Matt Yglesias writes that he “hear[s]” the Chilean earthquake is 1000x more powerful than the Haitian one. I get the feeling that a lot of people know that the Richter scale is logarithmic, but it’s not clear that they always know how to convert that back into raw units. The estimable Mr. Yglesias, for instance, shouldn’t need to “hear” that it’s 1000x more powerful; he should be able to figure it out on his own. (I get similarly vexed when people can’t compute tips at restaurants on their own.)

The USGS pages on the Chilean earthquake and the Haitian one mention their magnitudes (8.8 and 7.0, respectively) and give a helpful explanation of what that means:

> Seismologists indicate the size of an earthquake in units of magnitude. There are many different ways that magnitude is measured from seismograms because each method only works over a limited range of magnitudes and with different types of seismometers. Some methods are based on body waves (which travel deep within the structure of the earth), some based on surface waves (which primarily travel along the uppermost layers of the earth), and some based on completely different methodologies. However, all of the methods are designed to agree well over the range of magnitudes where they are reliable.
>
> Preliminary magnitudes based on incomplete but available data are sometimes estimated and reported. For example, the Tsumani Centers will calculate a preliminary magnitude and location for an event as soon as sufficient data is available to make an estimate. In this case, time is of the essence in order to broadcast a warning if tsunami waves are likely to be generated by the event. Such preliminary magnitudes, which may be off by one-half magnitude unit or more, are sufficient for the purpose at hand, and are superseded by more exact estimates of magnitude as more data become available.
>
> Earthquake magnitude is a logarithmic measure of earthquake size. In simple terms, this means that at the same distance from the earthquake, the shaking will be 10 times as large during a magnitude 5 earthquake as during a magnitude 4 earthquake. The total amount of energy released by the earthquake, however, goes up by a factor of 32.

So then the amount of shaking in a magnitude-7.0 earthquake is 10⁷, which is 10 million. A magnitude-8.8 earthquake will feature 10^1.8 times as much shaking as the magnitude-7.0 one. 10^1.8 is less than 10², which is 100. So the amount of shaking is nowhere near the 1000x that Mr. Yglesias heard.

But then the USGS also notes that the amount of energy goes up by a factor of 32 for every 1-unit increase in the Richter scale. So then there’s 32^1.8, or 512x, as much energy in a magnitude-8.8 earthquake as in a magnitude-7 one.

__P.S.__: I found an Ezra Klein piece that I was looking for before, where he suggests that he also doesn’t know what “logarithmic scale” means:

> The devastation in Haiti was not just because the earth shook, and hard. The quake there was 7.0. Harder than the 6.5 quake that hit Northern California a day before (remember, though, that the Richter scale is logarithmic, so 7 is many times harder than 6.5)

If we’re talking about the magnitude of the shaking, the Haiti quake was 10^.5 times as strong as the California one. You may remember that “x to the 0.5 power” is the same as “the square root of x.” To get your back-of-the-envelope-math muscles working, recall that “the square root of x” means “the number which, when squared, equals x.” The square of 3 is less than 10, and the square of 4 is more than 10, so the Haiti quake shook things somewhere between 3 and 4 times as hard as the California quake. As measured by raw power, Haiti’s quake was 32^0.5 times as powerful as California’s, meaning somewhere between 5 and 6 times as powerful.

“Many” has no exact definition, of course, but I doubt most people would say that “many times harder” means “between 3x and 6x as hard.”

Looking at the chart on Andy Gelman’s post about health-care expenses and outcomes, I wonder if there’s any way to put all of those data points in an order. You want to say that country A is better than country B in its health-care outcomes and expenses, and you want to be able to do that for all countries.

There’s an obvious *partial* ordering for all those countries: A’s health care is better than B’s if A’s health-care outcomes are better than B’s and if A spends less on health care. That is, if A is to the left of and above B, then A is better than B. But we’re unlikely to be so lucky that countries can be put into a line that slopes uniformly down and to the right.

If there were some widely accepted way to balance expenses and outcomes, then we could achieve a total ordering here. Let’s say, for instance, that we defined the “goodness” of a health-care system as 1/3 times its per-capita price, plus 2/3 times its health outcome. Then our two-dimensional chart would collapse into a one-dimensional line, and all countries would naturally be totally ordered. But unless I’m missing something, there’s no objective criterion for combining these two quantities.

What I’m asking, I think, mathematically, is whether there’s any natural total order on ordered pairs. Probably not, right?

__P.S.__: I wonder whether the ratio of quality to price has any claim to objectivity. One would expect, though, that the marginal gain in quality for every marginal dollar spent would decrease with the quantity of dollars. (Diminishing returns.) So if we’re not careful with this ratio, it will tend to reward those countries that spend hardly any money and have mediocre health outcomes. So I wonder whether the ratio of quality to price, limited to the set of countries with quality above a minimum threshold, would be an interesting metric. This does, however, start to get us into “how much money is an additional year of life worth?” territory, which is ethically contentious.

This particular ratio, too, depends on some possibly special features of the response function (i.e., the response of quality to increased cost). In particular, the response function probably has a positive first derivative (every extra dollar buys you *some* increase in quality) and a negative second derivative (…but the amount of extra quality attained for every dollar is decreasing). This is somewhat specialized, but decreasing returns of this sort are fairly common.

__P.P.S.__: Even without this specialization, it seems fair to say that country A’s health index is less than country B’s if they spend the same amount of money but A has lower quality.

I seem to be running into topics of conversation that return to mathematical logic in some form or another a lot lately. E.g., Adam Rosi-Kessel and I got to talking about [book: Gödel, Escher, Bach]-type topics recently, namely the connection — if there is one — between consciousness (whatever that is) and self-reference in formal systems. Then there was this blog post today about programs that can print themselves and other topics.

I need to learn me some mathematical logic already, extending (let’s say) all the way from propositional logic through predicate calculus, up to Gödel’s theorem. Anyone have any recommended readings here?