At the The Cynical Economist, I found this clip of mathematician Arthur Benjamin advocating replacing calculus with statistics in the public school curriculum:

Benjamin's reasoning for foregoing calculus for the sake of statistics is that statistics is a more pragmatic topic, since risk assessment and odds estimation are daily occurrences. Let's see how it works out for us. Say we go out and measure some continuous quantity like the error in the volume of Foster's in a standard bottle. In the name of science, we are compelled to repeat the experiment often and when we're done we plot the relative frequencies vs. the measured errors:

What we find is that we have a bell curve given by a probability density function f(z). What we really want is the area under the curve for some increment, as this yields the probability that a member of the Foster's population will fall in the increment. How do we get this area? Had we taken calculus, we'd know to integrate f(z) over the interval [a,b] of interest to get the desired probability, F(z)=F(b)-F(a).

You're probably thinking: Wait a minute, for a normal distribution, there is no closed form solution that provide F(z) directly and this is true. Instead, numerical approximations must be employed and this doesn't require calculus. In fact, via a simple transformation, a common Z statistic table or Excel function normsdist() can be employed to simply look up the probabilities sought, without no regard for the underlying math. This is precisely Benjamin's point, to perform statistical calculations, calculus is unnecessary, which is true. The watered down versions of statistics provided by psychology, biology, and business departments prove this daily.

However, there is a significant difference between plugging numbers into a standard formula and comprehending how and why the formula works. To actually understand the formula and its limitations, a student typically needs to see its derivation. This why math and engineering undergrads are required to take mathematical statistics, which is rather calculus intensive.

Benjamin states that this current economic crisis would have been avoided if public schools taught statistics, as opposed to calculus. This is entirely wrong. It was the use of statistics, without the understanding of their limitations, that lead to the gross asset and risk mispricings that contributed to this crisis. Clearly, allowing the dumbed down version of statitistics to become the norm is only encouraging more misapplications of the discipline.

## 7/1/09

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I'd offer that many folks who take statistics never learn to think probabilistically--which seems a key outcome of stats. Instead, people continue to think deterministically rather than across the spectrum of chance.

ReplyDeleteProposing that our current crisis would have been avoided had we taught more stats is laughable. Heck, many PhDs who take a boatload of stats have trouble applying lessons learned.

As for calc, agree that it provides a conceptual springboard for other operations. My current capacity for doing derivatives and integrals is a little rusty, tho ;-)

At my alma mater, probability was a one semester class and was a prerequisite for stats, as was calculus. And what's good for The Sally Struthers Truck Driving Institute has got to be good for the country.

ReplyDeleteHah! I can see the textbook now,

ReplyDelete*Assume that intelligence and work ethic is normally distributed across this imperiled planet and across all socio-economic strata...*

And, are they going to make Barney Frank et comrades first master basic arithmetic before throwing a Congressionally mandated statistics course at THEM?

I was just telling my wife yesterday that I don't think Calculus a *must teach* - even if my 4.5 year old homeschooled son will be ready for it in 2 years or so.

But I certainly wouldn't cover statistics without, as your post convincingly argues, hitting on limits, continuity, rates of change, integration, etc.

CN- As far as I can tell, the only math Barney Frank concerns himself concerns tallying his endless stream of campaign contributions.

ReplyDelete4.5 and he's proficient with math?!? Congratulations, it sounds like there will be plenty of time for dy/dx and the like.