“You guys have all given me different answers, and I don’t know what to do.”

The math tutoring room at the local community college has two parts, one for advanced math—trigonometry, pre-cal, and calculus; the other is for basic math, from carrying and borrowing up to college algebra.  I work in the latter, but as somebody with a Master’s in statistics, I am often the “go to” person for statistics questions.  The fact I have seldom used statistics in the last decade has made me rusty, but the material comes back, so long as one learns it well the first time.

When the individual came to me stating the conflicting opinions she had received, I should have either turned her down or told her she was going to have to decide up front whom to believe.  If I were not that individual, she should leave, and not waste her time.  The issue itself was a 1-sample proportion test, one of the M &M problems, where a certain proportion of different colors are put into the bag, people count out the number of each type to see if the proportion corresponds with the claimed proportion, within a reasonable margin of error.

The student had used the instructions given to her what to input into the calculator and found a probability that made no sense to me.  I looked at the question and came up with the correct probability.  The example she copied looked at the probability’s being greater than a specific number; the problem she asked me looked at the probability’s being less than a specific number.  She didn’t understand that the example given to input and the problem were asking opposite things.

I tried every way I could think to explain the issue to her.  I have become more adept at calculators, finding them fast and helpful.  This woman, as are so many students today, was faster with the calculator than I.  Her problem, however, was something that it took me some time to figure out.  I had drawn a diagram of the probability curve, the Bell-shaped normal or Gaussian distribution, and she had looked confused.  That led me to finally ask a simple question:

“Have you ever computed these probabilities using a normal probability table?”


I now understood her problem.  She was being asked to input data and push a lot of buttons.  Unfortunately, she had no idea what was being done to the data and why.  A lot of statistics is finding the difference between the sample and a postulated or known mean/average, then dividing by the standard error, a measure of variability.  The concept of variability is critical to understanding not only statistics, but everything statistics is used for, be it political campaigns or climate science.  Natural processes, like heart rate, body weight, stock market prices, or temperature, are not the same when measured over a period of time.  They fluctuate, and statistics helps us understand the fluctuation.

Dividing the mean by the standard error normalizes the data, allowing it to be compared to one standard, this instance to a table to find a probability.  By doing many problems where I had drawn a bell-shaped curve and looked at probabilities, I understood the concept well enough to teach it to undergraduates in Las Cruces for two years and in Tucson for another four.

This woman was from another generation of students, however, and in the decade where I have not been heavily involved with statistics, drawing a picture of how the data were distributed and having a sense of what the data were trying to say has atrophied, at least where I am tutoring.  The argument I was having with the student had a lot to do with the arguments I needed on the calculator;  she did not understand them, only that she was obtaining different answers.  Put simply, she did not have the background to be using a calculator.  I could say that about many students.  When I taught for a private for profit college, when a student saw a probability “6 E-4,” they wrote “6” as a probability, both impossible and showing no sense of what E-4 means, which is a power of 10 to a minus number:  6 E-4 =0.0006.  I don’t expect the average person to know that; I do expect somebody taking statistics and using a calculator to understand it.

That is only my opinion, from one who learned the material from first principles and is still slow to pick up a calculator, because I am often more comfortable performing my own calculations.  It remains to be seen whether we will continue to teach by calculator or teach by understanding the material, using the calculator as a tool to speed up the process.  I fear that in our rush to educate people, we are giving them instructions as to what buttons to push in a lot of subjects, without any idea of what is going on inside a calculator or more importantly, inside the system we are analyzing.

This is not idle philosophical musing.  When I taught, more than half the class did not understand what “the rate of increase in health care costs is declining” meant.  To them, the statement meant that the costs were decreasing, rather than the number was still increasing, but less rapidly than it was before.  This term is commonly used. The concept of statistical error to many people means that statistics is wrong, so it doesn’t matter.  Statistics is unable to tell us the exact results in a population, because from a poll, we do not know what the exact result is for the people in whom we are interested.  Where we differ from other fields is that we quantify the error in terms of confidence and probability, and we know the difference between the two terms.  We reject the concept that “anything can happen,” because we define a priori what “can happen” means.

We need to learn what calculators can do; equally importantly, what they cannot do.  Data that are not collected randomly have limitations what we can say about them.  Calculators do not have the ability to discern that. Calculators answer only what we ask them; they neither ask questions, nor do they tell us what we might want to know.

Calculators and computers are wonderful tools to get information that one needs, but education, critical thinking, and understanding remain timeless.

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