the harmonic mean (MPG standards)

One of the bits of math that's interesting to me is the "harmonic mean". It says that in things like fuel efficiency for cars, you often pay for the worst MPG vehicle, and it doesn't really matter how efficient you make the best one.

For a simple example, we'll consider a hypothetical Range Rover (10MPG) vs. a Prius (60MPG). You might think that a Prius "cancels out" a Range Rover in a sort of fuel-consumption way. But that's not true.

For instance, you compute a "simple average" like this: 60MPG + 10MPG / 2 = 35MPG. But this math is simply wrong. In fact, you have to use the harmonic mean (average the numbers in "gallons per mile") because you really care about gallons used, not miles/gallons.

Using the harmonic mean for this? 1/(1/60 + 1/10) = 17MPG on average.

How many Priuses does it take to "cancel out" one Range Rover (achieving the 35MPG above?)


It takes six Priuses to cancel out a Range Rover.

What's important here: improving the worst vehicles on the road is more important than making the best a little bit better.

The Hybrid Chevy Tahoe is much more important in the overall scheme of things than a plugin hybrid. Improving from 60MPG to 100MPG is diminishing returns until you get the 10-20MPG cars off the road. This includes old models! Improving from 10 to 30? Really significant.

Luckily the CAFE standards take the harmonic mean into account. But understanding this math explains quite well why "fleet averages" move so slowly upwards, even as technology gives us great ways to make the "best" MPG really quite good.

It also explains why our consumption levels are still going up. It will take years and years after the CAFE standards for new cars have improved for our actual consumption to decrease.

Practical ways to "reduce our dependence on foreign oil" should take a hard look at the worst vehicles on the road today, and not always try to optimize the best for slightly better gains.



Dec. 19 2007: The Energy Independence and Security Act of 2007 was signed into law. For the original text, go to p83 here:

This law progressively outlaws low-efficiency lighting (e.g., most incandescent bulbs) for "general service" lighting (310-2600 lumens, or approximately what you'd call a 30-150W incandescent bulb in a normal socket.)

There are special purpose bulbs that are exempted, but the bill appears to try its best to prevent exemptions. Notably excluded are 3-way incandescent lamps and reflectors, but there's quite a lot of provisions: if the sale of one of the types doubles, they'll make an effort to outlaw it, fast.

I have some concerns and comments:
  1. This law would be appear to me much better with a progressive tax on energy-expensive sorts of lighting, so people can gradually migrate to new form factors. I tend to dislike "prohibit the sale of X by date Y." There are always exceptions not granted by a law like this, and an economic solution would be much more appropriate. This is heavy-handed.
  2. It's not clear that the technology is appropriate to the problem (and let's hope the intervening years help it somewhat). For example, the bill specifies a minimum of 80CRI. This is rather insufficient for lighting high-quality artwork and photography (where 90+CRI is preferable). While high-CRI fixtures like these are available for purchase, it isn't clear to me that they will be available to the average consumer. (Incandescents are 100 CRI, which means that they represent all colors well, not just some.)
  3. Directional applications are not well covered by CFLs, as they do not have both compact and high-output configurations. A powerful CFL is a very large CFL.
  4. Low-intensity dimming is also not well covered by low-energy light sources today, and the warm candle-lit 1800 Kelvin output of an incandescent when dimmed may be replaced by the sort of sickly flicker that CFL's give at the same intensity. Dimming generally is not a strength of CFLs.

At home, we have replaced most of our "general" lighting in the house with CFLs. But it's the exceptions that make the place feel like home. We have very warm lights (yes, incandescents) in the bedroom that make you feel like sleeping. There are special lights for displaying artwork, and there are dimmers.

As I recall, most energy expenditure and CO2 emissions today come from heating and cooling (primarily heating), not from household lighting.

I've even been having the following thought this past few months: is it possible that the "waste" heat from Halogens and Incandescents actually cancel out lower heating bills, when used in cool climates?

It's nice to see some movement in this direction, but I wish in many ways that it weren't so heavy-handed. There are many market forces that would make this happen regardless: price, cost, etc. It makes so much more sense to me to find an economic solution.

I don't really want the "Attorney General" involved in "prohibiting the sale" of certain devices: they're not munitions and not imminently dangerous. The problem is solvable by great technology and prices in the market. The exceptions in life make it enjoyable, and I hope this bill spurs research in lighting to develop new solutions to many of these problems.


anniversary gifts

Lorna and I have been married for a year. That means (if you look it up) that we've just had our "paper" anniversary (meaning your anniversary gift should relate to paper). Technically we're allowed to substitute "clocks" if we're modern sorts of people.

While there were other gifts that I'm not going to go into, I thought it was worth sharing the paper one.

Over the past year we've had our honeymoon slashdotted and we've been on TV for not buying a car (because they wanted fingerprints first). So apparently we have some sort of privacy theme happening, at least in public.

So we bought ourselves a lovely 15-page micro cut paper shredder. (Frank Abagnale's picture is even on the box, promising that your identity won't be stolen as often if you use it.)

We had a really fun time cleaning the house of all those credit card offers, too.