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Perhaps the most natural measure of similarity between two close languages is the ease with which a native speaker of one can understand the other. (This might not be symmetrical in some cases, because of the background degree of exposure or other reasons.) For less-similar languages, the average time needed to learn the other could be assessed.

More-objective measures could be based on similarity between corresponding words and perhaps grammatical features, while possibly other types of measure have been devised.

My question is as in the title: What measures are there of similarity between languages? And where can I find data on such measurements?

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There are many approaches to similarity of languages, usually expressed as a small linguistic distance. There are several approaches to linguistic distance, including

phonemic distance measured, e.g., in the number of sound shifts that occurred for cognate words. For a reference and more details, see this answer poining to a work by Wettig et al. 2013.

lexical distance measured by the number of replacements of words from a given wordlist (say, the Swadesh list) by non-cognate ones. More on lexical distance can be found under this question.

syntactic distance that can be measured by various methods, for a comparison of some methods see this paper Heeringa et al. 2017: Measuring syntactical variation in Germanic texts. Incidentally, the authors of that paper conclude that English is a North-Germanic language syntactically.

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