Stephanie M. answered • 07/07/15
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Set up a couple equations to see that those two quantities are not equal. Let's say you have x1, x2, x3 and y1, y2, y3.
Dividing values of X by Y to obtain Z gives you:
x1/y1 = z1
x2/y2 = z2
x3/y3 = z3
The average of those Z values is:
(z1 + z2 + z3)/3 = (x1/y1 + x2/y2 + x3/y3)/3 = (1/3)(x1/y1 + x2/y2 + x3/y3)
The sum of X divided by the sum of Y is:
(x1 + x2 + x3)/(y1 + y2 + y3)
There's simply nothing you can do to rearrange that first equation into the second. They aren't the same. That's because, in the second equation, you're essentially dividing every X value by every Y value. In the first equation, you're dividing each X value by only a single Y value, then dividing each quotient by 3.
Stephanie M.
tutor
I'll try to explain it another way:
If you divide each X by its corresponding Y, you've actually found the number of pounds of that weight of casting you can get
per dollar. That's because when you divide X by Y, you're calculating x pounds / y dollars, which will give you
x/y pounds per dollar.
To find the price per pound, you'll want to divide Y by X: y dollars / x pounds will give you
y/x dollars per pound.
If you find y/x = z for each weight of casting, you'll have three price-per-pound values. Taking the average of those numbers will give you the
average price per pound.
Now, let's consider the other method. (I'll switch your X's and Y's since that's what I did with the previous method.) If you add up all the X's, you'll have found the total weight of castings bought. If you add up all the Y's, you'll have found the total price paid for the castings. Dividing ∑Y by ∑X will give you ∑Y/∑X dollars per pound
if all the castings weighed the same and cost the same price. That is, in this situation, you've bought 8+10+6 = 24 pounds of castings and you've paid a total of 5+4+2 = $11. So, if you bought 24 pounds all in one go and paid the same amount for each pound, you'd have the price per pound.
But, that's not what our customers did. They paid a different price per pound based on the number of pounds they bought. That means you ought to use Method 1, which takes into account the fact that there may have been more pounds bought at, say, $2 per pound than $4 per pound, but that you want to view the $2 castings as importantly as you view the $4 castings, no matter how many were bought at each price. By using Method 2, you put unnecessary importance on a customer who buys more pounds, so you come up with a less accurate answer.
Imagine, for example, that a customer buys 100 pounds for $100, and another customer buys 4 pounds for $20. That means one customer bought castings for $100/100 pounds = $1 per pound, and the other customer bought castings for $20/4 pounds = $5 per pound. That's an average of ($1+$5)/2 = $6/2 = $3 per pound. This method tells you that, of the two customers, they paid
an average of $3 per pound for their two groups of castings. The method doesn't give extra importance to the first customer for buying more pounds of castings. It gives the customers equal importance and therefore tells you that, no matter how many pounds of castings you buy, you can expect to pay around $3 per pound.
If you use the other method, however, you give the first customer much more importance because he bought so many more pounds. You'll find that a total of 100+4 = 104 pounds of castings were bought for a total of $100+$20 = $120. That's a price of $120/104 = $1.15 per pound. The price is much closer to the price per pound that the first customer paid, since he bought more castings. Using this method, you'd expect to pay around $1.15 per pound for any number of pounds of castings you bought, and you would be supremely disappointed if you bought something like 2 pounds or 10 pounds or 20 pounds and wound up paying closer to $5 per pound.
In other words, if you're interested in the average price per pound, you'll want to treat each customer separately and give them equal importance. You don't want to weight a customer more if they bought more pounds, since that's not at all what you're interested in. I hope this helps clarify!
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07/09/15
Jorge B.
Excellent explanation Stephanie, I do really appreciate the help. thanks for your time, I will for sure take you as the only resource in the future for a paid consult.
Thank you very much
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07/10/15
Stephanie M.
tutor
You're welcome! Happy to help. =)
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07/10/15
Jorge B.
07/08/15