"in or to the amount or extent of." Here, "by" indicates the extent or degree of the increase in tuition. The sentence communicates that the tuition increased to a certain extent, specifically by an average of three percent. It signifies the amount by which the tuition has increased, indicating the magnitude of the change.

" the phrase "an average" is used instead of "the average" because it is referring to a general or unspecified average across a certain population or dataset.

In the sentence "Tuition increased by an average of three percent," the word "by" functions as option 1: "in or to the amount or extent of." It indicates the amount of the increase, specifying that the increase was three percent.

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Note that using ‘by’ here can be omitted, but including it helps the reader more readily comprehend the meaning and therefore sounds slightly better. Consider: Tuition increased my financial difficulties. (Having to pay tuition caused my financial difficulties to increase) Tuition increased three percent. Tuition increased by three percent. (The ‘increased by’ suggests that we’re going to talk about something that happened to the tuition, not that something that tuition caused to happen.) English being fun, however, there is a sentence with another meaning of ‘by’. For example ‘Tuition increased by a vote of the Board of Trustees’ but that use is rarer and isn’t likely to confuse us!