Two different groups of students, each with 5 members, took a quiz. The average (arithmetic mean) score for each group was calculated and is shown below.
Group P: 82 points
Group Q: 78 points

Which of the following MUST be true?
I. Every student in Group P scored higher than every student in Group Q.
II. The highest score in Group Q is less than the lowest score in Group P.
III. There is at least one student in Group P who scored higher than at least one student in Group Q.