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All these three forms, high throughput, high-throughput or high through-put, are used in the scientific community by Google Scholar searching. Where is the hyphen should be? Is there a specific context for their using?

Thank you!

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I think the fact that you have a compound word (throughput) within the structure confuses things. I'll use another example to clarify the rule* then we go back to this

*it's not a "rule" strictly speaking, since - as you yourself realized through Google Scholar texts - there are many approaches.

Let's take the example sparkling white wall versus sparking-white wall The hyphen is used when sparkling-white is used as an adjective for wall (in other words, we mean that the color/shade of the wall is not merely "white", but "sparkling white". Without the hyphen you would essentially have two adjectives, sparkling and white - i.e. "the wall is sparkling and the wall is white". It's subtle, I know.

On your example, it depends on whether you use the word as an adjective or noun. Here are two examples to demonstrate:

I have noticed a high throughput in the production line

vs

I have noticed a high-throughput result in the production line

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