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I have built a new idea. To test this method, I applied it to several data sets. Then I would like to say that these data sets are only used to test the performance of the new method.

These data sets are only used to test the performance of the new method.

or

These data sets are used only to test the performance of the new method.

or

These data sets only are used to test the performance of the new method.

Which one is correct? I think the first one is the correct one, but not sure.

1

As per your statement, you have build the method and testing it with some data.

So, if you want to say, to test this method you need these data, then the word "only" is not required.

These data sets are used to test the performance of the new method.

Because, the same data can be used to test other methods as well.

Your first statement:

These data sets are only used to test the performance of the new method.

Your Second statement:

These data sets are used only to test the performance of the new method.

From this, you are saying, these data sets cannot be used for other methods.

So, both says, this data sets specifically designed for the method you have created newly.

So, decide based on your usage and requirement.

1

I would say the first two are the best ones, the first two have very little difference,

So that's why they are all good.

You could also use (this is maybe even more recommended):

These data sets are only used for testing the performance of the new method.

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The first two sentences are almost exactly the same, and perfectly fine.

The third sentence is understandable, but it does not have the correct position for the word "only".

There is are additional questions about "only" which may help you, on the English Language Stack exchange:

https://english.stackexchange.com/questions/5466/correct-position-of-only https://english.stackexchange.com/questions/5615/different-usage-of-the-word-only

There is a fourth position for "only", just for fun:

Only these data sets are used to test the performance of the new method.

This means that these data sets are the only things that will be used: nothing else will be used.

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