Nominal, ordinal, interval, ratio — as permitted operations

A ladder of what you may do

The categorical/numeric split is the first cut. The finer one is the level of measurement, and the useful way to learn it is not as four definitions but as four answers to what arithmetic is honest on this column?

• Nominal — categories with no order. You may count them. prov: ON, QC, BC, AB. Alphabetical order is not a ranking; Alberta is not "less" than Ontario.
• Ordinal — categories with an order but no fixed gap between them. You may count and order, so the median makes sense. A 1–5 satisfaction score: 4 is better than 3, but the gap from 3 to 4 is not known to equal the gap from 4 to 5.
• Interval — numbers with equal gaps but no true zero, so differences are meaningful and ratios are not. Temperature in °C: 20° is 10° warmer than 10°, but it is not "twice as warm", because 0 °C is not the absence of heat. Calendar year: 2025 is 25 years after 2000, but not "1.0125 times" it.
• Ratio — equal gaps and a true zero, so everything is allowed, including division. amount in CAD: $1,320 really is twice $660, and $0 really is no money. units likewise.

Each level permits everything the level below it permits, plus one more operation. That is the whole ladder.

Worked example — picking the summary for a satisfaction score

Maple & Birch adds a post-order survey: how satisfied were you? 1–5. Forty responses come back. The reflex is =AVERAGE(...) → 3.7. But the scale is ordinal: there is no guarantee that a customer who moved from 2 to 3 changed as much as one who moved from 4 to 5, so adding the scores treats gaps as equal that nobody has measured. The honest summaries are the ones ordinal permits: the median (the middle response — 4) and the counts per level (three 1s, five 2s, nine 3s, fourteen 4s, nine 5s). "Fourteen of forty gave a 4" is a statement the data supports; "the average is 3.7" is common practice and the quiz answer is that it is wrong in principle.

The same logic in the other direction: amount is ratio, so mean, median, differences and "AB's average is 2.3 times ON's" are all permitted. Nothing is lost by using the top of the ladder when the data is there.

The standing trap

On the paper the wrong answer is always the neighbouring level. A 1–5 scale is offered as interval ("the numbers are equally spaced"), and it is not — the numbers are, the satisfaction is not. Temperature is offered as ratio ("it's a measurement"), and it is not — no true zero. prov is offered as ordinal ("they are listed in order"), and it is not — the list order is a choice, not a property. Calendar year is offered as ratio, and it is interval: year 0 is a convention, not the beginning of time.

Place eight, choose three summaries

Place these: channel (nominal) · a product's size S/M/L (ordinal) · order date as a calendar date (interval — differences in days are meaningful, ratios are not) · units (ratio) · a customer's postal code (nominal) · a credit rating AAA/AA/A (ordinal) · store floor area in m² (ratio) · a 0–10 net-promoter score (ordinal). Then the summary for three: prov → counts (and the mode); size S/M/L → median or counts; amount → mean or median as the shape allows.

CRISP-DM: the level is recorded in the dictionary — data understanding → describe data — and it is what Module 4 reads when it chooses a statistic.