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Suppose you have a measurement $Y_i$. Now suppose that the differences between two measurements $\delta_{ij}=Y_i-Y_j$ are an ordinal measurement. From my understanding, $Y_i$ is more informative than an ordinal measure because the differences are meaningful. But it also is less informative than an interval scale because the intervals are not a ratio measurement.

What do you call $Y_i$?

An example that I was thinking of is a points system for a ladder in a sporting competition. Suppose the top three teams in some sport have $72,60,58$ points respectively at the end of a season. Then I would expect the first team to be "better" than the second team by a larger amount than the second team is "better" than the third team as $(72-60)>(60-58)$. But I wouldn't say that they are $6$ times "better" ($6=\frac{72-60}{60-58}$) using this measurement - I would just say (using poor grammar) "more better".

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    $\begingroup$ I think you actually want an interval scale. The fact that you do not want the difference between 72 and 60 to count as '6 times' the difference between 60 and 58 simply suggests that your definition of 'quality' is not a linear function of the attained score. $\endgroup$
    – MLS
    Commented Feb 25, 2013 at 14:26
  • $\begingroup$ @Macro -on a five point scale suppose that "strongly agree" is closer to "agree" than "agree" is to "nuetral". Just ordering them loses this property. $\endgroup$ Commented Feb 25, 2013 at 20:43
  • $\begingroup$ As a random aside on a six-year-old question, I independently "discovered" such scales myself, and referred to them as "ordinterval" since I couldn't find a term for them. They're interesting in that you can find (e.g.) the Median Absolute Deviation for them, which isn't possible for merely ordinal data. My (unproven) hunch is that in the limit -- if you keep taking successive degrees of differences-between-differences and so forth, and each new level continues to be ordinal -- that the original data ends up being interval scale. $\endgroup$
    – Bill Clark
    Commented Jun 24, 2019 at 21:12
  • $\begingroup$ How can you say 6 times "better"? If you are using ordinal, interval or even anywhere midway between them. This is the property of ratio scale. Also, if you are talking about ordinal scale then according to the definition they are categorical type of date in which if numbers are used then they only represent labels they don't have intrinsic mathematical properties as numbers in numeral scales (interval and ration) have. So, you have to decide either they represent ordered labels or they are numbers in numeral scale having intervals. $\endgroup$
    – Ubi.B
    Commented May 1, 2022 at 0:09

2 Answers 2

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Stevens' classification scheme can be useful, but it is by no means complete nor perfect. I wrote about this on Yahoo Voices.

Using Stevens' scheme you can only call your variable ordinal, since he doesn't a name for what you've got. I don't know of any particular name for a variable like yours. I propose "semi-interval" :-)

By the way, an interval level variable does not have to have ratio equivalence (that would be ratio scale), it has to have arithmetic equivalence. For example, temperature in degrees F or C is not ratio scale (50 degrees is not "twice as hot" as 25) but the difference between 50 and 25 is, in some sense, the same as that between 25 and 0.

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  • $\begingroup$ Thanks for the link. Note that my reference to ratio equivalence for interval level variables is in regard to the differences between values. So if $Y_i$ was a true interval value, then I thought that means that $\delta_{ij}$ was a ratio level variable. $\endgroup$ Commented Feb 26, 2013 at 20:04
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    $\begingroup$ This is a good answer, but I think what the OP describes is an interval variable under Steven's classification (so no need for another name). If you can apply arithmetic to the original variable certainly you can apply the same arithmetic to the differences. $\endgroup$
    – Andy W
    Commented Feb 28, 2013 at 21:14
  • $\begingroup$ The last point follows from the definition: intervals of the same length are equivalent if the scale is interval. $\endgroup$
    – Nick Cox
    Commented Feb 6, 2017 at 9:32
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Would you regard (72-60) = (60-48) = (48-36)? That would imply (72 - 48) = 2 * (48 - 36) which you don't want to accept.

You "expect"/would say that (72-60) > (60-58) implying some unstated quantification of the differences. I guess you would accept that (72-72) < (60-58). For what value of x would you say, at least approximately, that (72-x) = (60-58)?

If you can be explicit about your quantification of differences you can re-scale to a conventional interval scale. Otherwise I think your unstated, subjective, assessment of the relative magnitude of differences is not very useful, and your scale is no more than ordinal.

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