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# Statistics I · Week 1 — Data types and measurement scales Statistics starts with **what you measured** and **what arithmetic means** on that measurement — before any formula. ## Week map Case vs variable → categorical vs numerical → measurement scales → valid summaries → rows/columns in tables → common misclassifi...

Statistics I · Week 1 — Data types and measurement scales

Statistics starts with what you measured and what arithmetic means on that measurement — before any formula.

Week map

Case vs variable → categorical vs numerical → measurement scales → valid summaries → rows/columns in tables → common misclassification traps.

Core vocabulary

Case → one observed unit → one row often → student #42, Tuesday’s weather, one transaction.
Variable → characteristic recorded per case → column → height, major, rating.
Value → one observation of a variable for one case → 172 cm, "CS", 4.
Label → name of variable → not the same as a value.

Type split

Type	Values	Examples
Categorical	labels / codes	eye color, department
Numerical	numbers with quantity	height, count, income

Numerical subtypes: discrete (counts) vs continuous (measurements within interval).

Measurement scales

Nominal → categories, no order → gender codes, zip codes.
Ordinal → ordered categories → Likert “satisfied”, medal ranks.
Interval → differences meaningful, no true zero → Celsius temperature.
Ratio → true zero, ratios meaningful → mass, distance, income.

Mini-examples:

Shirt size S,M,L,XL → ordinal (order) but not ratio (L is not “twice” S).
Year 2024 → interval-like for calendar years (no year 0 in CE) — often taught as not ratio.
Kelvin → ratio (0 K means no thermal energy).

Valid operations (quick gate)

Scale	Order?	Mean?	Ratio “twice as much”?
nominal	no	no	no
ordinal	yes	caution	no
interval	yes	often	no
ratio	yes	yes	yes

Trap: averaging ordinal codes without justification treats them as interval.

Pattern families

Easy — Classify variable

Given description, state categorical vs numerical.
Pick scale for one variable with reason in one line.
Identify case count n from table rows.

Medium — Table orientation

Rows as cases, columns as variables (usual).
Distinguish variable name from value entry.
Several variables same cases — joint dataset.

Hard — News headline traps

ID numbers nominal, not “big number” numerical analysis.
ZIP code nominal — do not average.
Percent correct as ratio; letter grades ordinal.

Worked mini-examples

Example 1 — Classify.

Dataset: for 50 buses, record route_id (text), passengers (count), on_time (Yes/No).

route_id categorical nominal
passengers numerical discrete
on_time categorical nominal (binary)

Example 2 — Scale.

Variable: pain scale 0–10 where 0 means none, 10 worst. Ordered, differences somewhat meaningful — often ordinal; not ratio (10 is not “double” 5 in pain perception).

Example 3 — Celsius.

20°C to 30°C is +10°C difference. 30°C is not “1.5× heat” of 20°C in ratio sense — interval, not ratio.

Example 4 — Cases.

Table 120 rows, columns age, city, salary → 120 cases, 3 variables.

Example 5 — Invalid mean.

Average of zip codes 10001 and 10002 is meaningless — nominal.

Traps

Confusing variable with value (“variable is John”).
Treating student ID as numerical for correlation.
Averaging ordinal satisfaction codes blindly.
Ratio claims on interval scales (twice as hot in °C).
Binary categorical treated as numerical 0/1 without context — sometimes OK for modeling, but type is still categorical.

Diagnostic (try yourself)

Classify each variable as categorical or numerical: (a) number of siblings, (b) favorite fruit, (c) body temperature in °F, (d) race finish position.
For “years of education completed”, which scale (nominal/ordinal/interval/ratio) is most appropriate? Brief reason.
A spreadsheet has 200 rows and 15 columns of student records. What is n? How many variables?
Why is “mode of zip code” silly but “mode of eye color” OK?
Give one example variable that is numerical discrete and one numerical continuous.