Calculating a Grade Average

Date: 08/15/99 at 22:13:58
From: Rachel
Subject: Finding my exact average

This year my dad wants me to keep track of all my scores of 
worksheets, tests, etc. to find out what my average is. Well here's my 
problem: If some things are worth more than others are, how can I do 
that?

Thank you!
Rachel

Date: 08/16/99 at 04:42:43
From: Doctor Jeff
Subject: Re: Finding my exact average

Hello, Rachel. Thanks for writing to Dr. Math.

Let's assume that your average is calculated on a point system. That 
is, everything is worth a certain number of points, and certain things 
are worth more points than others are. A test might be worth 100 
points; a paper, 50 points; and a quiz, 10 points.

Here are a student's scores, using the above system, for the first 
quarter of the school year:

     Quizzes          Papers          Tests
      9/10 or  90%    43/50 or 86%    79/100 or 79%
      7/10 or  70%    41/50 or 82%    92/100 or 92%
     10/10 or 100%                    90/100 or 90%
      8/10 or  80%
      8/10 or  80%

If you want to calculate the student's average for the entire first 
quarter, the easiest way is to add up the total number of points the 
student earned and divide that by the total number of points the 
student could have earned. In this case, it's best to avoid dealing 
with the individual percentages, since, for example, a 90% score on a 
quiz does not count nearly as much as a 90% on a test.

This student's average would then be:

      9+ 7+10+ 8+ 8+43+41+ 79+ 92+ 90   387
     -------------------------------- = --- = .86 = 86%
     10+10+10+10+10+50+50+100+100+100   450

Of course, not all teachers use this simplified point system. Perhaps 
a more common method for determining a student's average goes 
something like this: "Quizzes are worth 20% of your grade; homework is 
worth another 10%; papers are worth 30%; the final is worth 30%; and 
class participation is worth the remaining 10%."

In this scenario, students might only be given percentages for their 
work. Let's say a student received the following scores:

     Quizzes     Homework     Papers     Final     Participation
     95%          84%         85%        88%       100%
     80%         100%         89%
     75%         100%
     90%          90%
                  91%

The first thing the student needs to do is come up with an average for 
each of the five categories above. Let's assume that each quiz is 
worth the same as every other quiz, that each homework assignment is 
worth the same as every other homework assignment, etc.

To find an average for a category, simply add up the percentages and 
divide by the number of scores there are. For example, the quiz 
average would be:

     95%+80%+75%+90%   340%
     --------------- = ---- = 85%
            4           4

The category averages are:

     Quizzes  =  85%
     Homework =  93%
     Papers   =  87%
     Final    =  88%
     Partic.  = 100%

Now, remembering that each category is worth a certain amount in 
calculating the average for the quarter, multiply the average for each 
category by the fraction of the quarter average that the category 
represents:

     Quizzes:  85% * .2 = 17%
     Homework: 93% * .1 =  9.3%
     Papers:   87% * .3 = 26.1%
     Final:    88% * .3 = 26.4%
     Partic.: 100% * .1 = 10%

Now, just add up these new percents to get the student's weighted 
average for the entire quarter:

     17% + 9.3% + 26.1% + 26.4% + 10% = 88.8%

Related to this question of calculating averages is how to determine 
what a student needs to score on, say, a test, to get a certain grade. 
Let's pretend that our student has not yet taken her final. The 
question is, assuming the student has the rest of the grades listed 
above, what does she need to get on the final to raise her average to 
a 90%?

We already saw that summing the products of each category's percent 
and its "category worth" will give us a student's average. We can use 
this same equation, but this time call the first quarter's average 
90%, since that's what the student is aiming for. Then all we 
need do is solve for the grade on the final, which we'll call f for 
simplicity:

     average = 90%   = 17% + 9.3% + 26.1% + (.3*f) + 10%
               90%   = 62.4% + (.3*f)
               27.6% = .3*f
               92%   = f

The student must, therefore, score at least 92% on the final to raise 
her average to the 90% level.

There are about as many ways of calculating averages as there are 
teachers. At the beginning of the school year, make sure to find out 
how your teachers go about their calculations so you don't wind up 
with completely different grades when report cards are given out!

I hope this helped you out. Please write back if you find anything 
confusing or if you have more questions.

- Doctor Jeff, The Math Forum
  http://mathforum.org/dr.math/