Average of Ratios vs. Ratio of Averages

Date: 10/24/2003 at 13:20:55
From: Laure
Subject: Average of ratios vs. ratio of averages

I write and maintain software for real estate agents, and we include a 
calculation called, "Average dollars per square foot."  We currently 
calculate this as the ratio of the average price divided by the
average square footage of all the homes in the list.  It seems to me
that it should be calculated as the average of the
price-per-square-foot ratio of each house. Can you think of any reason
why the ratio of the averages would be more useful than the average of
the ratios?  Is there a technical name for this ratio of averages?

Date: 10/24/2003 at 16:29:22
From: Doctor Peterson
Subject: Re: Average of ratios vs. ratio of averages

Hi, Laure.

Interesting question!

What you are currently calculating actually does make sense; you are 
just averaging over all the square feet of houses, rather than over 
all houses, and that may be just the right thing to do -- or it might 
not.  Here's what I mean:

Suppose that N houses are sold; the sum of all their prices is P, and 
the sum of all their areas is A.  (That is, if the individual prices 
are P1, P2, ..., Pn, and the individual areas are A1, A2, ..., An, 
then P is the sum of P1 through Pn, and A is the sum of A1 through 
An.)

Then the average price of a house is P/N, and the average area of a 
house is A/N; and you are calculating

  P/N
  --- = P/A
  A/N

as the average price per square foot.  And that is exactly what it is: 
the total price of all those square feet, divided by the number of 
square feet.

What you envision is

  P1/A1 + P2/A2 + ... + Pn/An
  ---------------------------
                N

which would average the price per square foot of all the houses.  This 
puts the focus on the individual houses, rather than the individual 
square feet.  How would this be different?

Well, let's take a simple case with N = 2.  Suppose we have a big,
well-built house of 10,000 square feet, and that it costs $2,000,000
($200 per square foot), and a little house of 1,000 square feet that
costs $20,000 ($20 per square foot). Then the total cost of the houses
P is $2,020,000, and the total area A is 11,000 square feet. The
average price per square foot is

  P/A = 2,020,000/11,000 = 183.6

(closer to the more expensive price) while the average of the two 
price-per-square-foot numbers is

  average(Pn/An) = (200 + 20)/2 = 110

(which is considerably lower).  What pulled the first number up is the 
fact that the bigger house had the bigger price per square foot; 
since we counted each square foot equally, the numerous high-cost 
ones won.  The second calculation treats all 10,000 of the expensive 
square feet equally with the mere 1,000 square feet of the little 
house, so the little house pulled the average down.

Both numbers are meaningful.  The first fully deserves the name you 
are giving it (though there is definitely some ambiguity in the 
English!); but the second may better reflect what the average 
homeowner (as opposed to the "average square foot of floor space") 
can expect.  Call it, perhaps, the "cost per square foot of the 
average house", where the number you are currently calculating is the 
"average cost of a square foot" or "cost of an average square foot".

So, again, both numbers can reasonably be called "average cost per 
square foot"; which is more useful to you depends on how you want to 
use it.  Do you want a number that is pulled up by big fancy houses, 
or one that shows what the average house is worth?  Or would separate 
numbers for different categories of houses make more sense?  Perhaps 
you can gather data that shows how costs per square foot are
distributed, and how each average reflects that.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/24/2003 at 19:35:45
From: Laure
Subject: Thank you (Average of ratios vs. ratio of averages)

Thank you for the quick response!  I feel a little better about
leaving our calculation as-is now.