Math Education and a Bit of Logic

Recently I’ve had several different things remind me of what I perceive to be a serious problem with numbers in this country. This can have a severe impact on one’s personal life, but also on church and social policy issues.

I recall when I argued some academic affairs committee into allowing me to count a probability and statistics course against a math-science requirement, even though it wasn’t on the list, and I have always been glad that I took that particular course as part of my own limited math education. Of the courses I took outside my own major, that one is easily the one that has contributed to my daily life.

Now the one course, and whatever reading I’ve done on the subject since, does not make me an expert. But you don’t need to be an expert to detect problems with the way people use numbers. You just need to know some basics, and then ask questions. Some of the questions don’t even require math. For example, if you read a newspaper story about sexual promiscuity that indicates that a certain percentage of teenagers are sexually active, you need to ask just how they know that. The answer can be found, though to get very specific you might have to go find the report. Reporters rarely give any of the methodology. (A second course in survey design, only partially completed though I read all the texts anyhow, helps me here.) In the survey you want to look at the questions asked to see just what the definitions are. Normally you will find that those who conducted the survey used good methodology and reported the facts in the appropriate context. It’s when the survey gets quoted that the problem starts.

Here are some of the interesting cases I’ve noticed. The Florida Lottery is advertising a new drawing. According to them, this gives you additional chances to win. Now this is one of those lines that can qualify as true, but only if you assume people will understand it in a certain way. The way consumers, especially those addicted to the medium, actually hear this is that they have a greater chance of winning. Unless you increase the number of winners while selling the same number of tickets, the probability of an individual winning does not increase. Similarly, a few years back the lottery advertised better chances of winning by placing five scratch-off patches on each ticket rather than just one. A moment’s thought will tell you that the probability of winning remains the same, since every ticket now provides five chances–every ticket.

Pepsi’s current commercial talks about the number of chances you can get. Here it’s more benign because you’re merely buying Pepsi products, which I presume you were going to buy anyhow (I could be wrong!) yet they work with the “billion” chances. If I give out a billion tickets to allow someone to win $10, but I only have one winner, then I have, truthfully, given out a billion chances. Of course Pepsi has many prizes, but the principle is the same. Here they are merely impressing everyone with large–and irrelevant numbers. The real number that should interest you is the probability of winning any prize or of winning a particular prize, a number that will be quite depressing.

Then there are the polls. Reporters have gotten much better at pointing out the margin of error, though they seem to miss the decimal portion of it frequently. A 3.7 margin of error is closer to four than to three, and I’ve seen a couple of cases where two candidates were actually within the margin of error but were reported as outside of it. Then people regularly miss (and are not told) the percentage chance that the result is outside of the margin of error. What I’ve noticed more in the last few days, however, is that reporters will note a trend when the difference between the previous figure for a candidate and the current one is still within the margin of error. I would point out, as well, that the margin of error is not a line inscribed in steel, in other words it doesn’t switch from “certain to be correct” to “certain to be incorrect” on the dot.

Then there is the division of demographic groups. I’m not really talking statistical measures here, but rather our need to divide and classify things. I don’t even object to this division. It’s necessary to analysis. But it’s useful to remember in thinking about these groups that people’s attitudes don’t undergo a radical shift on the line between 25 and 26, or at the point where they begin to make $50,001 annually. People are pretty analog. Analysis tends to be binary.

I want to mention one last church related issue. I remember a conversation with a pastor who informed me that most (I forget the particular number, but I think the percentage was in the 50s) people who were looking for a church in our neighborhood were looking for a traditional worship experience. The immediate assumption was that the road to church growth was by providing such a service and focusing on it. Now that might be true. But don’t forget the 40+ percent. Before those numbers have good context to provide a basis for decision making, we need to know how many churches are providing a traditional service and how many are providing something more free-flowing with contemporary music, amongst many other factors. Many business operate with the purpose of providing services to the minority in a community, those with specialized wants and needs.

For whatever reasons we place greater weight on an argument that has numbers in it. When I went to the emergency room a couple months back with abdominal pain, the nurse wanted me to rate it from 1 to 10. Now the fact is that I have experienced remarkably little pain in my life. How do I come up with a number? Painful enough to get me to the ER, but what number to assign? Once we have a number for the record, however, we feel that we have more accurate information. Those numbers, however, are only as good as the data collection method that produced them.

Statistical information could be extremely valuable, but it is also subject to abuse. That’s not because of an inherent weakness in the method, but because so few people take the time to take the numbers apart and understand what they’re saying. Thus the unscrupulous, or just the numerically challenged, can deceive us too easily.

(For those without math training, let me recommend a couple of books: How to Lie with Statistics, which is old but fun, and Lies, Damned Lies, and Statistics. I have seen some reviews that accuse the latter book of a conservative bias, and it may have one based don the selection of stories, but I think he does well in analyzing the data for each case he does cite.)

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  1. Don’t forget how the term “average” is abused, with rarely a mention of whether it is the mean, median, or mode that is being talked about.

    I teach a freshman college composition course, and I think that innumeracy is so much of a problem that I’m considering doing a week dealing with the issue (perhaps using one of the books you cite) so that students can responsibly use numbers to build arguments (and hopefully so that they’ll responsibly use numbers in life).

    1. You’ll need to be careful, using “How to Lie,” that your students use one of the newer printings; telling them to get used copies may be a career-limiting move.

      The original had some severely non-PC examples — e.g. you have to take a survey of “Negroes,” so you see one who is “clean and decently patched” and one who is “dirty and looks surly,” and you choose the first. There are better ways, these days, to illustrate selection bias in polling.

  2. On the supposed conservative bias of Damned Lies and Statistics, I’ve read bits and pieces of the book, and it basically argues that many of the problems we think are huge are really much more modest, but that it’s a media-saturated market that takes isolated statistics any mangles and magnifies them to produce political action. I lean left politically, and it really bugs me when fellow liberals reject arguments based on facts and hard analysis just because it doesn’t fit into their preconceived box (or, more probably, doesn’t allow for as good of a PR example to get things done). We like to style ourselves as being part of the “reality-based community,” yet we as a group are also highly susceptible to mis-interpretation and throwing out facts that we don’t like.

  3. I’ve found my Economics education is more comprehensive in asking questions. Sometimes it’s not just the statistics but the methodology or what’s actually being measured that is relevant.

    Reading the book Freakonomics or How to Think Like an Economist to get an Idea of what I mean.

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