Last week, we discussed the problems with having poor reading comprehension and the impact that has on consuming news. This week, I want to look at the lack of math skills.

America seems to becoming a dangerously innumerate society. Innumeracy is incompetence with numbers rather than words. This is a worrisome issue for the future competitiveness of the U.S.

I first encountered the word in a 2001 book, “Innumeracy: Mathematical Illiteracy and Its Consequences,” by Temple University math professor John Allen Paulos.

This has been an issue for quite a while, but it blossomed into view again earlier this summer in a New York Times magazine article, “Why Do Americans Stink at Math?” The deficiencies outlined are striking:

A 2012 study comparing 16-to-65-year-olds in 20 countries found that Americans rank in the bottom five in numeracy. On a scale of 1 to 5, 29 percent of them scored at Level 1 or below, meaning they could do basic arithmetic but not computations requiring two or more steps. One study that examined medical prescriptions gone awry found that 17 percent of errors were caused by math mistakes on the part of doctors or pharmacists. A survey found that three-quarters of doctors inaccurately estimated the rates of death and major complications associated with common medical procedures, even in their own specialty areas.

It is more than anecdotal: Fewer and fewer people are familiar with even the most rudimentary mathematics. People are too easily confused by simple figures. My favorite example is how many people believe that a 100 billion is more than 10 trillion (because, you know, 100 is bigger than 10).  Continues here



Category: Apprenticed Investor, Bad Math, Data Analysis, Really, really bad calls

Please use the comments to demonstrate your own ignorance, unfamiliarity with empirical data and lack of respect for scientific knowledge. Be sure to create straw men and argue against things I have neither said nor implied. If you could repeat previously discredited memes or steer the conversation into irrelevant, off topic discussions, it would be appreciated. Lastly, kindly forgo all civility in your discourse . . . you are, after all, anonymous.

31 Responses to “Got Math? Odds Are, You Don’t Understand Probabilities”

  1. > Got Math? Odds Are, You Don’t Understand Probabilities

    Probabilities? Most people struggle with basic percentages, or, say, the effect of compound interest rates in the calculation of credit card debt, etc, etc. Just last week a writer for VOX posted an article that contained some egregious math analysis. He went to Harvard . . .

    The fault here is with our education system IMO . . . by the time our daughters left high school they were totally turned off on math because the way it was taught.

    • MikeG says:

      Most people struggle with basic percentages

      The one I see frequently is the difference between a percentage increase and decrease, e.g. references to a “300% decrease” in some statistic.

  2. wally says:

    I disagree with your take on the “bup stop” error. Since there are simple ways to prevent that mistake (such as: proofread before painting; key the stencils so they only fit together one way, etc…) to not have taken one of those steps is incompetent.

  3. bigsteve says:

    I work in a technical field and use a variety of math in my work including statistics. I too have notice many people including some who hold technical degrees who are troubled by math problems. It is one thing to plug and chug and another to convert a real world problem into a mathematical equation that you can then solve. It also must mirror the real world. To be fair I really have a hard time empathizing with people math challenged as I am fairly talented. My grandfather was a child prodigy who was a professor at age 16. He was a genius at math. Most of his descendents inherited his gift in some degree. So you have to take the world as it is. Math talent is like art or athletic ability mainly inherited by a lucky few.

    • ottnott says:

      “Math talent is like art or athletic ability mainly inherited by a lucky few.”

      That is only true for ability in the elite range. The math sense Ritholtz is talking about doesn’t require any special gift. It is easily in reach of an average student and for many more students with some additional effort.

  4. VennData says:

    A story problem:

    Tax cuts for the rich means YOUR share of the debt goes up.

    Tax cuts for the rich have NEVER lowered the debt and deficit as Reagan, Bush claimed.

    The GOP claimed Clinton’s and Obama’s tax hikes on the rich would wreck the econony and didn’t.

    Who should you vote for?

    • wrongtrade says:

      Democrats (and, lately, Republicans too) will spend every penny and then some, no matter how high my marginal tax rate is.
      Is there a point where I will work less because I am not keeping enough of what I earn? Damn right- we’re there now. Is there a tax cut that will induce people who won’t work into working? I don’t think so, because, among other reasons like laziness and stupidity and entitlement, there are wealth transfers built in to our tax code.

      • Petey Wheatstraw says:

        So — guarantee everybody a job — it’s not like we don’t print cash, on demand.

        YOU have to pay off the deficits, which are actually deferred taxes. Now, get back on the treadmill — the squillionaires are getting worried.

      • Anonymous Jones says:

        Oh no, are you going to stop working because your marginal rate is too high?

        Let me call my broker. I don’t know what to do. Society is going to collapse.

        The graveyards are full of indispensable men.

      • Low Budget Dave says:

        Wrongtrade: There are several things wrong with this, so let’s just hit the high points:

        1. Recently, Republicans are much bigger spenders than Democrats. The reverse is true only at the state and local level, (and only for certain states and localities). If you are worried about who spends the most Federally, it is Republicans.

        2. It sometimes appears the opposite, because Democrats support some expensive programs, like Food stamps, Veterans benefits, Social Security and Medicare. In terms of actual recent increases, these pale into insignificance compared to war and tax cuts.

        3. Taxes, by definition, involve a certain amount of income redistribution, but not necessarily a form of wealth transfer. As a society, we tax “rich” people at higher rates, because a 20% tax on someone making $100 a week actually cuts into their food money. A similar tax on Paris Hilton does not. Also, high taxes on poor people don’t generate much money. It was not designed to punish rich people, it was designed to keep poor people from starving to death in the streets. The driving force for progressive taxes, like food stamps, was “Christianity”.

        The largest wealth transfer in the history of mankind is the “upward” transfer from the American taxpayers to bankers. This is not entirely the fault of the tax code, but much of it is. Oddly, the people who pushed the upward transfer also claimed to be Christians.

        4. If you want people to work, rather than rely on government programs, all you have to do is increase the minimum wage. Each dollar increase in the minimum wage removes about a million people from various government benefits, (more than a million, if you count unemployment insurance).

        5. If you are making less than about $320K per year, then your taxes went down under Obama, not up. If you are making more than $320K per year, then it is absurd to suggest that you are going to cut back on work because you only get to keep 80 cents out of each additional dollar.

        That would be (literally) like saying: “If someone walks into my office and offers me $153 to read his poem, I would do it. But if he offers me $122.40, I would refuse.”

    • victor says:

      Slight correction: Obama simply let W’s tax cuts for the “rich” expire but dared not do same with the ones for the non-rich which would’ve brought in 3 times the revenue gotten from restoring the tax rates for the “rich”.

  5. Low Budget Dave says:

    When Richard Feynman was working with the Rogers Commission, he pointed out that even NASA engineers did not understand probability very well. People who were literally “rocket scientists” estimated the probability of catastrophic failure at about 1 per 100,000.

    In actual fact, the probability of catastrophic failure has since been estimated as low as 1 in 10, meaning that there was only about a 6% chance of NOT losing a space shuttle. The reason why is simple math: The launch vehicle had over 2 million moving parts. Even though the odds of any one part failing were amazingly low, the odds that at least one part on the launch system would fail were about 100%.

    The engineers constructed many different models to figure out how many of which parts had to fail to bring down the individual sub-systems. What they did not do was combine the different models. If you have fifty different models (of different systems) that each show a 1 in 10,000 chance of failure, the combined odds could be as low as 1 in 200.

    Feynman used a few simple exercises to point out how few people really understand probability as applied to real-life systems.

    Using that same method of thinking, imagine all the models that central banks use to estimate the various forces that control national economies. Everything from money supply to consumer sentiment is based on estimates of estimates. Most published economic data is correct only in the general sense that it is more accurate than astrology.

    Once you start combining the results of different economic models, the odds of obtaining a completely random result approach 100%.

    • bigsteve says:

      Which is why the old saying KISS, (keep it simple stupid) works. The more complexity you have the greater the odds become of failure.

      • Frwip says:

        Or use redundancy, which also works pretty well, as long proper care is taken against common mode failures

        But in the domain of reliability and failure prediction, a big part of the problem lies in the way the data is usually presented, and people get all messed up because they don’t understand harmonic sums.

        “Harmonic” is a big word to say that to add certain measures, like … odds of failure or mean time between failures, the most common measures in failure prediction, you have to take the reciprocal of the sum of reciprocals.

        Which is why, 50 sub-systems with a 1 in 10,000 chance of failure have exactly (and not as low as, exactly) a 1 in 1/(50 x 1/10,000) = 200 chance of failing, once taken together as a system (only valid under the assumption of complete decorrelation of failure modes of the different sub-systems, applies to single failure only, void where prohibited, batteries not included, etc.).

      • rd says:

        The space program is a bit unique. Robust and redundant design would often preclude doing the activity itself within a desired time period because of the added weight. I have huge respect for all of the people who have been going into space because they are all riding experimental craft, even the ones that are used repeatedly. The re-used equipment will generally be used a couple of dozen times at most. That is not a lot of cycles, so there is a relatively high probability of a new, unexpected event occurring each time they use it. This is different from airliners which go through intensive testing, trial flights, certification processes etc. before they take to the air. Even then, there are unexpected glitches, like the batteries in the Dreamliners.

  6. Petey Wheatstraw says:

    We crashed a spacecraft into Mars because an engineer didn’t convert his calculations to the metric system.

    What are the chances of that?


    • MikeG says:

      An Air Canada 767 ran out of fuel over Manitoba in the 80s because the pilots had confused imperial and metric measurements in the fuel calculations. Disaster was averted only by superb airmanship and a lot of luck in gliding the big jet to an abandoned airstrip.

      • rd says:

        One of the things that I yell at junior engineers about is when they give me numbers without units. I have gotten too many results over the years where people handed me a number without telling me what the units were. Writing the unit down by hand (or at least typing it this days) greatly reduces the potential for error, both by the calculator and the person receiving the data who will pass it on.

  7. Wayne The Philosopher says:

    Some insights from Daniel Kahneman’s ‘Thinking Fast and Slow’ fit in here.
    Human beings typically react to statistical questions with their ‘fast’ heuristics first. The problem is that these default intuitive tendencies are often at odds with statistical analysis. Even Kahneman’s colleagues, who are all trained in statistical analysis, continue to make these kinds of mistakes because no amount of education will change how your ‘fast’ heuristics work.

  8. CD4P says:

    Finally got to use the hectogram!!! Price for the self-serve grocery buffet was priced per hectogram. But the BIGGER news was, unlike American counterparts which only include fruits and vegetables, the buffet in Sweden included crab meat, shrimp, salmon, cray fish, turkey, and chicken too!!! Talk about being able to eat like a king!

  9. jbegan says:

    So true. The issue with probability is that unlike most math (cutting a pie into 4 pieces, calculating the area of a rectangle, etc) is it isn’t intuitive. I’ve always had an automatic understanding of math and never really had to think about it…until I took “Statistics and Probability” in college.. Oy vey! I just couldn’t emotionally wrap my mind around “if you toss a coin 100 times, what is the probability it will be heads if it has turned up tails the previous 100 times?” I know for sure it will turn up heads! ..LMAO! I have to sit down and think about it every time.

    • ottnott says:

      “if you toss a coin 100 times, what is the probability it will be heads if it has turned up tails the previous 100 times?”

      Depends. Is it a homeopathic coin?

    • rd says:

      In the real world, a coin that has turned up tails 100 times in a row is more likely to have a bias that is unaccounted for in the theoretical problem than it simply being a low probability statistical streak. That is the type of approach that they use to catch cheaters in many fields.

  10. 4whatitsworth says:

    My favorite fun with math question is Monty Hall problem.

    Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

    • ottnott says:

      That is a good problem, if properly stated. Usually, though, the problem is presented with unstated assumptions about the host’s behavior — especially the assumption that the host always (or randomly) shows one unselected door — that doesn’t match Monty Hall’s behavior on the show.

  11. Clif Brown says:

    Don’t tolerate innumeracy for yourself. Get over to Khan Academy (its free) and get started on “The World of Math” that will take anyone from simple addition to calculus. I was a terrible math student who with all the effort I could muster failed repeatedly at algebra. Now, at age 63, I am into calculus and having fun with it thanks to Sal Khan who is out to bring math to the world online. The problem is deeper than math, people shy away from learning in general. There’s nothing like the feeling of “I’ve got it!” but the high it provides only comes after effort.


    ADMIN: See this

  12. faulkner says:

    Your well-intended article leaves out that probability and statistics are fairly recent mathematically developments that even math and engineering types have difficulty applying correctly and consistently. The reasons for this include: the unconscious substitution of System 1 heuristics for System 2 reasoning, the Executive Function is not inhibiting the substitution, as well as the possibility there may not be sufficient Working Memory to compute a result. In everyday language, you need to know to inhibit your quick response, be able to do so, and then have the logical/mathematical horse power and experience to actually think the problem through. These multiple mental skills make working with probabilities and statistics advanced intellectual skills in neo-Piagetian developmental stages (Formal Operations). According to some studies, a quarter of the US population cannot think at this level at all. Most seldom do. Everyday life either doesn’t require it and/or actively discourages it. (E.g. consumerism, credit cards, car loans, mortgages, etc.)

    Applying probability theory and statistics is much more about thinking through the assumptions of a situation than finding a straight-forward solution. This shouldn’t be a discouragement to learning them. In fact, Charlie Munger writes a man without these skills is in a “one-legged ass-kicking contest.” Just realize the efforts at doing so could be considerable as the skills themselves.

  13. slowkarma says:

    I was never much good at math, beyond a certain point (I was good through trig) because I was pretty sure that whatever I did in life, wouldn’t much involve mathematics, and so studying was largely a waste of time (for me.) And I was right. But when I was in my 40s, the company I worked for sent me off to study political polling. I did well with the statistics and probability theory involved, because it was concrete, practical, much in the way that trig was. However, one thing really struck me as I went through the polling course, was how often we were working with bad data — you could be a mathematical genius, but if the data was bad, you were screwed anyway. And in the social/economic realm, the data is almost always bad to some extent. It’s estimating the badness of of the data that’s often the problem. For example, in Barry’s example at of the bus stop painters, he throws a number at us. I think he made it up. But what do I know? Maybe he didn’t. But if he did make it up, then that 99% accuracy estimate is not relevant to anything. Furthermore, as somebody else noted, there are extremely simple techniques that would guarantee 100% accuracy if the bus stop was painted at all (unitary stencils,) so that even a 99% accuracy rate would be a shocking failure, rather than the triumph he suggests. So in addition to the fact that you have to be *extremely* wary of bad data, you also have to be *extremely* wary of the conclusions somebody reaches, and tries to sell you, if you are not aware of the quality of the data, or the honesty of the person translating it for you. Barry has a taste for graphics; so do I. But I would estimate (to pull a number out of a hat) that 97% of graphics on complicated subjects are largely wrong, because complicated subjects do not lend themselves to simple graphics, and graphics almost by definition are simple.

  14. rd says:

    I don’t think the US was ever particularly good at math. The focus in education was universal education through high school, an admirable goal unto itself. The math problem is becoming much more obvious now because of the advent of computers and data gathering. It didn’t matter as much before because there wasn’t much data to work with.

    A big part of the problem is that math professors write most of the text books and usually there is little context provided for the student. Very few math problems encountered in high school or the first couple of years of university were developed for the sheer beauty of math. Most of them had very specific problems that were trying to be solved, and a genius put in a lot of hard work and invented the math tool to solve the problem. Framing math in this type of context could make it much more interesting for many people.

  15. Robert M says:

    You want to kindergartners learn probablity go to a POKEMON tournament. There has to be a comic book store where you live. A five year old will teach you the game..

  16. victor says:

    The owner of the ski resort, deploring lack of snow, deposited at a shrine the Virgin
    Mary a $100 wishing for snow. Snow came, with such abundance, and avalanches, with people stuck in the cars, and the resort was forced to close, prompting the owner to quip “I should have only given $25”. What the owner did is discover the notion of nonlinear exposure and extreme events. Under so-called “fat tails”, there is no such thing as a “typical event”, and nonlinearity causes even more severe problems.