can not find correct grammar for the use of : the... the [duplicate]

Traditional grammar defines the sentence as 'a complete thought containing a subject and a predicate'. So by this definition the second 'sentence' is incorrect, but of course writers constantly break the rules of grammar to achieve a particular effect.

The problem I see with the 'sentence' is that it may lead the reader astray. He or she gets to the end of the string of words (a long noun phrase) thinking that they comprise the subject, but then the predicate is missing. This will possibly cause a backtrack to reread the first sentence in order to make sense of the whole. It's generally inconsiderate of your readers to make them do this.

(And no, I would not recommend breaking this traditional rule in an application letter.)

The main problem is that "the different ways in which you could look at data and infer knowledge from it" is an appositive that explains the noun statistics, and, as such, shouldn't be in a sentence of its own:

I have always been fascinated by statistics, the different ways in which you could look at data and infer knowledge from it.

If you're attempting to convey why you're fascinated with statistics, it's best to set off the second part with dashes, per this guideline:

I have always been fascinated by statistics -- the different ways in which you could look at data and infer knowledge from it.

The second 'thing' in your quote is not a sentence, by most generally accepted notions of sentence.

Linguistically, in English (not necessarily so for some other language) a sentence is -defined- (I'm not saying 'should be') as having a noun phrase and a verb phrase. That is an academic definition, which may be more or may be less restrictive than grade school grammarians, and which may be more or may be less restrictive than styles acceptable for publication/email/formal speech/informal speech/etc, etc

Stylistically, a 'sentence' that is missing a verb is not considered good style in written publications, and would show a lack of writing experience which would get you thrown out of a job for a newspaper, or would stand out as 'poor grammar' in an essay for a university application.

Of course, people (even English speakers) go around making utterances that don't include verbs, or verbs without nouns, or quite often just plain interjections. And they're not berated for that.

So whether you call your things sentences or not, in a serious narrative (fiction or non-fiction) especially for an educational institution, missing a verb is considered 'bad grammar'. For other things (more loose narratives, poetry, fill in the blank answers on tests, etc) a verb might be optional depending on what you want to get across.

IMHO, a sentence without any verb could be correct.

Yes!
No!
Away with flying saucers!

I.e., any slogan.