Wed, 29 Apr 2009

Driving The Numbers

Because of my geek nature and technical training, I have an uncontrollable compulsion to check numerical relationships. It makes no difference if the relationship is obvious or subtle, or if it's such a platitude that we all have it memorized as axiomatic fact. For example, if somebody says, "That's six of one, half dozen of the other", I'll compulsively verify in my mind that the division of 12 by 2 equals 6, exactly. Since this was typically offered as a cliche for approximate equivalence, I usually quip "Maybe more like six of one, two pi of the other," hinting that the equivalence may not be precisely exact.

This number checking habit is, no doubt, one of my less endearing qualities. But, despite the fact that I know I'm being a dork, I can't help but do it.

Not very endearing to my mates, perhaps, but definitely useful in practical ways. I find this habit particularly useful when I'm trying to make up my mind on something I'm not an expert in. Yes, I know that there are lies, damn lies, and statistics, and that numbers can just as effectively hide the truth as reveal it. And no, I'm under no illusion that in numeranalysis I'm tapping any sort of foolproof, universal cleaner for stripping away obfuscation to get at the shiny essence on something. It's simply this: when I rope together a collection of unrelated numbers, I feel safer using them to scale the precarious pinnacle of some challenging new idea. If I find that they cooperate well, don't pull in opposite directions too badly, and mostly pull together in coherent, rational ways, I get a warm feeling that I may reach some moderate enlightenment through their aggregate assistance.

I certainly don't want to rely on some "expert" on a TV show to decide a question for me. If I can do a little arithmetic on my own and draw my own conclusions based on facts that I can cross check for myself, then I feel a lot better about my decisions.

That's the thing. Pundits, politicians, advertisers, religious figures regularly splatter out numbers to sell soap, but rarely show any regard for how they fit together. I'm certainly not the first person to notice this. Nor am I the first person to notice that most people listening to those pundits and such, are oblivious to the inconsistency in the math and numbers they are being splattered with.

The general topic of Innumeracy has been widely explored, most notably by Douglas R Hofstadter and John Allen Paulos. My purpose here is not to rehash their observations. Rather, I'd like to share a few case studies of how my mind seems to work – work in a way that appears to be a bit different than what I observe in the math abuse producers, and math oblivious consumers.

Take automobile gas mileage, for example. A friend (also somewhat mathematically astute) once observed that the units of mileage are actually inverse area. Miles – a distance – when divided by gallons – a volume – results in the reciprocal of area. Thus it would be better to say gas peracreage than mileage. This tickled me to no end and I set me off to compute what peracreage my Civic achieves: about 60 billion per acre.

Such quirky (and arguably frivolous) unit conversions hardly reveal any truth, but I've found that a network of conversions and comparisons can be far more important than any given one. Tie enough different ideas together with rigid mathematics and some Fullerean synergy is bound to emerge.

Case in point with gas mileage, I started wondering how 60 billion per acre stood in relation to other related units. For example, my car has this 60 giga/acre fuel economy while traveling 60 miles per hour. Coincidence? Or is this all part of the big oil conspiracy to suppress that legendary 600 billion per acre carburetor?

This is all nonsense, surely, to actively seek coincidences in numbers and expect they tell you anything of value. Numerology books abound, chock full of numeric patterns that imply all sorts of conspiracies. Complete bunk. We are highly evolved pattern recognizers. Finding patters in numbers is a form of masturbation: using a biological function for frivolous pleasure rather than it's intended purpose. All can result is a brief cheap thrill paid for by precious lost time.

This same tendency to gratuitously recognize patterns can be used against us. Take the recent (2009) bank bailout. How many reports conjoined the $150 billion bailout number with the $150 million bonus number. There's no question that this was done intentionally to distort the relative sizes of these numbers. The executive bonuses represented a fraction less than 0.001 of the total bailout. Randall Munroe's great cartoon xkcd commented nicely on this cheat.

Now, I'm not saying that I support, or don't support, the populist rage directed at seemingly egregious executive bonuses at failing, publicly supported companies. I'm merely pointing out that the pitchfork and torch crowd are readily duped by the numbers.

A similar thing happens all the time with whining about "pork barrel spending" and "earmarks". Yes, these are probably bad things, arguably corrupting the appropriations process, but I feel mathematical dizziness when some politician indignantly asks why we need to be spending $315 Million on a bridge to nowhere when we are trillions in debt. People say "it all adds up", but does it really? A quick look at the US budget show spending dominated by mandatory entitlements, defense, and debt service. Bridges to anywhere, let alone nowhere, are so low down in the numerical noise as to have no practical relevance. By the way, when I say "numerical noise" I may lose some readers. There are some people (often accountants) that think there is such a thing as an "exact" number in the real world. Sadly, this is not so. Anything that is measured is subject to error. That means that digits to the left in measured results tend to be more significant, and the digits on the right are less so. The digits of lesser significance are scrambled a bit by measurement errors. That makes these digits "numerical noise" and earmarks tend to be down at this level.

Things that are significant stand out, by definition. We observe that that since WWII ended, federal spending and taxation has sat relatively steady at 20% of GDP with only a few percent deviation. I suppose we can get out a magnifying glass and study the numerical noise to find a tiny ripple in a particular year and try to attribute it to somebody or something, or we can observe subtle, and possibly random, trends over decades, particularly with the relatively significant rise of State and Local spending.

Don't get me wrong. Yes, I agree that budgets should balance and all that, but given all the hyperbolic hot air about "profligate spending" or "taxing us to death" in Washington, I have trouble discerning correspondingly wanton or fatal trends in the actual numbers. I mean, look at the spike in federal spending to 35% of GDP during WWII. Now that is a significant rise in spending.

Speaking of hot air, a lot of the same folks that are horrified by a 0.001% of GDP line item in the federal budget will dismiss as insignificant adding 0.001% to the atmosphere's CO2 concentration by coal power plants and cars. One case is an abomination, the other is a natural right of man. Or visa versa upon changing political aisle. Again, I'm not out to sell soap here – I'm simply grinding numbers. Maybe there are cases where 0.001% really is an abomination.

Regardless of where I sit on these particular questions, I remained compelled to check the numbers. A good way to do that in these cases, I've found, is by numerically analyzing rhetorical analogies. People like to make analogies to help frame an issue and thereby steer the decision about it. For example, global warming deniers will liken mankind's greenhouse gas emissions to a drop in a bucket. This will give them some sort of "evidence" backing up their claim.

The problem, of course, is these analogies are not shored up by the necessary physical analysis. They rely more on emotional connotation than empirical correlation. As such, these abstract analogies don't serve as evidence for gaining knowledge. Given their propensity to generate abstract analogies, I often wonder if some of the ideologues putting forth these analogies even HAVE a criteria for being able to distinguish what they know from what they don't.

But I'm getting ahead of myself. First I'm obliged to ask the following three questions:

1. How much is a drop in a bucket?
2. How much CO2, say, is mankind dumping into the atmosphere relative to the atmosphere.
3. And assuming 1 is relatively similar to 2, are drops in buckets generally easy to dismiss?

I'll take the last question first. To dismiss, or accept, bucket pollution, I think it depends on what you're dropping in the bucket and what you are doing with the resulting mixture. Let's say it was a bucket of milk you just got from the household cow, and you were planning on feeding the milk to your infant child. What sort of drops would you permit in the bucket? A drop of rainwater? A drop of motor oil from the tractor? A drop of snot from the cow's nose? From your nose? A drop of cow pee? A drop of rat poison?

Another factor is who's doing the dropping. One of the objections to industrial dumping of greenhouse gasses is that they are externalizing their carbon disposal costs without universal consent. They are dropping their carbon in the world's bucket without compensating the world. They only compensate customers of their products who pay lower prices as a result of being subsidized by the non-customers accepting externalities.

Thus, if your neighbor dropped something in your bucket, you'd probably be a whole lot more upset than if you dropped something into it yourself.

For the sake of argument, however, let's assume that one drop in one bucket is the standard by which all hazardous pollution should be measured. Two drops, no good. One drop, safe. How much is that.

Both the "drop" and the "bucket" are essentially fuzzy units, but that hasn't stopped people from making them rigorous. Wiki lists several precise definition of "a drop". Since this is a conservative rhetoric supporting analogy, I'll stay away from metric based definitions and only consider the official "US Drop": 1/60 of a teaspoon or 1/360 of a U.S. fluid ounce.

OK, now what's a bucket? Again Wiki knows. A bucket is equivalent to four liquid gallons. There are, of course 128 ounces in a gallon, so a drop in a bucket is 1/(4*128*360) or a little over 5.4 parts per million by volume.

Excellent. A drop in a bucket is 5.4 PPMV. How does this compare to mankind's carbon emissions?

First of all, the concentration of all CO2 currently in the atmosphere is currently about 385 PPMV. Measurements first taken at the Mauna Loa Observatory in Hawaii under the supervision of Charles David Keeling are the longest continuous direct measurement, dating back 50 years to my birth year, 1958, when the measurement was 315 PPMV. Nowadays, lots of sites measure this. It's soon expected to cross 400 PPMV, or 0.04%. At about 10,000 PPMV, people start going to sleep.

Considerable science shows that all this rise has pretty much been in the last hundred years. Global warming deniers dismiss the scientific evidence that CO2 has risen only recently. For some reason, or lack thereof, they believe that this rise is somehow "natural". Maybe they're right, but if six of one is half a dozen, the 60 PPMV rise in my lifetime is not a drop in a bucket – in fact, it's over ten drops in a bucket.

There is, however, an annual, natural variation in CO2 concentration of about 5 PPMV caused by the different amount of landmass in the northern and southern hemispheres, but we can safely ignore that as it's hardly a drop in a bucket.

I'm still curious about one more thing: how many drops of CO2 are added to the atmospheric bucket by man every year?

According to the USGS and several other sources, mankind put 27 billion tons of CO2 into the atmosphere in the year 2004. Unfortunately, that's mass, not volume.

The size of a unit of gas
is not to be measured by mass.
It is measured in moles.
Not the kind that make holes!
Would you please pay attention in class.

Grinding through some additional calculations left as an exercise for the reader, one can show that the mass of atmospheric CO2 is around 3 trillion tons, so man's annual contribution of roughly 30 billion tons is about 1% of the nominal – way, way more than a drop in a bucket.

An interesting coincidence is that the federal budget is about 3 trillion dollars. Would a "wasteful spending" project costing $30 billion be considered a drop in the bucket by the same conservatives that think 30 billion tons of CO2 isn't significant in a 3 trillion ton atmospheric budget? Of course not.

Anyway, back on the subject of automobile peracreage. Let's suppose society feels that the drops of CO2 into the atmospheric bucket are bad drops and decides to act against CO2 pollution by phasing out gasoline powered cars, what replaces gasoline cars? Some think it will be electric cars. Well, I wonder what sort of peracreage does an electric car have?

If the fact that fuel economy for gasoline cars is measured in inverse area tells us anything, it should tell us how strikingly arbitrary the number is. EPA mileage estimates on window stickers are carefully scrutinized by buyers. These critical numbers are weighed alongside other critical numbers, like price and seating capacity. But a 16 per-millimeter-squared perarea rating is an island of meaning. It doesn't connect with other numbers. The sole practical reason for the number in the usual form, miles-per-gallon, is that it makes it relatively easy to figure the cost of driving in dollars per mile, given a fuel price.

If calculating dollars per mile is the only reason for the perarea number, then maybe dollars per mile should be on the sticker. There are some good reasons why this isn't done, but for the sake of argument suppose I saw a table of operating cost estimates on a sticker in terms of dollars per mile.

It's been said that money is power, but if you think about the physical meaning of watts per mile you'll conclude that this can't be correct. It takes some work to move a car from point A to point B. Work is energy. Money must be energy. A gallon of gasoline is purchased for its energy content. You can get more or less power from that same four bucks worth of gas (2008 price) depending on how fast you burn it, but you pretty much get the same energy.

In fact, the heat of combustion of a gallon of gasoline is a measure of the fuel's energy capacity the way it's normally used. In terms of mass, gasoline contains 48,000 joules per gram, or 11.5 kilocalories per gram, roughly twice the energy, per gram, that's in peanut butter, and just about 20% more than you'd find in french-fry oil.

But gasoline is sold by the gallon, which makes its energy sold vary significantly with temperature. At room temperature, you get about 36 kw-hrs of energy from a gallon. And on a good day my Civic can go 36 miles at comfortable highway speeds on that same gallon. Finally! This a memorable, pattern-recognizer masturbatory coincidence we can use to gain insight: One kilowatt-hour per mile.

Now, physics and engineering nit pickers will no doubt point out that an auto engine is inefficient and that much of that kw-hr is wasted in heat. It's certainly something less than the full kilowatt in an hour powering my Civic, which covers the mile in a minute. Maybe it's only 746 watt-hr (1 horsepower-hr) that actually makes it to the wheels at a rate of 60 horsepower. Maybe it's something less. Telsa Motors claims that their roadster uses about 180 watt-hr per mile. I find it hard to believe that my Civic, after 40 years of optimization, is that inefficient relative to the upstart Tesla. Tesla is probably exaggerating downward, if we take them at their word, it bounds the range of possibility somewhere between the memorable kilowatt-hr per mile down to 180 watt-hr per mile. On my bike I can ride twenty miles in an hour using 250 watt-hr of energy, which is 215 kcal, or an ounce and a half of peanut butter.

In any case, a raw kw-hr costs $0.11 if bought from my utility that burns coal mined nearby to make cheap electricity. My Civic, running on $4/gal gas and needing a kw-hr to go each of the 36 miles it can go on that gallon also costs me $0.11 per mile. That makes $4/gal gas a sort-of critical number. Below that price, it's cheaper to burn gas than electricity to drive cars. Above that price, electricity is better. If you can argue that electric cars can be more efficient, with regenerative breaking, continuously variable transmissions, etc..., then maybe the crossover price is lower – say $3/gal.

Since gas has cost more than $3-$4/gal in recent days, making electricity quite competitive, why don't we see a bunch of electric cars? If the numbers say that we could save money driving electrics, why aren't there electrics? Why hasn't some enterprising company latched onto this fact and gone to market. After all, electric cars were common 100 years ago, it's not like they are some sort of new technology. Is there a great conspiracy on the part of oil companies? Are auto makers suppressing the technology so as to milk profits out of gas cars? Do consumers simply not like the idea of an electric car? What's the deal?

One fact that can shed light on this is battery capacity. Again, it's fun to do the numbers based on what I can find directly. I can buy NiMH batteries as AA size cells that have about 2.4 w-hr energy capacity and weigh about an ounce. Lead-acid cells have less than 1 w-hr per ounce capacity. Some of the more modern lithium chemistries can achieve 5 w-hr per ounce. So, we're talking somewhere between 1-5 watt-hours per ounce for practical batteries.

Gasoline contains about 375 watt hours per ounce. Even peanut butter has almost 200 watt hours per ounce!

Now, remember the 1 kw-hr per mile number? Without even a calculator we can estimate 200-1000 ounces of battery are required to go a mile. That's 12.5 lbs per mile, at best. The 300 mile range most of us like to see in cars requires almost two tons of batteries! At best! Yes, the Tesla folks claim they can do better than this. I hope they can, but still I wonder how to avoid at least one ton of batteries.

After all, a ton of batteries is hardly a drop in the bucket.

Posted Apr 29, 2009 at 20:37 UTC, 3274 words,  [/dan]  Permalink