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This is math 101 but I am struggling to fully comprehend the difference between these 2 calculations.

I have 50 observations of the value of sales and volume of sales (in lbs). Price per unit = value/volume.

If I calculate the price for each observation, sum, then divide by the number of observations, is this a weighted average?

If I sum all observations of value of sales and volume of sales, then divide the two, am I calculating a simple average price?

What am I calculating in each instance and why are they different?

406LQE
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  • A good way to see it is to do only two observations. Make up the data with very different volumes and rather different prices. Do the two calculations and think about what is being divided by what. – Ross Millikan Sep 10 '15 at 18:17

1 Answers1

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Let's write out what you are computing. Suppose your data is $$\{S_i,V_i\}_{i=1}^{i=50}$$ Of course $P_i=\frac {S_i}{V_i}$

What are your two procedures? Assuming I have understood you correctly we have:

Method I: $$\frac {1}{50}\sum_{i=1}^{50}\frac {S_i}{V_i}$$

Metod II: $$\frac {\sum_{i=1}^{50} S_i}{\sum_{i=1}^{50} V_i}$$

Phrased this way, it's clear that they are not equal in general, no?

The big difference, to me, is that method II is dominated by the big volume days whereas method I is not. As an example, suppose you only had two observations: $$\{1,1\}\;and\;\{2^*10^6,10^6\}$$ The price for the first day is $1$ and for the second day it is $2$, hence Method #1 gives $\frac 32$. But method #2 gives:$$\frac {2,000,001}{1,000,001}\sim 2$$. As I say, the low volume days simply don't register.

Conceptually: The first method gets you an average Price (it literally computes the average of the price realized on each day). The second method gets you "the average realized value of a unit of your product". These will be comparable if the volumes from day to day are roughly constant, but not otherwise.

lulu
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  • Never mind, figured it out. – 406LQE Nov 02 '15 at 13:04
  • How does Method 1 give you 3/2? What happened to the 1/50 it is multiplied by? – 406LQE Nov 02 '15 at 13:10
  • Are your comments out of sequence? I expect you've worked it out, but just in case: in my example I used $2$ observations, not $50$. Easy to see what's going on with small data sets. – lulu Nov 02 '15 at 13:14
  • Thanks for replying so quickly. I just wasn't thinking clearly and asking questions before I thought them through. – 406LQE Nov 02 '15 at 13:24
  • Oh, no problem. I note that the other commenter also proposed working with $2$ observations as a way to see things clearly. Hard to hold $50$ observations in your head. – lulu Nov 02 '15 at 13:26