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Please forgive me if this is naive. I am only an aspiring educator after all. Why do we still teach estimation when there are easily accessible/teachable exact techniques? Why do statisticians teach $p$-value estimation instead of exact computation. Why do calculus professors teach Riemann sums alongside integration?

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    $\begingroup$ How do you even define an integral if you haven't first introduced Riemann sums? It seems much more intuitive to first introduce Riemann sums, then pass on to the fundamental theorem. As for $p$-values, when are exact techniques even possible? In most examples that undergraduates are going to see, you are going to be working with functions that don't have elementary antiderivatives (the normal distribution, Student's $t$, etc). The best you can hope for is numerical integration... $\endgroup$ – Xander Henderson Dec 4 '17 at 22:41
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    $\begingroup$ Data can be mis-keyed in technology. If one seeks to compute $56 \times 2.5$ and misses the "2" key, then one should immediately realize that the answer cannot be $28$ and fix the mistake. Etc. $\endgroup$ – Daniel R. Collins Dec 4 '17 at 23:27
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    $\begingroup$ What's $\pi$? Can we write down $e$? How about $\sin(45^\circ)$? There are no exact decimal answers to these questions, only estimations. And it's very, very important that students understand that. $\endgroup$ – Nate Bade Dec 5 '17 at 16:56
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    $\begingroup$ To elaborate on what @Xander Henderson said, at least regarding Riemann sums, even if one isn't interested in rigorous aspects of Riemann integration, knowing about Riemann sums allows you to understand how to use integration to find areas, volumes within surfaces of revolution, centers of mass, arc length, fluid pressure, moments of inertia, work done by a variable force, and all the other standard applications of integration. $\endgroup$ – Dave L Renfro Dec 5 '17 at 17:02
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  1. Number Sense: At the elementary level, estimation helps students to develop number sense. As Daniel R. Collins notes, order of magnitude estimates can be quite important. Anecdotally, I once rented an apartment. I moved out of the apartment about halfway through the month, and the landlord offered to prorate the rent. I said "Great!" However, when she operated her calculator, she determined that the prorated rent was more than the monthly rent. When I asked how this could be right, she conceded that it was confusing, but the number on the calculator don't lie. Better number sense could have saved us all some time.

  2. Emphasizing Uncertainty: At least one of the "exact" techniques that you mention is rarely exact. To wit, most of the probability functions used in statistics don't have nice antiderivatives in terms of elementary functions. This means that any attempt to integrate them is almost certainly going to have to rely on numerical integration. Getting estimates for $p$-values (for example, via the "empirical rule") once again provides a check against technology. It also can be used to emphasize that we lack certainty.

    For example, if I have a table for a $t$ distribution that only gives entries for multiples of ten degrees of freedom, then I am going to have difficulty coming up with an "exact" $p$-value for a hypothesis test involving 47 degrees of freedom. I have to make a choice: should I use 40 or 50 degrees of freedom? In this context, I want use the number that gives me less certainty, so I work with 40 d.f. This is still an issue with computers. If a computer gives me a $p$-value that rounds to my level of significance, what should I do? Understanding that there are estimates throughout the process can help me to make that decision.

  3. Emphasizing Theory: I believe that we should be teaching mathematics not just in service to other fields, but also in service to the field of mathematics itself. You may not have that many future pure math Ph.D.s in your class, but you still need to make sure that you are preparing those students as well as all of the others. If integration or linear algebra is taught as a bunch of computational techniques, the math majors are going to be in a very poor position when they start taking "real" math classes.

    To use your example, Riemann sums are emphatically not an estimation technique, but the actual tool that is necessary to build the Riemann integral. If you can't work with Riemann sums, how are you ever going to work through a proof of the fundamental theorem of calculus?

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  • $\begingroup$ Wow, thank you for the great answer. Admittedly, I posted origi in frustration $\endgroup$ – euclidiography Dec 5 '17 at 1:35
  • $\begingroup$ *originally in frustration with a professor who uses estimation liberally. My query was narrow in scope; of course I see the importance of Riemann Sums when teaching introductory calculus. And you're right, we shouldn't think only to teach the future PhD students in the classroom. $\endgroup$ – euclidiography Dec 5 '17 at 1:37
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    $\begingroup$ Great answer. In addition to (2) would add that learning linear interpolation (and not just theoretically but a willingness to use it often) was one of the best things I got out of baby thermo ("steam" for naval officers). Many many problems in business world, engineering, even just operating a shop involve some need to do linear interpolation of listed values. (yeah, yeah you can fit fancier curves, but often linear interpolation between two close to medium points is enough). $\endgroup$ – guest Dec 5 '17 at 23:23
  • $\begingroup$ @guest -- Your comment would make a good answer. $\endgroup$ – Jasper Dec 6 '17 at 2:36
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    $\begingroup$ "When I asked how this could be right, she conceded that it was confusing, but the number on the calculator don't lie." This is frightening. $\endgroup$ – Brendan W. Sullivan Dec 6 '17 at 17:37
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I'd say

a good approximation is often better that an exact result.

This may sound counterintuitive, but as the phrase is vague anyway, here is a longer explanation what I mean: An "exact result" is often a formula, and often it's a complicated formula and one needs at least a calculator to evaluate is. Since calculators use finite arithmetic there is, most likely, some approximation going on in the background (imagine, that the answer is $\sqrt{\pi}$…). An approximation often comes a sequence of approximations and thus, a kind of algorithm alongside with an error bound. For example the series representation $e = \tfrac{1}{0!} + \tfrac{1}{1!} + \tfrac{1}{2!}+\cdots$ gives you 1) a way to compute an approximation of $e$, 2) gives an idea that $e$ is somewhere near $2.5$, 3) can be augmented with an error bound to prove that $e\approx 2.7181 \pm 0.0001$.

Some further aspects:

  • Statistics (in contrast to stochastics) is not about calculating certain probabilities, but inferring them from experiments. And inference is approximation/estimation.
  • In analysis, a good estimate if often good enough for an exact result. An example is the "epsilon of room"-strategy: Instead of proving $a=0$, prove that $a\leq \epsilon$, and $a\geq -\epsilon$ for any $\epsilon>0$ (by using estimates) and then conclude $a=0$.
  • In applications (real world applications, I mean), approximation are much more important than exact computations. This is due to many reasons, one is that data is often uncertain, and an exact result based on uncertain data is worse that a good approximation (no one would do an exact data fit for noisy data) and another is, that approximations are often much quicker to get and still good enough for all practical purposes.
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Most real-world problems are only approximately described by nice mathematical formulas.

Depending on the situation, it can be either silly or dangerous to assume that an "exact" result of a mathematical calculation exactly describes a real-world situation that caused you to perform the calculation.

For example, to what extent does an "exact" result of a p-value calculation predict what would happen the next time you randomly tried to use the baseline hypothesis to get a value?

When performing real-world integrations, often an approximate answer (which resembles a Riemann-sum) is both more practical to determine, and more accurate, than an "exact" calculation based on a function that more-or-less describes the real-world situation.

Furthermore, teaching the approximate method teaches an easy-to-remember rule-of-thumb, which has several advantages:

  • Some people find it easier to remember.
  • It might help the student know when the exact method is likely to be useful.
  • It might help the student remember the logic of how the exact method was derived.
  • It provides a way to sanity check results from the exact method.
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From the perspective of a professional statistician, I often find myself asking the opposite question: why do we spend so much time learning about closed form solutions in our calculus classes?

The reason I say this is that for the majority of applied statistics problems that require any sort of integration (such as the 95% of the field of Bayesian statistics), there is no closed form solutions and numerical methods are required. Similar for optimization problems (i.e. maximum likelihood estimation): some simple problems have closed form solutions (i.e. linear regression)...but that vast majority of problems are solved iteratively.

So from my professional perspective, we spent a lot of time learning closed form solutions to various problems...but when it comes time to do the math behind any new statistical methods, 99.9% of the time we end up using iterative methods instead.

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    $\begingroup$ The closed form solutions are helpful in many physics and engineering derivations. $\endgroup$ – guest Dec 20 '17 at 15:11

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