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The data in Tables 1 were assembled from studies in North America...
Fluxes shown in Table 1 are different from Tables 2 and 3 because...
Results of the calculations indicate that ....
These three sentences are from one paragraph from a published research paper. But the first sentence keeps 'the' whereas the second and third choose to neglect 'the'. I wonder why is the case.
There are some general rules about when a "the" is or is not required.
Plural nouns referring to specific things
In the specific three examples you provided, the inclusion or exclusion of the "the" is largely just a matter of personal preference, or personal style in writing, for two reasons. First, all the nouns are plural. Second, they are all referring to specific examples (e.g. data, fluxes and results in 'Table 1').
Since the "the" is optional where both the above conditions are met, all of the following pairs are acceptable.
Example 1
Example 2
Example 3
Plural nouns referring to general things
By contrast, where the noun is plural AND it refers to a general not a specific thing, then it is conventional to exclude the article. For instance, the following sentence would always be used without the article.
Singular nouns need a "the"
If the noun is singular, a "the" is almost always mandatory, because it is referring to a specific thing (which is why the term for the word "the" is "definite article").
For instance, if your examples were modified to have a singular noun, the only correct expressions would be as follows.
You will note that in my reworded version of your example 1, the word "data" has been changed to "data points". The purpose of the change was to avoid the debate about whether the word "data" is a singular or plural noun. Most authors would treat it as a singular, but there is still some debate (see, for instance, this article: https://www.theguardian.com/news/datablog/2010/jul/16/data-plural-singular).
The distinction is important, because if "data" is singular (like the word "datum") then:
(a) it must have a "the" preceding it, and also
(b) it needs a singular verb.
In your first example above, the author has used the verb "were", which is a plural verb. This means that the author is treating "data" as plural noun, in which case the "the" is optional.
But it's way more nuanced & complex ....
The above general rules are just a starting point ....
Not always. A counterexample where omitting the "the" does not make sense is
The day I went to London.
Keeping the "the" in cases where the meaning does not change if omitted is a matter of personal preference for the author.
However "the" is sometimes used to emphasise or denote uniqueness or to distinguish from any other and is often stressed when spoken. As in (assume John Doe is a famous performer) .
The John Doe took to the stage.
not any other John Doe
In a similar vein the build up introductions by comperes often uses
The one and only John Doe!
Bold denotes stress when spoken.
In the example sentences you quote, using the definite article "the" is indeed optional. However, the meaning of the sentences is subtly different between when "the" is used and when it's not.
The basic distinction between "the" and no article for plural nouns is the same as between "the" and "a"/"an" for singular nouns: it indicates the definiteness of the noun that follows it:
Basically, a phrase like "the <things> in <location>" implies that there's a definite set of these things in that location, and that we're talking about all of them (and no others). In contrast, the phrase "<things> in <location>" without a preceding "the" can simply refer to some things in the location, not necessarily all of them.
For example, your example sentence:
The data in Table 1 were assembled from studies in North America...
makes it clear that all of the data in Table 1 were assembled from studies in North America. Without "the", it could mean that only some of the data were from studies in North America, while the rest of them might be from some other sources.
Conversely, in the sentence:
Fluxes shown in Table 1 are different from Tables 2 and 3 because...
the lack of a definite article leaves open the possibility that some of the fluxes shown in Table 1 might not actually be all that different from those in Tables 2 and 3, as long as at least some of them are different. With "the" in front of "fluxes", the sentence would imply that all of the fluxes are different.
Similarly, the sentence:
Results of the calculations indicate that...
might refer to only some of the results of the calculations; other results could well be completely irrelevant and indicate nothing like what the rest of the sentence says, or, just possibly, they might even appear to be contradictory. Writing "the results" with a definite article would imply that the author was talking about all of the results of the calculations, and that they all indicate the same conclusions; it would also at least implicitly suggest that none of the results indicate anything else worthy of note, at least not in this particular context.
But in practice, those distinctions can be very subtle, since often there's not much difference between the definite and indefinite versions, and since usually the intended meaning is clear from context anyway. In your first example sentence, even if author had omitted the definite article, we could still reasonably assume that they were referring to all of the data if they didn't mention any other possible source for the rest of them. Meanwhile, in the other two example sentences, it probably doesn't really matter whether all of the fluxes are actually different between the tables, or whether all of the results of the calculations are really needed to support the author's conclusion (at least as long as none of them actively contradict it, in which case an honest author really should explicitly note it anyway).
Grammar has its place; so does style
Start an idea in a paragraph or sentence with something specific:
The data in Tables 1 and 3 were gathered in 2015. [for example]
"The" is used in contrast to what could or might be something like:
Once the specificity of the data has been established, you no longer need to use "the":
Example: The accidents [on some date] involved many vehicles. Cars and trucks were piled up on the highway for miles and miles.
Example: The games played by the team were very exciting. Players were very motivated to win.
Summary: The use of articles can be determined by grammar but also by context.
