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Say there is a group of OCEAN profiles. How do I calculate how well another profile fits into this group of profiles?

Is https://en.wikipedia.org/wiki/Congruence_coefficient the way to go or are there alternatives?

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As https://en.wikipedia.org/wiki/Similarity_measure states:

Cosine similarity is a commonly used similarity measure for real-valued vectors, used in (among other fields) information retrieval to score the similarity of documents in the vector space model. In machine learning, common kernel functions such as the RBF kernel can be viewed as similarity functions

Also see: Congruence coefficient

Other measures: Euclidean distance

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