How to calculate weighted average when the weights do not equal 100 percent?

Question

So, I'm in a bit of quandary. I have to calculate a weighted average but the weights do not equal 100%. What potential adjustments can I make to address this issue?

Thanks!

Answer 1

You have items $x_1, \ldots x_n$ with weights $w_1, \ldots, w_n$. The weighted average would be $$ \frac{\sum_{k=1}^n x_i w_i}{\sum_{k=1}^n w_i}. $$

In other words, let $W = \sum_{i=1}^n w_i$ and you can produce new weights $$ u_i = \frac{w_i}{W} $$ with the property that $$ \sum_{i=1}^n u_i = \sum_{i=1}^n \frac{w_i}{W} = \frac{1}{W} \sum_{i=1}^n w_i = 1 $$ and then compute $$ \sum_{i=1}^n u_i x_i = \sum_{i=1}^n \frac{w_i}{W} x_i = \frac{1}{W} \sum_{i=1}^n w_i x_i = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} $$ as noted above.

Answer 2

If the weights are absolute and thus supposed to add up to 100%, then check to see if you added wrong or missed something. For example, if exams are 60% of the grade and homework is 30% of the grade, you might be missing what the remaining 10% is.

If the weights are relative and thus not supposed to add up to 100%, just divide by the sum of the weights. For example, if exams count twice as much as homework in the grade, then double the exam score, add the homework score, and divide by 3.