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I have two sample groups with the same variables. For example, age and blood pressure for two groups. I would like to say that these two variables are correlated either at both corners (first group) or at the upper corner only (second group)

Variable A and B show strong correlation only at either both upper and lower corners or at one corner only.

I fee my sentence is very difficult to understand. How to use either and both (if I can) in my sentence.

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(To start, I don't understand what correlations at corners are, but I assume you have explained it beforehand in the document you are writing.)

Now, your first statement,

these two variables are correlated either at both corners (first group) or at the upper corner only (second group)

is understandable, but when you rephrase it as

Variable A and B show strong correlation only at either both upper and lower corners or at one corner only.

it's not understandable. Since you removed the qualification "upper", it doesn't make sense.

It might be as well not to compact the information too much. Maybe this would express it:

The first group showed strong correlation of variables A and B at both upper and lower corners, while the second group showed correlation at only the upper corner. (Neither group showed correlation at only the lower corner.)

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It's easier to understand if you mark out the phrases and see the roles they play in the sentence:

Variable A and B show strong correlation only at either (both upper and lower corners) or (at one corner only).

You can now clearly see that both is not related to either, but rather part of the first phrase of the pair connected by the either...or... structure. Both in its turn belongs to another similar connecting structure (both...and...).

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