(Geeky math stuff ahead. You've been warned!)
Get Rich Slowly posted on how to learn to love a house that isn't so big. They found out that they fit better in their first house, which had over 400 sq ft less than their current house.
As additional reading he talked about an NPR article entitled “Behind the Ever-Expanding American Dream House” that was published last July. He includes a chart from the article which compares the average square footage of new single-family homes in 1950, 1970, 1990, and 2004. The numbers — 983, 1500, 2080, and 2349 sq ft, respectively — are printed next to house icons whose sizes, one would reasonably expect, reflect the size of the house in order to reinforce the comparison.
I saw immediately that there was something wrong with the scaling, and JD did too. The 2004 house really appears huge compared to the 1950 house. It has over twice the square footage, but is it really that much bigger? Here's what JD said in the title attribute of that image (which you can read by hovering over the image on his post or by reading the source of the page):
Yes, this graphic is very misleading. The big house isn't 2.5x the size of the small house, but more like 5x the size.
Well, the lengths of the lines used to draw the big house actually are about 2.5 times larger than those used to draw the small house. By my calculations, it's about 2.3 times larger. And the ratio of the square footages of the large house to the small house is 2349 / 983, or 2.39, which is reasonable.
But that's precisely why the chart is misleading. You don't take the ratio of the square footages and enlarge the picture by that ratio. You enlarge it by the square root of this ratio.
Let's take an example. You have two square rugs. One is 100 sq ft and the other is 400 sq ft. The length of the side on the small rug is 10 ft, and the length of the side on the large rug is 20 ft. The ratio of the areas is 4:1 and the ratio of the sides is 2:1. Two is the square root of four. In order to expand the small rug to the same size as the large rug, we need to enlarge it to 200% of its original size, not 400%. Make sense?
But the designers of the graph that was published in the NPR article did just that. The 1970 house icon, instead of being enlarged to 1500 / 983 = 1.53 times the size of the 1950 house icon should have only been enlarged by the square root of that number, or about 1.24 times. Likewise, the 2004 house should only be about 55% bigger than the 1950 house, not 139% bigger.
The picture here shows the original NPR chart on the left, and my poorly-altered but properly-scaled version on the right. Notice that the 2004 doesn't really look quite as gigantic compared to the 1950 house. Or, the 1950 house doesn't look quite so tiny compared to the 2004 house.
Just a reminder that chart graphics can be used to sway the interpretation or enhance the spin of the article.
Ok, you're totally right on the math… but I've got to say that if you step into a 1950's 983 sq. ft. house and then into a 2004 2,349 sq. ft. house it feels like the first graph. Fer reals, dude.
Anything over 2.000 sqft is still large to me for a family of 4 people or less. Having grown up with 4 in my family under a roof that was less than 1500 sqft, it seemed ‘right-sized’ to me.
It figures, you can trust NPR will do things to distort facts to match their view. In this case I'm sure they didn't do it on purpose, this type of analysis is probably beyond their comprehension. Thanks for doing the footwork to correct this. (Of course with square footage the number of levels in the house also plays a factor, I've been surprised by the "size" of some townhomes that have four levels)
I think the NPR graph gets the point across pretty well. Yes, they're not scaled properly, but the fact is that houses are twice as large today as they were barely 50 years ago. Scaling the way you lay out (which I'm not saying is wrong) doesn't convey that message as effectively.
Nice article. I agree with you, but you know that no story is any good unless it is sensationalized! 😉
You still have to be a pig to want such a big house.
But a house is three-dimensional, not two-dimensional like a rug. There is an increase in ceiling heights as well, though perhaps not as large an increase. Still, there's an overall increase in volume as well, not just square footage of houses today. What happens when you add in the numbers for the increase in ceiling height from what used to be 7 or 8 feet to 10 feet or even more now?
Oooh, I understand the math. McMansions do feel a lot bigger. But in my experience they also feel emptier.
We live in an old 2 story house in the historic district – 2300'ish ft. My mother-in-law lives with me and my wife so we can help her with day to day life.
Although it feels a little big at times for the most part a big house comes in handy with the daughters and grandchildren visits. In the back of my mind I do think of selling and going for a smaller home. Only when the time comes and we're alone.
We live in a two story house with 1240 sq feet built in 1860. The downstairs ceilings are 9 feet tall. Everything is relative.
Nice work on the math, though.
Dear Jason, I don't think NPR is as devious as you think it is.
For all readers, I think it's simply a design aesthetic by the graphic designer to make the chart look pretty. Notice how their house are very evenly graduated on the chart on the left. I don't think there was any intent for it to be an accurately scaled representation of anything. I think someone thought a little house would be cute next to the numbers and that's it.
However, I do agree, MBH, you have a point in that it's not a mathematically accurate representation of volume, but it could be. Take for instance this, how many old homes have cathedral ceilings? Your footprint in liveable square footage might be the same, but your volume might skyrocket.
People need to get a grip. There are rowhomes/townhomes in Fells Point, Baltimore that are highly prized pieces of real estate in a flood plain. These homes are at most 18ft wide, but most of them are about 12-15 ft wide. WIDE. They were built hundreds of years ago and they're still very desirable.
Thanks for the comments everyone!
Feirfiz, you're right that houses are 3D, but the measurements are given in floor space, which is 2D. And scaling by the square root increases the area covered by the icons appropriately (pretend that the entire living area of the house is in the bottom floor).
Mapgirl, a chart can look pretty without distorting the facts. You might be interested in reading some of Edward Tufte'e work (edwardtufte.com). He devotes an entire section (chapter?) of one of his books to these kinds of scaling errors.
On the contrary, I do think the designers of the chart tried to scale the sizes appropriately (as I pointed out in the post), but they still got it wrong, and as a result the chart lied. They dilated the length proportionally to the area of the floors inside, and that's not the way to do it. Did they do it on purpose? I don't think so. But when you slap numbers on a figure, the expectation is that any figures accurately reflect those numbers. If the figures don't clarify the data, why are they there?