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I'm wondering what the difference between the Pseudo F Index and the F Statistic is.

The Pseudo F Index is used in clustering analysis as an index to determine the the right number of clusters in a dataset. Its defined as follows:

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Link to the source

The F Statistic is from the common ANOVA:

F-Statistic

Link to the source

What is the difference in the two calculations? If there is none, why is the index for clustering called "Pseudo F Index" and not simply "F Index" or "F statistic"? It is because in the case of clustering, you cannot perform hypothesis testing based on this statistic?

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