# Is cubic footage equal to squared footage times average height?

I am reading a book by economist Thomas Schelling, when talking about mathematical identities he says the following:

To say that the cubic footage of housing space in the United States is equal to the square footage multiplied by the mean height of the ceilings , cannot do much more than remind us of the definition of cubic footage.

So this is how I interpret this statement: $$\underbrace{\sum^n_{i=1}l_iw_ih_i}_\text{cubic footage} \equiv \underbrace{(\frac{1}{n}\sum^n_{i=1}h_i)}_\text{mean height}\underbrace{(\sum^n_{i=1}l_iw_i)}_\text{square footage}$$

Where $$(l_i,w_i,h_i)$$ are respectively (length,width,height), of the ith room and n is the total number of rooms. It is not clear to me that this statement should hold by definition. I would appreciate it if someone could show me why it does, or explain to why my interpretation of the statement is wrong.

• The mean height should be weighted by the size of the rooms, not averaged over the number of rooms - a large room will affect the mean ceiling height more than a closet. – Nuclear Wang May 6 at 16:24

That identity is not correct, but it becomes correct if you weight the average by area $$V = \sum_{i=1}^n l_iw_ih_i = \frac{\sum_{i=1}^n (l_iw_i)h_i}{\sum_{i=1}^n l_iw_i}\sum_{i=1}^n l_iw_i = \bar{h}\sum_{i = 1}^n A_i$$

• Thank you for your answer, I didn't expect 'mean' to mean 'weighted average'. – user18214 May 6 at 16:44
• @user18214 All averages are weighted. Your average is weighted by rooms instead of by floor area. Which is correct depends on what sense of "average" captures the meaning you're looking for. – eyeballfrog May 6 at 17:01
• you're right of course – user18214 May 6 at 19:37

As you correctly note, that calculation is wrong. Imagine two rooms, one with area $$1$$ square meter and height $$1$$ meter, the other with area $$100$$ square meters and height $$11$$ meters. Then the volume calculation using the average height is wrong: $$6 \times (100 + 1) \ne 1100 + 1$$.

That said, heights of rooms in residential housing don't vary all that much, so the formula will provide a good enough estimate for economic analyses.

The context for the quotation is missing. Schelling does seem to be criticizing the assertion, though it's not clear why.

• I'm pretty sure you're supposed to weight the average by area. – eyeballfrog May 6 at 16:19
• The average ceiling height in your example is 10.9m, not 6m. You need to weight the average by the room size. – Nuclear Wang May 6 at 16:20
• imagine what happens when house has a triangular roof and no attic... – Vasya May 6 at 16:22
• I interpreted this as an example Schelling gave of something which holds by definition. The problem was my interpretation of the word 'mean', so the equation that I gave misrepresents the statement. – user18214 May 6 at 16:48
• @eyeballfrog Of course weighted by area rather than by room is the right way to rescue Schelling's statement. But it's not at all clear to me that that is what he means to say. – Ethan Bolker May 6 at 18:32