Receiving a degree with honors shows that the individual achieved either a high grade point average, or had a grade point in the top percentile of a class. For example, some universities award a degree with honors to everyone who achieves a 3.5 grade point average (on a 4.0 scale). Other universities grant honors to the top, say, 15% of the class.
The rankings have several levels: typically, cum laude (i.e., "with honor"), magna cum laude ("with great honor"), and summa cum laude ("with highest honor"). When I attended Harvard Business School, degrees awarded to the top students were granted With Distinction (top 15%) or With High Distinction (top 3%). My own degree was awarded With High Distinction, and Harvard translates that in its records as magna cum laude. However, as I graduated in the top 0.5% of the class, in any of the other Harvard colleges that degree would have been awarded summa cum laude.
There are other ways of evaluating merit and quality, than by grades earned in college classes. A high grade point average does not alone indicate superiority in any greatly meaningful way. However, high grades (and especially an honors degree) does show that a person is likely more intelligent than most of his peers, and has applied himself more diligently to his studies, having learned and retained more than most. Hence, an honors degree does deserve some degree of respect, a bit above that one might accord to most college graduates.
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