Can I consider any continuous quantitative variable as super ultra finely categorised ordinal variables?

Question

Data can be quantitative (eg. 'how much?') or categorical (eg. 'what type'?). Categorical data can be nominal (unordered eg. blood group) or ordinal (ordered eg. cancer stages). Quantitative data can be continuous ('measured' eg. height, shoe size) or non-continuous ('counted' eg. how many children in family).

My question is, let's take height for example - which is a continuous quantitative variable. Can't I also say height is a infinitely categorised ordinal categorical variable? So basically we have height of eg. 180.0000000000...000cm, 180.0000000000...001cm, 180.0000000000...002cm, 180.0000000000...000 + or - dx cm where dx approaches zero. That way we have the whole height spectrum covered in a ordered, categorical manner. Am I right?

Then I can basically say I can consider any continuous quantitative variable as super ultra finely categorised ordinal variables. Sounds like nonsense to me. Please explain to me where my misconception is. Thanks.

Answer 1

In quantitative variables you have as many categories as ten to the power of the number decimals in your quantity (a 3 digit number would be like having 10 to the third power, categorized ordinals). The difference is that those 10 exp 3 categories are equally spaced when you use a quantitative variable (like when you are measuring weight, time, dollars using three decimals), but may not be so if you use the numbers as categories, like giving 1000 people a number based on their height: shortest 1, tallest 1000. When you do that, the difference between 1 and 301 -which is 300- may not be the same if you take the difference in inches. That's why you say some variables are ordinal and some quantitative, in ordinal variables you can only use a>b and a


I don't think Rudin's book will be of much help though.

Answer 2

Yes, in the example you gave the variables have real values, and R can be ordered in a natural way

R^n can be ordered with the lexicographic order

Even if we suppose the quantitative data has non-real values, we can still order it. Though it may be abstract, without much practical value.

Answer 3

Oh dear....... It sounds as if you're floundering with the concept of 'rational' vs 'irrational' values.



Go find any good introductory book on elementary analysis (I used to use Rudins, 'Principles of Analysis' when I taught it) and get to the part on the 'supremum' of a rational set.



Doug