For some people, average case analysis is difficult because they don't have a very flexible idea of what an "average" is.
Last semester, Professor Cord gave out the following grades in his CS361 class:
A, A, A-, B+, B, B, B, B-, C+, C, C-, D, F, F, F, FTranslating these to their numerical equivalent,
4, 4, 3.7, 3.3, 3, 3, 3, 2.7, 2.3, 2, 1.7, 1, 0, 0, 0, 0what was the average grade in Cord's class?
According to some classic forms of average:
the middle value of the sorted list, (the midpoint between the two middle values given an even number of items)
the most commonly occurring value
Computed from the sum of the elements
The mean average is the most commonly used, but even that comes in comes in many varieties
Example: Last semester Professor Cord gave the following grades
Grade | # students |
---|---|
4.0 | 2 |
3.7 | 1 |
3.3 | 1 |
3.0 | 3 |
2.7 | 1 |
2.3 | 1 |
2.0 | 1 |
1.7 | 1 |
1.3 | 0 |
1.0 | 1 |
0.0 | 4 |
The weighted average is
Example: When one student asked about his grade, Professor Cord pointed out that assignments were worth 50% of the grade, the final exam was worth 30%, and the midterm exam worth 20%. The student has a B, A, and C-, respectively on these.
Category | Score | Weight |
---|---|---|
Assignments | 3.0 | 50 |
Final | 4.0 | 30 |
Midterm | 1.7 | 20 |
So the student's average grade was
The expected value is a special version of the weighted mean in which the weights are the probability of seeing each particular value.
If `x_1, x_2, \ldots,` are all the possible values of some quantity, and these values occur with probability `p_1, p_2, \ldots,`, then the expected value of that quantity is
Note that if we have listed all possible values, then
so the usual denominator in the definition of the weighted sum becomes simply "1".
Example: after long observation, we have determined that Professor Cord tends to give grades with the following distribution:
Grade | probability |
---|---|
4.0 | 2/16 |
3.7 | 1/16 |
3.3 | 1/16 |
3.0 | 3/16 |
2.7 | 1/16 |
2.3 | 1/16 |
2.0 | 1/16 |
1.7 | 1/16 |
1.3 | 0/16 |
1.0 | 1/16 |
0.0 | 4/16 |
So the expected value of the grade for an average student in his class is
The expected value is the kind of average we will use throughout this course.
In the Forum: