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We are analyzing the ranking changes of countries based on an economic index to determine whether these variations are significant. For example, Country A moves from rank 5 to rank 3, representing a 40% improvement, while Country B moves from rank 50 to rank 48, showing only a 4% improvement. Although both countries have moved up by two positions, the percentage change reveals that the improvement is far more significant for Country A than for Country B.

Using rankings to compare performance is broadly used in social sciences suchs as economics, sociology and psychology. It makes sense because it simplifies complex data, and facilitates comparative analysis. Sometimes the exact numerical score of indices might be less important than understanding where a country stands relative to others, especially if the goal is to identify leaders and laggards. Economic indices often aggregate multiple indicators (e.g., Human Development Index ), making direct comparisons of the underlying data complex. Rankings distill this complexity into a single, digestible figure that facilitates comparison across multiple dimensions. The non-linear nature of rankings in this context (not equal intervals between ranks) might justify that rank percentage changes can better reflect the relative importance of shifts across different positions.

This raises the question: Is it more meaningful to assess ranking changes using percentage changes rather than absolute changes when dealing with ranking data? Are there academic references that discuss this issue, and what conclusions do they provide?

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    $\begingroup$ If your countries are ranked by an economic index, why don't you look at the changes in the economic index itself instead of the ranks? Whether the increase from rank 5 to 3 is more significant than an increase from rank 50 to 48 depends on the values and changes of the underlying index. $\endgroup$
    – quarague
    Commented Sep 2 at 13:29
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    $\begingroup$ Gini score - if you know the definition of a metric, or of measure, does it qualify? $\endgroup$
    – EngrStudent
    Commented Sep 2 at 16:33
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    $\begingroup$ It sounds like a XY problem. Can you elaborate on why you're using ranks in the first place? What makes ranks more relevant to your research than the underlying index? (This is not a rhetorical question.) $\endgroup$
    – J-J-J
    Commented Sep 3 at 6:31

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This question reminds me of this xkcd strip. All humor aside, comparing percentages which do not share the same basis (i.e. denominator) can be very misleading (as xkcd points out).
In addition, computing differences of ordinal variables is not mathematically sound. The distance between ranks is not defined; in fact it is quite different for different ranks. All you can say about the 3rd and 5th ranks is that 3rd>5th. But you do not know if it is by a whisker, or by 2 country miles. Then comparing 2 differences of ranks compounds the issue. It makes even less sense to compute a ratio of ranks, let alone 2 of them and compare them. Arhitmetic (+,-,*,/) on ordinal values is not appropriate; all you can do are comparisons (>,<,=).
Now you can rank the economic indexes, and show that country A has a higher index than country B; or look at the economic index's growth in time, and show that country A grew faster than country B. But there is no ground to say that a change from 5th to 3rd place is "better" (in any way you can define "better") than a change from 50th to 48th place.

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