REVISED 9/11/08

I just recently got an email from my father that showed to me once again that the apple doesn’t fall far from the tree. I guess geekgirl is truly the progeny of geekdad. (He’s a retired physical chemist, BTW). He saw an interesting article in Science about how we calculate fuel efficiency in quite a misleading fashion. We calculate **miles per gallon, **which tells us if we’ve got a gallon of gas in the tank then we can go X number of miles. But, says Richard Larrick of Duke University, this ratio should instead be turned on its head. If we want to go, say, 1000 miles, how many gallons will it take us? **After all, the amount of gas consumed by a car does not decrease linearly as the mpg of the car increases.** My Honda Civic (40 mpg) will take me twice as far on the same amount of gas as, say, a Ford Explorer (20 mpg). And if I want to drive 1000 miles, my Honda Civic will take me there on half the amount of gas that it would take the Explorer. Great, that all makes sense. But now if I start looking at making *improvements* to the mileage of either car, there’s where the “mpg” measure is misleading. It turns out that adding 10 mpg to the low-mileage car will save you a lot more gas and money than a 10 mpg improvement in the high-mileage car.

So he went and graphed it (gotta love geek dad) by just dividing 1000 miles by the miles per gallon and multiplying by $4 per gallon. The 1000 miles part isn’t important for the argument, it just scales up the final answer.

You can see that improving the efficiency of a 10 mpg car to 20 mpg has a much larger effect in the cost (and the # of gallons used) than does improving the efficiency of a 40 mpg car to a 50 mpg car. I’m not feeling so bad about not getting a hybrid car now.

The essential message is that we can’t do calculations like 1/x – 1/y in our heads. In Science Magazine’s podcast, Larrick says:

And, to kind of understand why MPG tricks people it’s useful to do a little bit of math. And so, you could think about a problem that a, a family might face of deciding whether to get rid of an SUV that gets 10 miles per gallon, or a sedan that gets 25 miles per gallon. And let’s say that they’re both driven about the same distance, roughly like 100 miles a week, and with the SUV they need another big car, so they’re thinking about a minivan that gets 20 miles per gallon. And with the sedan they’re thinking about replacing it with, let’s say a hybrid sedan that might get 50 miles per gallon. Well people are very attracted to the idea of replacing a car that gets 25 miles per gallon with one that gets 50 – that’s a big jump of 25 miles per gallon. And, getting rid of the SUV that gets 10 miles per gallon to a car that gets 20 miles per gallon – that just

isn’t as big of a jump, it doesn’t look as impressive….So, let’s just think about how many gallons the SUV uses – the car that gets 10

miles per gallon. So, if we’re driving a hundred miles, that’s going to use 10 gallons to

go the hundred miles. If we replace that with the minivan that gets 20 miles per gallon,

we’re only going to use 5 gallons to drive the same hundred miles – we’ve now saved 5

gallons just by replacing the SUV with the minivan, going from 10 MPGs to 20 MPGs.

Let’s do the same calculation for the other car that could be replaced – which was the

sedan, that does get 25 miles per gallon – replace it with a small hybrid that gets 50 miles

per gallon. Well, at 25 miles per gallon that car’s only using 4 gallons to go a hundred

miles, and the hybrid’s only going to be using 2 gallons to go a hundred miles. That’s

just a 2-gallon savings.The big savings comes from getting rid of the most inefficient

car, the SUV that gets 10 miles per gallon with one that’s more efficient – the one that

gets 20 miles per gallon.

Or, as my dad says, in more abstract language:

The dependence of the cost (or gallons consumed) is not a linear function of mpg with a constant negative slope, rather it is a reciprocal function of mpg with a decreasing negative slope as mpg increases.

The cost is not a linear function of the mpg as most people think and base car buying decisions on. Instead, as the simple calculation shows, it is a curved function. What is clear from the graph is that small gains in mpg for a low mpg vehicle have a relatively large effect on the cost to drive 1000 miles. The effect becomes much less important as the mileage of vehicle improves. Bottom line, you save much more money if you ditch the SUV getting 15 mpg for a car getting 25 mpg (savings $107 for a 10 mpg increase) than if you change from a vehicle getting 25 mpg to one getting 45 mpg (savings only $71 for a 20 mpg increase!). Our dependence on

foreign oil would be greatly reduced if we were to focus on improving the mpg of the very low mileage vehicles on the road.

I also learned recently that European’s rank their vehicles according to how many liters of gas are required to drive 1000 km rather than rating them according to kilometers per liter (or mpg as in the U.S.). The European’s ranking is a more realistic way to compare vehicles and we should adopt it in the U.S..

You can see more on this at the Everyday Scientist, who says:

The real problem with MPG is that the same change in the MPG correspond to a huge change in fuel used at the low MPG end, and almost no change if a car already has a high MPG rating. Going from 20-25 MPG saves more gas that going from 35-50 MPG; going from 12-14 MPG saves more than either. This isn’t intuitive, and you really need to calculate the savings per mile in order to make a rational decision.

The take-away message is that we can’t do calculations like (1/a – 1/b) in our heads.

Richard Larrick tells us more about his research on the topic:

So, our actual research posed a series of questions about if you wanted to replace one vehicle with another one, which change is going to be most beneficial, in terms of reducing – and we, we couched it largely in environmental terms – the gas that’s used and therefore the effect on the environment. And people rate, for example, a change from 42 to 48 as being more beneficial then a change from 16 to 20 miles per gallon. And, without working through that math I hope that it’s obvious now that that 4 MPG improvement on 16 really reduces the amount of gas used quite a bit, and 42 to 48 doesn’t make, it’s still beneficial, but it isn’t nearly as large a change.

Interviewer – Robert Frederick

Right, they’re thinking it as a linear scale.Interviewee – Richard Larrick

Exactly. And it really is a curvilinear relationship where the steep drops, in gallons that are actually used, occur among the MPG in the teens, and it gets flatter and flatter as you approach the kind of high end of what we see now – which is about 50 MPGs. So, the small MPG steps, on inefficient cars, have a big impact on reducing the amount of gas that’s burned.

UPDATE 11/24/08 – I just heard from Rick Larrick, who saw this post and wanted to share his websites with us:

I just ran across your sciencegeekgirl blog. Great stuff.

And thanks for the mention of our Science article. I’m definitely trying to get the word out, and, to be completely honest, really want to see the EPA, consumer reports, or both change to GPM. That has become my mission!

I wanted to let you know about two webpages I’ve been running where I keep updates on the GPM argument:

http://faculty.fuqua.duke.edu/~larrick/bio/Reshighlights.htm

http://www.mpgillusion.blogspot.com/

Thanks again for the mention. Best, Rick

{ 6 comments… read them below or add one }

I found your site on technorati and read a few of your other posts. Keep up the good work. I just added your RSS feed to my Google News Reader. Looking forward to reading more from you down the road!

Firstly, I believe that certain persons (person) spent a couple minutes trying to explain this to you a few weeks back, but that’s neither here nor there.

You ask for some help explaining how to put into words the concept “You save $100 per 1000 miles when you upgrade to a 40 mpg car from a 20 mpg car, but you save twice as much ($200) when you upgrade to a 20 mpg car from a 10 mpg car!” Here goes:

Doubling your MPG will cut your fuel costs in half. This means that the more you spend on fuel now, the more you will save by increasing your MPG.

If you are spending $400 a month in gas, doubling your MPG will cut it to $200, a savings of $200. Obviously, another doubling cannot cut it by another $200, because then you would be paying no money at all (and another doubling would mean you earn $200 a month?)

So, the biggest bang for your buck is in improving the MPG ratings of extremely fuel inefficient cars. As a matter of public policy, the single biggest change we could make would be to prevent car companies from considering SUVs to be trucks for the purposes of fleet fuel economy calculations.

If SUVs had to be included in fleet fuel economy calculations (CAFE standards require a fleetwide average economy of 28 MPG, I believe, but trucks are not included in that calculation), the Big 3 would have to make smaller, lighter, more efficient SUVs the norm, which would, as Geekdad says, make a huge difference in our national petroleum consumption.

This reminds me of a trick question from radioactive decay and half-lives, which I will adapt — you have a car that gets 10 mpg and trade it in for one that gets 20 mpg. (any doubling will do) This saves you X amount of gas for your favorite trip.

If you wanted to upgrade to an even better car, what mileage improvement do you need to again save X amount of gasoline?

You need infinite mileage; as Dave pointed out, you’ve already saved 50% of the gas, so to save that amount again, you have to use no gas whatsoever.

Ooh, Tom, that’s a great example. My *own* difficulty with thinking about mpg versus gpm really highlights how tough it is for us to think about things that don’t behave linearly. I mean, I’m good at math, I can do calculus, got an A in statistics, but this problem keeps twisting my head around. Thanks for the illuminating comment!

First, great site. Lots of good stuff here. Second, related to this issue, I created a “optimal commuting speed calculator” that determines how fast you should drive. If you drive too fast, your efficiency goes down. If you go too slow, you lose money that you could be making from working. So, how fast should you go?

http://www.dotphys.net/page1/page10/efficiency/calculator.html

Hmm, my optimal speed appears to be 105 mph. But that’s only because I value my time so highly. If my time is cheap, then I do find a local minimum at 78 mph. One thing I notice is that if the gas prices double (to $6 per gallon) then it starts to be much more economical to drive slower, because the price of gas becomes more expensive than the price of our time. I heard recently that, considering how many man-hours are saved through running machinery on petroleum products (rather than using our muscle power), either gasoline should be many hundreds of dollars per gallon, or conversely, our time should be valued at pennies per hour. So, when you plug in very high values into this calculator, you start to see more realistic money/time/gas equivalencies. Very nice calculator, Rhett!

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