Expert: Josh - 12/2/2010

Question
Hello:

I am curious to know how to determine the weighted average of the following by using a simple average. I am just curious to know whether or not it is possible.

Here is a simple average example:

1 pound of coffee @ $0.90 per pound
1 pound of coffee @ $1.20 per pound
1 pound of coffee @ $1.60 per pound

3 pounds -----------$3.70

$3.70/3 lbs. = $1.2333/lb

Here is a weighted average example:

100 lbs. of coffee @ $0.90 per pound = $90.00
70 lbs. of coffee @ $1.20 per pound = $84.00
30 lbs of coffee @ $1.60 per pound = $48.00

200 lbs.-----------------------------$222.00

$222/200 lbs = $1.11/ lb.

I divided 100, 70, and 30 pounds by 3 lbs. from the simple average from above.
100/3 = 33 1/3, 70/3 = 23 1/3, 30/3 = 10. These amounts total 66 2/3. I multiplied 3 lbs by 66 2/3 and got 200 pounds the total from the weighted average example, but I cannot determine a method to use $3.70 to get $222.00. Is there a way?

I thank you for your reply.  

Answer
Hi Kenneth,

The word "averaging" refers to the process of adding up terms and dividing by the number of terms. As an example, consider the operations written on the left hand side of the following equation.

(1/3) x [ 1x0.9 + 1x1.2 + 1x1.6 ] = 3.7

In the case of a weighted average,

(1/3) x [ 100x0.9 + 70x1.2 + 30x1.6 ] = 222

Once we have determined the average (or weighted average), we end up with a number such as 3.7 (or 222) which does NOT tell us about the relative weights or composition of the sum. The number 3.7 is related to 222 by a ratio which we may write as a fraction in 3.7/222.

What you have done earlier(divide everything by 3), is later canceled by its inverse operation (when you multiply everything by 3 again). So, it achieves nothing.

The important point is that while a simple average is a special case of a weighted average, it is no substitute for a weighted average calculation. If a weighted average is required, we cannot do a simple average, and hope that by manipulating the resulting number, we can somehow produce an equivalent result -- without actually computing the weighted average.

That is, if you cover up the second line (above) which describes the weighted average calculation, we cannot know the result of the weighted average. Here, it turns out to be 222; but of course, it varies case by case. Without doing the weighted average, we simply do not know what to multiply (the answer of the simple average) 3.6 by, to obtain the result produced by the weighted average. We need to know (and cannot ignore) the composition of each ingredient (i.e., 100, 30 and 70)