# 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?!? • Do you need to give a techincal explanation? This phenomenon is an instance of the Simpson's Paradox. Perhaps you will find this and/or this and/or this helpful. – an4s Feb 24 '20 at 17:45 • I would agree with you. The problem with the administrator's approach is that each square foot in the five properties do not count for the same percentage of their overall average. This link talks about a similar example regarding the height of$4\$th graders in different classrooms. Now if each property were to have the same number of square feet, then the average of averages would end up equaling the total average. Feb 24 '20 at 17:48

If I understand your question correctly, you are right: averaging the averages is almost always a terrible thing to do.

It will probably help to show a simple extreme example. Suppose I first "sell" $$9,999$$ square feet for $$\0$$ total, and then sell one remaining square foot for $$\10$$. In total, I've sold $$10,000$$ square feet and spent $$\10$$, but the average of the averages would suggest that I've been spending an average of $$\5$$ per square foot (since $${0+10\over 2}=5$$).

It should be obvious that this is silly - especially since we could replace $$9,999$$ with literally any other number and get the same "answer" when taking the average of averages!

• And the term "weighted average" can now be fruitfully introduced: the average of the whole aggregate can correctly be computed by taking the weighted average of the averages, where the weights are just the respective number of square feet involved in each case.

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.