I’m trying to prepare for statistics but every time I try to wrap my head about what makes these different from each other I get more confused. It feels like the only difference between nominal and ordinal is just how they’re format in graphs and tables.
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Consider people being polled for their favorite colors. Some people say blue. Others red. One guy says puce because he’s a fan of Monsters Inc.
Is blue greater than red? What do you get when you multiply puce by two? These questions are nonsensical. Color does not give rise to these sorts of relationships. All you can say is that blue is a different color from red. Color is thus a nominal scale: it gives a name to different elements, but goes no further than that.
You then look at the counts of people who picked each color. Blue was the most popular, followed by red, then green, yellow, and finally puce. From this information you can start to make comparisons: more people selected green than yellow. However, just from this listing you can’t say things like “red was half as popular as blue” (perhaps seeing 2nd place as “twice” 1st place); that’s just not what positions tell us. This scale is ordinal: it gives an order to the listings, but no further information.
Finally you look at the actual counts of votes. 20 people picked blue, 15 red, 10 green, 5 yellow, and 1 puce. From here you can say things like “twice as many people picked blue as green.” Multiplication is allowed on these numbers and the results are sensible. When you multiply with these numbers you’re describing a ratio, hence this is a ratio scale. Scales that describe real quantities are nearly always ratio scales.
You’ll note that I skipped one scale type, and that’s because it’s seldom actually seen: interval scales. The one common example of an interval scale is temperature (in C or F). Consider trying to use this as a ratio scale: “10 F is twice as hot as 5 F” may seem to make sense, but then convert the numbers to C and those same values become “-12.2 C is twice as hot as -15 C” and we’re left with a claim that makes no sense. There is nothing that 10 F is twice as many as when compared to 5 F.
That shows that temperature (in F or C) isn’t a proper ratio scale, yet we can also see that it goes above and beyond a simple ordinal scale. We *can* reason about how 31 C is just a little bit hotter than 30 C, while 80 C is much hotter. More formally, on an interval scale you can perform addition and subtraction and come up with sensible results.
Putting this all together, the allowed basic operations on each scale are:
* Nominal: **=, ≠**
* Ordinal: =, ≠, **<, >, ≤, ≥**
* Interval: =, ≠, <, >, ≤, ≥, **+, -**
* Ratio: =, ≠, <, >, ≤, ≥, +, -, **×, ÷**
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