Explain Why This Average of Averages Is Incorrect Five properties are sold at varying prices and number of square feet.
SQ FT    PRICE       Price/Sq Ft
1635     $630000     $385.32
2045     $675000     $330.07
1900     $685000     $360.53
2045     $700000     $342.30
2305     $715000     $310.20
====    ========     =======
9930    $3045000     $345.68

The report shows the Average Price per Square Foot as \$345.68. Which I think is incorrect. I think it should be \$3045000/9930 = \$342.90 per sq ft.
How do I explain to the administrators responsible for these reports that they cannot use an average of the averages to give the Average Price per Square Foot of these homes. They must total the square footage and total the prices and then divide the total price by the total square footage to determine the actual average price per square foot, right?!?

The word "average" here is ambiguous.  They are using it to mean the average across properties, while you are using it to mean average across square feet.  To see that these are different, use a more exaggerated example. Suppose we have two properties:
SQ FT    PRICE       Price/Sq Ft
10000    $1000000    $100
  100          $2    $.02
====    ========     =======
10100    $1000002    $50.01

If you are talking about the "average" price per square foot across these properties, the answer is $\$50.01$. This is the expected value of how much  you will pay per square foot if you choose one of the two properties at random to buy. But this is misleading because if you buy both properties, the vast majority of the property you are buying will be at $\$100/$sq foot so the average square foot of land is worth just a bit less than $\$100$: it comes to $ \$1000002 / 10100 = \$99.01$.  But I would not say their answer is wrong - it just has a different meaning, which might be easily misinterpreted.
