Based on accounts of various ancient writers, scholars have painted a

Hi mvikred,

Based on accounts of various ancient writers, scholars have painted a sketchy picture of the activities of an all-female cult that, perhaps as early as the sixth century B.C., worshipped a goddess known in Latin as Bona Dea, “the good goddess.”

After reading your doubt, I feel that you have misinterpreted the meaning of this sentence somehow. Now the real meaning might not have significant impact in choosing the correct answer choice in this sentence. However, it is very important that you understand the correct intended meaning that, in most cases, has a direct bearing on choosing the correct answer choice.

The sentence means that the scholars studied the accounts of various ancient writers and have painted a sketchy picture of the activities of an all-female cult worshipped a goddess called Bona Dea, the good goddess. Now, the scholars did not paint any literal sketch of the activities of the all-female cult. Here, this expression has been used as metaphor to mean that scholars presented their not-very-perfect understanding of the cult. Notice, the sentence says that the scholars painted the sketchy picture of the activities of the cult and not the goddess.

Error Analysis:

Let’s look at a set of simple examples first:

1. Kim has kept the lamp in the casket designed by her grandmother.
2. Designed by Kim’s grandmother, the casket got sold in the auction at a very high price.

In the first sentence, “designed” is preceded by the noun “casket”. Hence it modifies the preceding noun “casket”. The modifier suggests that the casket was designed by Kim’s grandmother.

In the second sentence, “designed” appears in the beginning of the sentence. In this case, it modifies the subject of the following main clause “casket”. The modifier suggests that the casket was designed by Kim’s grandmother.

Now let’s apply this rule on the OG sentence:

Based on accounts of various ancient writers, scholars have painted a sketchy picture of the activities of an all-female cult that, perhaps as early as the sixth century B.C., worshipped a goddess known in Latin as Bona Dea, “the good goddess.”

The verb-ed modifier “based” appears in the beginning of the sentence. As per the meaning, it modifies the subject of the following clause. The subject is “scholars”. This modification suggests that “scholars” were based on accounts of various ancient writers. This modification is absolutely illogical. That’s why OG says that the modification is incorrect. Here we need a modifier that must correctly and logically modify “scholars”.
Now let’s analyze the sentence with the correct answer choice E:

Using accounts of various ancient writers, scholars have painted a sketchy picture of the activities of an all-female cult that, perhaps as early as the sixth century B.C., worshipped a goddess known in Latin as Bona Dea, “the good goddess.”
This answer choice is correct because the verb-ing modifier “using” correctly modifies the scholars. It suggests that scholars used accounts of various ancient writers to do something.

Now let's consider your example:

Based on the assumptions listed, the professor calculated the rate of dispersion.

Now in day-to-day language, this form might be acceptable but this sentence will stand incorrect for the same reason – illogical modification. The modification in this sentence suggests that the professors were based on listed assumptions. So on GMAT, this sentence is incorrect. The correct version would be:

Based on the assumptions listed, calculations of the rate of dispersion were made by the professor.

For more on verb-ed and verb-ing modifiers, please refer to this article:
https://gmatclub.com/forum/verb-ed-modif ... 25611.html

Hope this helps.
Thanks.
Shraddha