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Suppose I have a variable $X$ with samples $x_i$ for $i=1,\ldots,N$ where $N=10,000$ (just something big).

Posit that a sample subset looks like so: $x_{1:10} = [4,2,3,4,6,3,8,9,1,3]$.

How can I tell, or what tests can I do, to determine if these $N$ samples are ordinal? I.e. what tests are available, or techniques for differentiating between this ordinal variable and say just random integers in the range 1 to 9?

Or say I have another variable say $Y$ which has a subset sample that looks like so $y_{1:6} = [\text{'fat', 'thin', 'average', 'fat', 'fat', 'thin'}]$ - this is clearly a discrete ordinal variable, but is there a way to statistically establish this?

Similarly for interval variables, what testa can one make, or models run, to determine the nature of said variable?

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The definition of each variable should provide clues as to whether the variable can be treated as an ordinal variable. Remember that an ordinal variable is one whose values can be grouped into a (relatively small) number of unique categories, such that the categories follow some natural ordering.

For the variable in your second example, the unique categories appear to be "thin", "average" and "fat". If you could argue that these categories follow a natural order, then you would feel comfortable claiming that the variable is ordinal. But you would also need to refer to the actual definition of the variable. What does that variable mean? Is it intended to measure someone's level of body fat?

For your first variable, in the absence of a clear definition of the variable, it's impossible to say whether the variable should be treated as ordinal or discrete. If the values assumed by this variable are counts of some kind (e.g., typical number of hours of TV someone watches over a week), then the variable is discrete. If the values assumed by this variable are simply numerical codes for some underlying categories, then the variable could be either nominal (if the underlying categories don't follow any natural ordering) or ordinal (if the underlying categories follow a natural ordering).

To sum up, knowing how a variable is defined and measured is crucial to helping determine whether the variable is nominal, ordinal, discrete or continuous.

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