [0, 1]:

B2: =(A2-MIN($A$2:$A$13))/(MAX($A$2:$A$13)-MIN($A$2:$A$13))

Copy down to B13.

To normalize to a different range [new_min, new_max], say [0, 100]:

B2: =(A2-MIN($A$2:$A$13))/(MAX($A$2:$A$13)-MIN($A$2:$A$13)) * (100 - 0) + 0

Z-Score (Standardization)¶

C2: =(A2-AVERAGE($A$2:$A$13))/STDEV($A$2:$A$13)

Copy down to C13.

Note: Excel's STDEV uses sample standard deviation (divides by n-1). For population std dev use STDEVP.

Euclidean Distance Between Two Rows¶

Suppose point x is in row 2 (A2:D2) and point y is in row 3 (A3:D3):

=SQRT(SUMPRODUCT((A2:D2 - A3:D3)^2))

This works as an array formula — SUMPRODUCT handles the element-wise squaring automatically.

Stratified Sampling¶

Excel does not have a built-in stratified sample function. The standard approach: 1. Sort your data by the class/group column 2. Use RAND() to add a random column within each group 3. Sort by group first, then by RAND() within each group 4. Take the top N rows from each group