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May 29

Is index (statistics) ordered?

Is "index" as described in index (statistics) an ordered value? In case I'm using the wrong word, I mean "ordered" as in there is a strict order to the values. You can sort them from least to greatest. If you have an index of 10, it is always considered more than an index of 5? I can see that 10 is not necessarily double 5, but I'm not certain, based on the article, that 10 is always more than 5. I'm curious to know if this is the definition of "index" used in Jaccard index. Looking at that article and the references, it states that an index of 1 is always more than an index of 0.5, but an index of 1 is not necessarily double 0.5. 97.82.165.112 (talk) 16:44, 29 May 2021 (UTC)[reply]

Short answer: No; order statistics, rank statistics, and Jaccard index are completely different things. Long answer: An index, based on the examples that the article gives, is basically a composite statistic that is supposed to summarize a complex phenomenon (such as "gender gap" or "stock performance") into a single number such that a certain kind of desirable/interesting phenomenon (like "more gender equality" or "higher profits") roughly corresponds to an increase/decrease in the statistic, so as to serve as a single metric of the complex phenomenon. In terms of numerical values, indices can take any values whatsoever a priori. Duckmather (talk) 21:00, 29 May 2021 (UTC)[reply]

Somewhat longer answer. (1) The Jaccard index is not an "index" in the sense described in our article Index (statistics). To avoid confusion, let me call the latter type of index a "composite indicator". (2) Indicators assume numerical values; if you are comparing two groups using, an indicator designed, say, to represent their tendency to be inclined to use violence to achieve their goals, then a much higher value for this indicator should indicate a markedly stronger acceptance of violent means. Otherwise the indicator is poorly designed. (3) But if the value for group Red is found to be 24.6 and that for group Blue is 12.3, you cannot say that the research indicated that group Red is "twice as violent" as group Blue, just like someone with an IQ of 150 is not "twice as intelligent" as someone with an IQ of 75, and a room at 10 °C is not "twice as warm" as a room at 5 °C. That is not an issue with such measures; the issue is that it is not clear what "twice as ..." means in these domains. For the temperature domain this should be obvious, since the room temperatures 50 °F and 41 °F are the same pair expressed using a different scale. (4) "Jaccard index" is just a term; it does not involve a precisely defined meaning of the term "index". IMO, "Jaccard similarity coefficient" is a better term, but also there the term "coefficient" has no precise meaning by itself. --Lambiam 22:56, 29 May 2021 (UTC)[reply]