Ben Orlin writes the delightful Math With Bad Drawings blog. He recently posted examples of “What Headlines Would Look Like if We Lived in a Mathematically Literate World,” which I recommend highly.
1. In one instance, Orlin makes a point about regression.
Our World:Mathematically Literate World: Market Rebounds After Regression to the Mean
To a very good first approximation, stock prices are a random walk. There is some momentum and some mean reversion at different time scales, but both are subtle effects that can be teased out only by careful analysis of large amounts of data. They are useless for explaining single events. They are also attractive analogies that lead to vast investment errors. Therefore the best headline usually would be two different ones attached to unrelated articles, “Market Went Up Yesterday,” and “Assurances from Fed Chair.” On the Math With Bad Drawings blog, the rewritten headline has been changed to “Market Rebounds Without Clear Casual Explanation,” which is halfway to my version.
On a geekier note, “regression” and “reversion” to the mean are essentially different concepts. Regression is a statistical phenomenon. If yesterday was an exceptionally good day for the stock market, today is likely to be less good than yesterday. Reversion is stronger and occurs (if it does) for economic reasons, not statistical. If yesterday was a good day for the stock market, today is more likely than average to be a bad day. The usual mistake is to observe regression and assume reversion. Orlin does the reverse, positing reversion but labeling it regression.
Our World: Controversial Program Would Cost $50 Million in Taxpayer Money
Mathematically Literate World:The question here is getting the denominator right. A dollar figure is meaningful for a one-time project whose scope is easily contemplated, say building a bridge. But when discussing a proposal to improve job training programs, the cost is only meaningful in relation to the number of people helped. I think Orlin has in mind macro proposals that affect the entire budget, for which dollar figures should be related to total government spending over the same period.
For example, someone might suggest tinkering with the cost-of-living adjustment methodology in federal programs, and add up the total dollars saved over the entire budget over the next 60 years to make savings look big. Reporting this number as if it were a one-time expenditure for a bridge is silly. But just as often, someone takes a real cash expenditure and divides it by some absurdly large number to make it look small. In my experience, non-quants are apt to err by focusing too much on dollars, while the quant error is to ignore hard dollar figures in favor of model-dependent denominators.
Our World: Proposal Would Tax $250,000-Earners at 40%
Mathematically Literate World:The main point is that tax reporting seldom distinguishes properly between marginal and average tax rates. However I think the specific example is not well-chosen. The rewritten headline applies strictly only to someone making exactly $250,001. In fact, the average taxable income of people earning over $250,000 is over $500,000, so the 40% actually applies to more than half of the income earned by this group (the top 2.3% of earners).
The difference between marginal and average rates is more significant for middle-income households. Their marginal federal income tax rate is 25%, but their average federal income tax paid is 5% of taxable income. If you add in payroll taxes, the figures rise to 33% and 11% respectively.
, using taxable income to decide who is rich and who is not is misleading. A person can have low income because he's finishing up at Harvard Business School with a $250,000/year offer for next year in his pocket, or she's a retired billionaire with a lot of municipal bonds and good tax lawyers. A person can have a high income because he just sold his business he spent a lifetime building, or signed a one-year contract in the National Football League.
Our World: Market Share for Electric Cars Triples
Mathematically Literate World:Growth rates are important for things likely to maintain reasonably constant growth rates for an appreciable period, or in other words, for exponential things. When AIDS cases increased 385% in the US from 1981 to 1982, it was an event worthy of a scare headline. It would have been irresponsible to report, “AIDS Cases Rise to 0.0003% of the US Population.” On the other hand, there was a 23% increase in fatal dog bites in the US from 2011 to 2012, but the relevant fact is that the number went from 31 to 38. Change “dog” to “vampire” or “zombie” and we're back to something exponential.
5. In the next case, issues of definition are key:
Our World:Mathematically Literate World: Still 90% Scientific Consensus on Global Warming
It's more complicated when you ask questions like, “What is the probability that the 50 years from 2014-2063 will be warmer than 2013 temperature levels?” or, “Are human behavior and human impacts on the environment understood well enough to say with confidence that policies attempting to reduce carbon emissions will lead to a better overall climate for the next 100 years?” For these, you get different results depending on how you define scientists and how you gather the data. Orlin now regrets including this example due to the number of complaints he got from scientists.
Similarly, “consensus” can be defined for straightforward questions like, “Does the earth revolve around the sun?” But in a field with lots of complex, nuanced opinions of varying degrees of confidence, you cannot easily define the common denominator beliefs, and you certainly cannot measure them with surveys.
6. In our final example, the emphasis is on using reliable models.
Our World:Mathematically Literate World: Economist: Eliminate Minimum Wage, then Pray Our Model Has Some Basis in Reality
First of all, economists who are free to speak for themselves and who are not pure ideologues, generally have a rational model that is not contradicted by empirical data that supports their views. That is, their positions are self-consistent and mathematically possible. Yes, that is a very low standard, but it's higher than most non-economists. “Politician: Eliminate Minimum Wage, Consequences Not My Interest or Problem,” is a fairer criticism.
Third, you don't need a model to argue for the obvious first-order effects of a change. If we passed a law that no car could be sold for less than $20,000, we would expect car sales to go down. Some low-quality and used cars would become unsalable, and some potential car buyers would be priced out of the market. Fewer cars would be produced, because one segment of the purchaser market would go away, and cars would be less desirable without the ability to resell them. The economy would suffer.
Now people aren't cars. A $10,000 car can't go back to school to become a $30,000 car. Cars don't spend their purchase prices on rent and food. Cars don't run better if owners pay more to buy them. Therefore it's possible to argue that second-order effects of minimum wage laws increase total wages paid (a small or zero decline in employment combined with increased average wages) and help the economy. But you need a model and empirical evidence to do it. If your policy is adopted, you then have to pray that your model has some basis in reality, because otherwise you will have caused a lot of harm from the first-order effects.
I encourage you to read the original article, as well as other posts from the Math With Bad Drawings blog. I claim the drawings aren't bad; they are playfully minimal with striking composition, expression, and energy, although they lack realism and detail. Orlin insists they are actually bad, since he cannot draw anything more complex.
No positions in stocks mentioned.
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