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Weighted Averages

```
Date: 11/02/98 at 21:09:00
From: Jacob Smith
Subject: Help with "weighted Averaging"

Dr. Math,

I am in the 9th grade, and our math teacher is explaining "weighted
averaging." Could you help me by giving a simple but detailed
description of this?

Sincerely,
Jacob
```

```
Date: 11/03/98 at 12:18:47
From: Doctor Peterson
Subject: Re: Help with "weighted Averaging"

Hi, Jacob. I think I'll start at the beginning with an explanation of
what weighted averaging means in a simple case, and then look at a more
general case.

Suppose that your teacher says the final exam counts as much as three
tests. Then if your scores are:

tests: 70, 80, 90   final: 100

your average will be just as if you got:

tests: 70, 80, 90, 100, 100, 100

70 + 80 + 90 + 100 + 100 + 100   540
average = ------------------------------ = --- = 90
6                   6

If we want to calculate this directly, we can just multiply the final
score by 3 when we add them up, but we have to remember also to count
it three times in the denominator, not just divide by 4. You can do
this by writing it out this way:

score    weight    value
-----    ------    -----
70        1        70
80        1        80
90        1        90
100        3       300
---      ---
6       540  --> average = 540/6 = 90

That is, you divide the sum of the weighted values by the sum of the
weights themselves.

A similar problem arises if you've calculated the average of some set
of things, and want to figure out the new average. It's natural to
want to just average the old average with the new value:

tests: 70, 80    average = (70 + 80)/2 = 75
new test: 90     new average = (75 + 90)/2 = 82.5 ?

But this is wrong, since the new average really is:

new average = (70 + 80 + 90)/3 = 240/3 = 80

What you have to do in this case is to weight the old average
proportionally to the number of scores it represents, since this
situation is just as if you had scores of 75, 75, and 90:

2 * 75 + 90   240
new average = ----------- = --- = 80
2 + 1       3

Now, in general, you can assign any weights you want, not necessarily
integers. Often we choose weights that add up to 1. In our first
example, we can say that each test counts for 1/6 of the grade, and
the final counts for 1/2 of the grade. Then we calculate this way:

score    weight    value
-----    ------    -----
70       1/6     11.66
80       1/6     13.33
90       1/6     15
100       1/2     50
---     -----
1      89.99  --> average = 90/1 = 90

I hope that helps out.

Here's an explanation of weighted averages I  found in our archives, if
you want another perspective:

http://mathforum.org/dr.math/problems/riggins11.16.95.html

Incidentally, do you wonder why it's called "weighted"? An average can
be thought of as the place where a set of identical weights placed at
different locations on the number line would balance:

X   X   X   X
---+---+---+---+---+---+---+---+---+---+---+---
0  10  20  30  40  50  60  70  80  90  100
^

If the weights are different, you get a weighted average:

1   1   1   3
---+---+---+---+---+---+---+---+---+---+---+---
0  10  20  30  40  50  60  70  80  90  100
^

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Statistics

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