Paul and his friends average their test grades and find that the average is 95. The teacher announces that the average grade of all her classes is 83. Why are the averages so different

The averages did not match because of the number of grades calculated into the total average. Let's say for example that the teacher in this case has 25 students in her class. She would take the average of all 25 students' test grades. Let's say that Paul has 4 friends with whom he averaged his test grade. He has taken the average of 5 tests, whereas the teacher has calculated the total amount of student test scores (25). Paul and his friends may have done better on that particular test than the other 20 students in the class, explaining the average of 95 that he got. Other students who got B's or C's on that test pulled the total class average down, explaining the teacher's average of 83.