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I frequently use the expression the correlation between A and B. And I wonder whether commas are necessary in the following sentences.

The correlation between A and B, and that between C and D are similar.

The difference between the correlation between A and B, and that between C and D is insignificant.

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A comma separates a list of more than 2 items. Your examples only have 2 items in each sentence, so can't use a comma.

Visually, the problem with your examples is that without a comma, it is confusing with too many "and"s. I usually try to use "&", so it becomes "the correlation between A & B and that between C & D are similar."

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The correlation between A and B, and that between C and D are similar.

The difference between the correlation between A and B, and that between C and D is insignificant.

Example 1 does not need comma. I bracket the 2 phrases for easier reading:

[The correlation between A and B] and [that between C and D] are similar.

Example 2 also does not need comma. I bracket the phrases similarly:

The difference between [the correlation between A and B] and [that between C and D] is insignificant.

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