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.


  1. The big question in my mind is geothermal. It's the most attractive answer I can think of. Don't have to burn food, don't have to worry about another Cherynobyl, don't have to be dependent on foreign oil. As far as I can tell, the biggest obstacle is the initial capital investment, but the market doesn't tend to factor in things like the cost of war. If we put our best minds on the most efficient means of tapping the energy embedded in all the molten magma under our feet and spent $500B on that, I wonder how far we could go.

  2. I just want to point out that the harmonic mean of X and Y is defined as 2/(1/X+1/Y), not 1/(1/X+1/Y). Fortunately, it looks like this was just a typo and you used the right formula in calculations.

    More generally, the "n-th power mean" of two numbers X and Y is defined as ((X^n+Y^n)/2)^(1/n). n=1 corresponds to the Harmonic mean.

  3. Oops: n=-1 is the harmonic mean. n=1 is the regular arithmetic mean.

    There's a beautiful inequality saying that if n is less than m then the nth power mean is less than the mth power mean.

    For n=0 the definition doesn't make sense, but it can be shown that the nth power mean converges to the geometric mean (square root of X times Y) as n goes to 0.

    This all can be generalized to arbitrarily many nonnegative quantities, with the same inequalities holding among their power means.

    Way to go me, turning a topic of important social interest into abstract mathematical curiosa!