The Data

Attribute: HairColor    Domain: {Black, Brown, Blonde, Red, Gray}

• Person A $\rightarrow$ Black    Person B $\rightarrow$ Blonde    Person C $\rightarrow$ Red

• We can say: A $\neq$ B   

• We cannot say: Black $>$ Blonde, or Average(Black, Red) = ?   

• Most frequent value (mode) $\rightarrow$ valid summary statistic

Attribute: Occupation    Domain: {Doctor, Engineer, Teacher, Lawyer, Artist}

• Encoded internally as: Doctor=1, Engineer=2, Teacher=3, Lawyer=4, Artist=5

• Even though stored as numbers, $1 + 2 = 3$ (Doctor + Engineer = Teacher?)   

• Only meaningful question: “How many people are Engineers?” $\rightarrow$ frequency count

• Ordering like Engineer $>$ Doctor is completely invalid

Attribute: PinCode    Example values: {110001, 560001, 400001}

• Looks numeric — but arithmetic is meaningless

• $560001 - 110001 = 450000$ has no geographic meaning

• Used only to identify a region (label), not to measure anything

• Correct use: grouping, frequency, equality checks only

Attribute: HIV_Test    States: {Positive=1, Negative=0}

• Most patients test Negative (0) — this is the common, uninformative state

• A Positive (1) result is rare and highly significant

• When comparing two patients, a 1-1 match (both positive) is very meaningful

• A 0-0 match (both negative) is ignored in similarity computation — it adds no information

• Jaccard coefficient used instead of simple matching coefficient

Attribute: Gender    States: {Male=1, Female=0} (or vice versa)

• Both states are equally important — no state is rarer or more critical

• A 0-0 match (both Female) is just as meaningful as a 1-1 match (both Male)

• Simple matching coefficient is appropriate for similarity

• Recoding Male=0, Female=1 does not change the analysis

Attribute: Bought_Item    States: {Yes=1, No=0}

• In a large transaction database, most entries are 0 (item not bought)

• A 1-1 co-occurrence (two customers both buying the same item) $\rightarrow$ meaningful pattern

• 0-0 co-occurrence (both did not buy) $\rightarrow$ uninformative, ignored

• Used heavily in market basket analysis and association rule mining

Attribute: Grade    Order: F $<$ D $<$ C $<$ B $<$ A

• Student X: B    Student Y: A    Student Z: C

• We know: A $>$ B $>$ C   

• We cannot say: the gap between A and B equals the gap between B and C   

• Median grade of {B, A, C, B, D} = B (after sorting: D, C, B, B, A)   

• Mean of {B, A, C} = ?    undefined and meaningless   

Attribute: Satisfaction    Order: Very Dissatisfied $<$ Dissatisfied $<$ Neutral $<$ Satisfied $<$ Very Satisfied

• $M = 5$ states, mapped: Very Dissatisfied=1, …, Very Satisfied=5

• Normalized: $z = \{0,\ 0.25,\ 0.5,\ 0.75,\ 1.0\}$

• “Satisfied” is better than “Neutral” — but by how much? Unknown

• Report median satisfaction, not average — averaging ordinal labels is statistically incorrect

Attribute: Rank    Order: Private $<$ Corporal $<$ Sergeant $<$ Lieutenant $<$ Captain

• Clear ordering of authority and responsibility

• Sergeant is higher than Corporal   

• (Captain $-$ Lieutenant) = (Sergeant $-$ Corporal)?    No such claim valid   

• Useful in sorting, ranking comparisons, and ordinal regression models

Attribute: Temperature_C

• $30^\circ C - 20^\circ C = 10^\circ C$    difference is meaningful   

• $40^\circ C$ is NOT twice as hot as $20^\circ C$   

• Proof: $40^\circ C = 104^\circ F$,    $20^\circ C = 68^\circ F$,    $104/68 \neq 2$    — ratio changes with scale

• $0^\circ C$ does not mean absence of heat — water still has thermal energy

Attribute: Year

• $2024 - 2020 = 4$ years difference   

• Year 2024 is not “twice” as recent as Year 1012   

• Year 0 is an arbitrary reference, not the start of time

• Differences (durations) are meaningful; ratios of years are not

Attribute: Weight_kg

• Person A: 80 kg    Person B: 40 kg

• A is exactly twice as heavy as B    $80/40 = 2$   

• 0 kg = truly no weight   

• Difference: $80 - 40 = 40$ kg meaningful   

• Mean, variance, standard deviation all valid

Attribute: Annual_Salary

• Salary of $0 = no income (true zero)   

• $80,000 is four times $20,000   

• Meaningful to compute: average salary, salary ratio, percentage increase

• Transformations (log scale) commonly applied for skewed salary distributions

Attribute: Age_years

• Age 0 = just born — true absence of elapsed life time   

• Person aged 60 is twice as old as person aged 30   

• Difference: $60 - 30 = 30$ years, fully meaningful   

• Mean age, age ratio — all statistically valid operations

Attribute: Items_Purchased    Domain: {0, 1, 2, 3, …}

• A customer can buy 3 or 4 items — never 3.5 items

• Set is countably infinite — no upper bound, but each value is a whole number

• Represented as integer; bar charts and frequency tables are natural visualizations

• Mean is technically computable but result (e.g., 2.7 items) must be interpreted carefully

Attribute: PinCode    Example: {110001, 560001, 400001}

• Finite set of valid pin codes across the country

• No pin code exists between 110001 and 110002 — they are adjacent, distinct labels

• Although stored as numbers, this is a nominal discrete attribute

• Frequency analysis by pin code region is a valid operation

Attribute: Height_cm    Example: 170.5, 171.2, 165.87

• Between 170 cm and 171 cm, infinitely many real values exist: 170.1, 170.11, 170.111…

• Measured by instrument — precision limited (e.g., 0.1 cm), but the attribute itself is continuous

• Histograms (not bar charts) are the appropriate visualization

• All statistical operations valid: mean, variance, percentile, standard deviation

Attribute: Temperature_C

• Can be 36.6, 36.61, 36.612… — infinite precision possible

• Thermometer records to 1 decimal, but true value is continuous

• In data preprocessing: often discretized into bins
  e.g., $[<20^\circ C =$ Cold$]$, $[20$–$35^\circ C =$ Warm$]$, $[>35^\circ C =$ Hot$]$

• Discretization converts continuous $\rightarrow$ ordinal for certain algorithms