Grade Calculation
Let's take this in small steps, since many students know how to compute an
average, but few know how to compute a weighted average.
Weighted Average
Suppose you have a class with only a midterm and a final exam. The
midterm is 40 percent of the grade, and the exam is 60%. You got an
80 on the midterm and a 100 on the exam. What's your overall grade?
Certainly not a 90! If it were a 90, that would be giving the midterm
the same weight as the final exam. It seems like you deserve better
than a 90, since you got 100 on a test that was worth more than the
test you got an 80 on.
Here's the exact calculation:
80 * (40/100) + 100 * (60/100)
32 + 60
92
So, your overall grade is a 92; that's more like it. In general, if
you get "M" points on the midterm, and the midterm is "m" percent of
the total, and similarly you get E on the exam, which is worth "e"
percent, your overall grade is:
M * m + E*e
Note that that formula works as long as m+e=1.0. In other words,
they are fractions of the whole. If they don't add up to 1, we have
to divide by their total. This is called "normalizing." Here's the
more general formula:
(M * m + E*e) / (m+e)
Let's re-do the first one using this idea, only this time we'll have
m=40 and e=60:
(80 * 40 + 100 * 60) / (40+60)
(3200 + 6000) / (100)
9200/100
92
Try the following example. The grade is based on four categories:
| category | percentage |
|---|
| tests | 40 |
| homework | 10 |
| programs | 20 |
| exam | 30 |
Suppose that these are your grades:
| category | scores |
|---|
| tests |
80 |
90 |
| homework |
90 |
95 |
100 |
| programs |
70 |
100 |
100 |
| exam |
90 |
To find your overall average, you have to compute your test average,
your homework average, and so forth. These are each simple
averages, without weights, since, for example, all the tests count
the same:
| category | Simple Average |
|---|
| tests |
(80+90) / 2=95 |
| homework |
(90+95+100) / 3=95 |
| programs |
(70+100+100) / 3=90 |
| exam |
90 |
Now, we can compute the overall average, which is a weighted average
of these numbers:
95*0.4+95*0.1+90*0.2+90*0.3
38+9.5+18+27
92.5
It couldn't be any easier, once you see how it works.
Missing Grades
This is all fine, but suppose you're trying to calculate your current
grade halfway through the semester, so you don't yet know what you
will get on the final exam? There are two approaches.
- One is simply to estimate what you will get on the final exam,
and then calculate your overall average using the estimate. The
resulting number is the grade you'll get if you get that score on
the final. A really nice way to do this is with a spreadsheet
program, such as Lotus 123 or Microsoft Excel. If your formula
takes the scores from cells, you can just type in different
estimated scores, and the spreadsheet will automatically recalculate
your grade based on those different estimates. Very convenient.
- Another way is to leave out the unknown scores and normalize the
result by dividing by the total percentage that you do know.
Suppose we don't know the exam score, but we know all the other
scores above. Here's the calculation:
(95*0.4+95*0.1+90*0.2) / (0.4+0.1+0.2)
(38+9.5+18) / 0.7
93.57
Notice that this is higher than the 92.5 we got before, but that's
to be expected. We got a 90 on the final exam, and that brought
our average down a little.
Scott D. Anderson
Last modified: Tuesday, August 29, 2000