Paul wrote that it was a very small thing to him to be judged by others (1st Corinthians 4:3-6), and that he would not even judge himself since such judgement belongs only to God.

 Later, however, he wrote telling the same saints to examine themselves to see whether or not they were in the faith and so as not to "fail the test" (2nd Corinthians 13:5-6).

 Is there a difference between "judging" and "examining"?

When you take an examination in school, you likely go over your answers again and again to make sure that you have answered each question in the exam as correctly as possible.  You examine yourself in this way, judging as best you can that each answer is as correct as you can make it before you give it to the teacher.  The teacher, however, is the judge.   

We don't get to grade our own papers.  You certainly don't get to grade it according to your own opinion of the answers.  That authority belongs solely to the teacher.

You can judge your answers with accuracy beforehand, if, and only if, you know exactly what answers the teacher expects.  A good teacher provides this information long before the examination by means of assignments, exercises, lectures, field trips and other teaching methods.  A good teacher will so inform you of what is expected that you have the potential to examine your work and thereby know how you scored before the "official" judgement of the teacher is ever made.

For example, if your teacher has taught that George Washington was the first president of the United States of America, then you will know how to answer a question to that effect. Just as you will know that "George Washington" is the correct answer to this question, you will also know that to put down any name other than that is to invite a deduction in your grade.

If students were allowed to help one another during the exam, surely they would want to correct anyone who wrote a name other than George's, or to be corrected themselves by fellow classmates had they written the wrong name.  Since all want to pass the exam, surely none would be offended by being corrected.  In fact, gratitude would be the likely as well as the proper and reasonable response.

Why would anyone demand to answer an examination question according to their own opinion when they know that their answer is neither correct nor what the teacher has taught?

When I was in school, there were times when I thought my whole life hung in the balance awaiting the result of a teacher's judgment of answers I had provided during an examination. None of those exams were as momentous as I then believed.

 But there is one coming that is.