Am I the only one who sees a problem with yesterday's "average"?

As I said in another post, the VWAP formula is the discrete version of the first moment of the price probability density function. For a continuous probability distribution, the moments are given by,



Where n=1, x is the price, and dF(x) is the differential cumulative probability distribution. That's what the average price is measuring -- the mean of the distribution.

Your calculation isn't taking into account the shape of the underlying distribution.

Maybe using a physical example will help this make more sense. What we're basically calculating is the "center of mass" of the price distribution. So, in your example, you would have 100,000 lbs at a point 300 ft. from the end of your measuring tape and another 100,000 lbs at a point 100,000 ft. from the end. Where would you have to put a pivot point so your weights balance? You'd have to put it at a point 50,150 ft from the end of the tape.

Hope this helps!

Ricky: is it not the case that your weighted average is the weighted average of the exchange rate and not the weighted average of the price? Why should I care about the average rate of exchange at which the lindens were sold?

You buy two steaks. One costs $10 and the other costs $20. What is the average price?

$15

( 1 * $10 + 1 * $20 ) / ( 1 + 1 ) = $15

Except by your method you would use the price expressed in steaks per dollar instead of dollars per steak:

( 1 * 0.1 S/$ + 1 * 0.05 S/$ ) / ( 1 + 1 ) = 0.075 S/$

0.75 S/$ = $13.33 $/S

So by your method the answer is $13.33.

I'm wondering why I read all of this and what possible relevance any of it could have to anyone, other than to inflate their own ego by being able to put sigmas and deltas and other such obtuse symbols, albeit pretty ones, in a post. But then again I hate lawyers for much the same reason.
Or it could be because I lost $20,000 L because the group land system is seriuolsy flawed, Hey I have an idea how about using rhose brain cells for what would be useful to the average citizen and come up with a system that doesnt hurt we unemployed mathmatically illiterate types.

The average citizen should aspire to being mathematically literate if they are not already.

You buy twenty-one steaks. Ten cost $10, ten cost $20 and one costs $1000. If someone asked you how much they might expect to pay for a steak, what would you tell them?

The average would be $61.90.
The volume-weighted average price would be $13.99.

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01110010011011100110100101101111011011100010000001101101011000010111010001101000
00100000011101000110111100100000011000100111010101101001011011000110010000100000
01110100011010000110100101101110011001110111001100101100001000000111010001101000
01100001011101000010011101110011001000000111010101110011011001010110011001110101
01101100001000000110000101110110011001010111001001100001011001110110010100100000
01101101011001010110000101101110001000000110111101100110001000000110110001101001
01101110011001000110010101101110011100110010000001101001011100110110111001110100

In your opinion. Other people have other interests and find other thing important. More importantly, in the Land and the Economy part of the forums it is on topic.

/award ceremony

This thread is totally geeked out. I award this thread the #1 geeked out thread of all SL history.

I'm a logical guy, that's why i dropped economics very early. It was that or kill the teachers.
Here again people managed to make something simple into something very convoluted... so i'll gladly contribute.

Isn't Francis right? If you use VWAP (which i never knew before this thread). That's multiply the volume of Lindens by the US$ rate per Linden (or volume of dollars by the L$ rate per US dollar ?

Basically, what you want to end up with is an average rate, that if you multiply the total sold L$ volume with, you end up with the actual resulting amount of US$, not?

To take keiki's example:
- 100.000 L$ at L$300/1U$ (0.0333 US$/1$L)
- 100.000 L$ at L$100,000/1US (0.00001 US$/1$L)

Keiki now calculates the VWAP:
Total amount of US$
( 100,000 * 0.00333 + 100,000 * 0.00001 ) = 334.33 US$
divided by total volume ( 100,000 + 100,000 ) = 0.001675 US$/1L$
This is 598.21 L$/US$

Which makes the VWAP of 998 L$/US$ very strange...

The volume weighted average (which as far as I can tell is just another name for "average" is:

10 steaks * $10 = $100
10 steaks * $20 = $200
1 steak * $1000 = $1000

$1300 / 21 = $61.90

I'm not sure what $13.99 is supposed to represent in this example.

You're right. LL VWAP calculation is probably wrong in the sense that they multiple volume of L$ with L$/U$ rate. This completely skews the average when small volumes are sold for rediculous prices. Example:

1,000,000 L$ sold for 300 L$/ U$
10 L$ sold for 1,000,000,000 L$ / US$

actual VWAP:
( 1,000,000 * 0.0033 + 10 * 0.000000001 ) / 1,000,010 = 0.0033 (300 L$ / US again).
The rediculous sale completely disappears out of the average, as intended.

LL 'VWAP' (as i suspect):
(1,000,000 * 300 + 10 * 1,000,000,000 ) / 1,000,010 = 10,300 L$ / US$ (wrong)

One thing that has occurred to me is that if you want to use Lindens/$ as the price in the VWAP formula is that you should use dollars as for the volume part of the equation.

Example (which is close to what happened on Thursday):

7,000,000 Linden sold at 300 L/$ at a cost of $23,333.33
71,000 Linden sold at 71,000 L/$ at a cost of $1

( $1 * 71,000 L/$ + $23,333.33 * 300 L/$ ) / ( $23,333.33 + $1 ) = 303.03 L/$

I'm not sure how using long math words help, but OK, let's try it your way.

What you're suggesting is that we should be interested in the probability that a L$ is traded at some rate expressed in L$/USD.

I think this is completely nonsensical. But I'm not an economist or any sort of trader. Can you point out an stock/currency exchange (not counting LindeX) which observes your interpretation of "average" but not Keiki's? I'm just skeptical that anyone in their right mind would do so.

I think we should be interested in the probability that a L$ is traded for some rate expressed in USD/L$. (or alternatively, the probability that a USD is traded for some rate expressed in L$/USD) Under this interpretation, the formulas that you've posted would be consistent with the definition of "average" that Keiki understands.

<math>Mathematicians describe this transformation of arguments as an isomorphism, where both the mapping from one argument to another, and the inverse are bijective. That is to say that arguments that involve long math words are equivalent to arguments using the same words that normal humans use.</math>

Sorry, if this is annoying anyone, but I feel like I should respond to this example...

This example is very different than the other examples in this thread. It's quite obvious that if you had 100,000 lbs at 300ft and 100,000 lbs at 100,000 ft, that the center of mass would be 50,150 ft. Look at the units in this example: lbs and feet. Neither unit is a rate or price where you are dealing with units/$ or $/unit. You would never even consider inverting the feet in this example to 1/300 ft and 1/100,000 ft. If you did it would greatly skew the results in a way that would be completely illogical.

Looking back on my posts they seem quite repetive. Sorry, if they are, but I don't feel you ever responded satisfactorily to my examples. I guess this has been frustrating because the way that people commonly figure averages is so incredibly simple that I was quite surprised this turned into a debate. Aren't these things mathematically self evident?

1. How do you figure average miles per gallon? Take the total number of miles you have driven and divide by the number of gallons you used.
   
2. How do you figure your average miles per hour? Take the total number of miles you drove and divide it by the number of hours it took.

I see no reason to look at currency exchanges any other way. If the number of lindens on Thursday is 7.35 Million and the number of dollars exchanged is around $24,000, then the average should be around $3.03 L/$. In fact, the formula you posted suggests the exact same thing:Afterall, what would the SUM(Number sold at a given price * that price) equal? In the case of the Lindex, this would equal the amount of dollars spent that day. So average (and this is a "volume-weighted" average) would be ( Total dollars spent on Thursday ) / ( Volume of Lindens Traded ), or if you wanted to express that in lindens/dollar, just flip the formula. Either way you get an average price around 303L/$.

The trade in question on Thursday which caused this huge leap in the average was around 71,000 lindens sold for 71,312 L/$. It was a trade worth $1 USD and 71,000 Lindens. That trade represented less than 1% of the daily volume in Lindens. It represented about 0.00412% of the daily volume in USD. And yet, this one tiny trade, caused the daily average to increase by over 300%. It just doesn't add up. It's not logical at all.

You must not be an economist or financial trader

The issue in this debate is not whether Ricky is using the correct formula to compute an average -- he is using the correct formula to compute an average.

The trouble several of us are having is the fact that Ricky is computing the average exchange rate at which Lindens were sold, while we think the statistic of interest should be the average price.

I would like to know from Ricky, please, the following:

1. Does he agree with my claim that his formula computes the average exchange rate, not the price?

2. If so, does he think the average exchange rate is an interesting statistic?

3. If so, could he rise to Francis Chung's challenge and show us an example in some context other than Linden Lab where it is routinely published and commented on?

2. Or, does he think that the Linden Lab just happens to be computing the average exchange rate, for no good reason, or for no reason.

In this case, it's a matter of semantics. People have been using the terms "exchange rate" and "price" interchangeably.

What Keiki is estimating as the average exchange rate is actually "1/avg(1/x)", which is certainly a measurement of the distribution, but is not a correct way of calculating the average. I steadfastly maintain that my method of calculation is the correct one to use in order to determine the average exchange rate.

And, as for my "balancing weights" example, it is *exactly* the same situation as we have here. The math is formally identical and even the interpretation is consistent -- we have differing amounts of "stuff" placed at various positions, with the equilibrium position given by the integral of the density over all positions.

Actually the 1/avg(1/x) is a recognised process for the harmonic mean - most commonly used by electronic engineers (for resistance in parallel) and formerly used by immunologists and some biochemists when calculating mean dilutions for zero-points.

And would an engineer call the harmonic mean an average?

EDIT: based on prak's link, I'm concluding that harmonic mean is the same as what we commonly understand as average, at least averages of rates.

A better estimate of the price of a steak than $61.90.

In some cases.
http://en.wikipedia.org/wiki/Harmonic_mean

http://en.wikipedia.org/wiki/Average