✍ 04: Data Quality & Preprocessing — Exercise¶
Tip
Practice — try each question first, then expand the answer to check your reasoning.
Data for Q1–Q4: 31, 38, 42, 29, 46, 23, 83, 43, 51, 55, 27, 35 (min = 23, max = 83, range = 60, n = 12)
📊 Q1. Discretize the data above into 4 bins using equal-width partitioning, then using equal-depth partitioning.¶
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**Equal-width** (bin width = 60/4 = 15): Bin 1: 23–37 · Bin 2: 38–52 · Bin 3: 53–67 · Bin 4: 68–83 | Data | 31 | 38 | 42 | 29 | 46 | 23 | 83 | 43 | 51 | 55 | 27 | 35 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Bin | 1 | 2 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 3 | 1 | 1 | **Equal-depth** (12 values ÷ 4 bins = 3 values/bin, sorted first): Bin 1: {23,27,29} · Bin 2: {31,35,38} · Bin 3: {42,43,46} · Bin 4: {51,55,83} | Data | 31 | 38 | 42 | 29 | 46 | 23 | 83 | 43 | 51 | 55 | 27 | 35 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Bin | 2 | 2 | 3 | 1 | 3 | 1 | 4 | 3 | 4 | 4 | 1 | 2 | Equal-width bins have equal *ranges* but can hold very different *counts* (bin 4 here only has the outlier 83). Equal-depth bins have equal *counts* but ranges that stretch to absorb outliers.Data for Q2–Q3: a Food/Distance/Company table, first 5 rows: Chinese/far/good, Italian/very close/good, Italian/close/good, American/far/bad, Chinese/very far/good.
🔢 Q2. One-hot encode the Food values in the first 5 rows (unique order: American, Chinese, Italian, Japanese).¶
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| Row | Food | One-hot [American, Chinese, Italian, Japanese] | | --- | --- | --- | | 1 | Chinese | [0, 1, 0, 0] | | 2 | Italian | [0, 0, 1, 0] | | 3 | Italian | [0, 0, 1, 0] | | 4 | American | [1, 0, 0, 0] | | 5 | Chinese | [0, 1, 0, 0] | One-hot encoding gives each category its own binary column with no implied ordering — appropriate because Food is **nominal**.🔢 Q3. Convert the Distance values in the first 5 rows to 3-bit Gray code, given the natural order: very close < close < far < very far < too far.¶
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Gray code sequence: very close→000, close→001, far→011, very far→010, too far→110 | Row | Distance | Gray code | | --- | --- | --- | | 1 | far | 011 | | 2 | very close | 000 | | 3 | close | 001 | | 4 | far | 011 | | 5 | very far | 010 | Gray code changes only **1 bit** between adjacent ordinal values (000→001→011→010→110), which preserves the ordering better than plain binary counting for ordinal attributes.📊 Q4. Apply min-max normalization to the Q1 dataset so all values fall in [0, 1].¶
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Formula: x′ = (x − min) / (max − min) = (x − 23) / 60 | Real | 31 | 38 | 42 | 29 | 46 | 23 | 83 | 43 | 51 | 55 | 27 | 35 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | [0,1] | 0.133 | 0.250 | 0.317 | 0.100 | 0.383 | 0.000 | 1.000 | 0.333 | 0.467 | 0.533 | 0.067 | 0.200 | The min maps to 0, the max maps to 1, and everything else is a proportional position between them.📏 Q5. What is the Euclidean distance between x = (1, 3, −2, 5) and y = (2, 4, 1, 6)?¶
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Difference (y − x) = (1, 1, 3, 1) d(x, y) = √(1² + 1² + 3² + 1²) = √(1+1+9+1) = √12 = 2√3 ≈ **3.464**📚 All Exercises · Next: Chapter 05 — Clustering ➡️