• Lately, I've been hearing people say "based off" instead of "based on," and my research shows that the use is increasing rapidly. Since I’ve started teaching, one thing that has jumped out at me is how many of my students, and also people I just hear in the halls, say “based off” instead of “based on.” As in, I believe we’re all doomed based off what I saw last night instead of I believe we’re all doomed based on what I saw last night.

• I can’t tell you why people started using the wrong phrase. English prepositions are often idiomatic, which means that there’s no logical reason that we use one instead the other. When people ask why New Yorkers say they stood on line when almost everyone else says they stood in line, the only answer I have is that it’s a regionalism, and when people ask whether they should say they are in a restaurant or at a restaurant, all I can tell them is that either is fine.
  

• I can tell you that the correct, Standard English phrase is based on, and when you compare the Google Book Ngram results for based off and based on, you reassuringly find that based on is still vastly more popular. Stick with saying your assumptions are based on what you saw last night, but don’t be surprised if you start hearing people use based off. It’s on the rise.

这还只是回答了Base On 和Base Off 的问题。两个回帖才真正回答Base Off Of 的问题：

• My students not only use "based off" all the time, but it is usually extended into the even less logical or grammatical monstrosity, "based off of." Yeesh.

• They actually say "based off of" more often now, adding another incorrect preposition.

另外的一个网站，也说Based Off Of仅限于口语用法，在其他场合，只能用Based On。

• You can build something off of a starting point, but you can’t base anything off of anything. Something is always based on something else.

• I checked in the Corpus of Contemporary American English and there are only 7 instances when based off of is used, limited to spoken style. On the other hand, based on occurs with very high frequency in all sorts of discourse, particularly academic but also spoken.