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Is the variable at ratio or interval level if it can take any value between 1 and 100 but not 0? I cannot determine using the definitions of ratio and interval variables.

Nick Cox
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xyx
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1 Answers1

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This doesn't depend on the possible values but on interpretation and what is measured. If it's counts of something, it's ratio, as if one value is twice as high as another, it really means "twice as much" in a properly interpretable way (actually proper counts are absolute scales, which means that they fulfill the conditions for all lower scale types).

Many sources say that a "ratio scale" requires an "absolute zero point". Note that according to the "definition" that I have given, the zero has a special role, because one can't divide by zero. In fact "interpretable ratios" imply (at least in all practical situations, if not strictly mathematically) that the measurements to be compared all have the same sign (normally they are all positive). Zero is then a borderline value that is treated differently from all the other values, as it is not involved in any meaningful ratios. Usually there is a specific interpretation for this, with zero meaning the absence of anything to be measured (zero counts, zero length, zero weight etc.).

On the other hand, if it is for example a rating scale without properly defined meaning of the numerical values (like for example used by a film critic to rate films), it is not even interval (which requires differences to be properly interpreted, meaning that in a well defined sense the difference between 94 and 96 has to be the same as between 12 and 14, say) but only ordinal (all we can say is 96 is better than 94 and 14 better than 12, but there is no quantitative meaning to "how much better").

Christian Hennig
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  • thank you for the response. I understand the type is related to the interpretation and usage of the variable. Let's assume the values are equally spaced so that, for example, the difference between 1 and 2 is the same as between 98 and 99. According to your response, it is a ratio variable? I agree it is more likely a ratio variable than an interval variable. But I am concerned of interpretation of 0 in this case. Could you please clarify? Thanks. – xyx Oct 26 '20 at 02:12
  • What you describe is the definition of an interval scale, meaning that the same "intervals" (differences) have the same meaning. The ratio scale on top of that requires that ratios ("50 is related to 25 in the same manner as 20 to 10") are meaningful. With counts, for example, a ratio of 2 always means "taking twice the lower number and putting them together gives the higher one". – Christian Hennig Oct 26 '20 at 10:37
  • What might be called judgment scores -- where a personal judgement is what lies behind a score -- are indeed ordinal insofar as there are no grounds for supposing that e.g. 96 $-$ 94 $=$ 6 $-$ 4. But these are often averaged nevertheless, and indeed in many academic systems they are given in full knowledge of their being averaged. Otherwise put, the pragmatic view is that they are as good as interval scales as you can get. (And marks or grades are often mixed with those from a marking scheme, a correct answer on Question 1 getting 10% of the total and so forth.) – Nick Cox Oct 26 '20 at 14:22
  • Note that even ranks -- supposedly the quintessential ordinal scale -- are often averaged, e.g. Spearman correlation treats ranks exactly as if they were interval scale. Some purists say that this is precisely what you shouldn't do. More discussion at https://stats.stackexchange.com/questions/67551/calculate-mean-of-ordinal-variable – Nick Cox Oct 26 '20 at 14:26
  • Other way round, something like Fahrenheit or Celsius temperature is often held up to those outside physical science as an exemplary interval scale but the history of temperature measurement over some centuries is a messy mixture of approximations, some practical and some theory-guided. – Nick Cox Oct 26 '20 at 14:30
  • Whether one can take averages and ascribe some sense to it and whether a variable is interval scaled are two different questions though. Averaging a variable doesn't make it interval scaled. – Christian Hennig Oct 26 '20 at 14:45
  • "Averaging a variable doesn't make it interval scaled". Naturally I agree. The point is pragmatic; refusing to take averages may deny a researcher a useful technique. The limiting case is binary scale, 0 or 1, where despite the utter difference between alive and dead, present and absent, and so on, an average has an easy and valuable interpretation as a probability. Is binary scale nominal or ordinal so that is nonsense? The nominal-ordinal-interval-ratio scale prejudice makes the fair general point that what the numbers represent affects what makes sense, but "affects" is not "determines". – Nick Cox Oct 26 '20 at 18:57
  • Fair enough. By saying "variable X is not interval scaled" I'm *not* saying "you shouldn't compute an average." The first is about the definition and what "interval scaled" actually means. The second is as you say about being pragmatic. – Christian Hennig Oct 26 '20 at 20:50
  • Thank you both for the very helpful information. @Lewian, I understand the difference between interval and ratio in the definitions you provided. However, my concern is that many other resources distinguish interval and ratio using the meaning of '0'. I just want to see, if the example I gave fit into ratio level, how would we explain '0'? Thank you. – xyx Oct 28 '20 at 21:19
  • @NickCox, Thank you for your input. I understand that from the pragmatic point of view, this is a numeric variable. Do you consider it an interval variable over ratio? Thanks. – xyx Oct 28 '20 at 21:21
  • @xyz: I added something on zeroes to my answer. – Christian Hennig Oct 28 '20 at 21:37
  • I don't think you say enough about your zero to add anything helpful. Zero can be a point unattainable but well defined, so no person is zero height, but zero height is a well defined origin. – Nick Cox Oct 28 '20 at 23:51
  • Again, although not wrong, I don't quite see how this points to a problem with my answer. Why don't you write your own answer if you think you have something valuable to say, rather than criticising mine? – Christian Hennig Oct 28 '20 at 23:55
  • My previous comment was aimed at @xyx. Comments can be for criticism and discussion, which is entirely fair practice, but I am trying to complement your answer and add nuance. I am not trying to undermine it. I have already written at length on this topic, as linked previously. I haven't voted to close, but I don't think this is a very clear question without more detail on the variable concerned, whether as the focus of the question or as an example of a wider group of variables. – Nick Cox Oct 29 '20 at 07:35
  • Fair enough, I hadn't seen this was @xyz. – Christian Hennig Oct 29 '20 at 09:35
  • @NickCox Thank you for the suggestion and input. I asked this question because I am considering which category does a score ranged between 1 and 100 fits into (in reality many test grades exclude 0 e.g. GRE scores). I agree that 0 being a point defined but unattainable doesn't automatically disqualify a variable from ratio-level measurement. – xyx Oct 30 '20 at 01:42
  • @Lewian Thank you for the response. – xyx Oct 30 '20 at 01:43