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Natural languages are known to follow some statistical laws, the most famous among them is Zipf's law.

Are there published studies of some conlangs with respect to statistical laws of language, not necessarily restricted to Zipf's law, and what are the results of them?

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Zipf's Law and other similar laws regarding the statistical behaviour/properties of languages are studied in Quantitative Linguistics. Using QL as a search term together with the name of a conlang (eg Esperanto) will likely lead to some relevant results:

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