Statistical thinking is one of the most important formal methods of engaging reality available to human beings. Sadly, it is also one of the more difficult, because human beings, in general, have absolutely no intuitive sense regarding probabilistic claims or statistical analyses. The people who do such things – even the ones that do them poorly – only reach such a stage of analysis after a significant amount of disciplined education. For the rest of us (and I must perforce include myself in this list) our statistical guesses only rise to the level of the merely appalling on those rare occasions that they are not completely idiotic. Quite usable texts can be had for the downloading (although the interested reader might consider supporting the Open Intro foundation), but one still requires no small measure of determination to “climb Mount Statistics” on one’s own. It is a challenge I’ve never completed at any substantive level, making this post more than a trifle daunting. However, even lacking any measure of expertise on the subject, there remain a few intelligent things that can be said, even by someone like me.
It might first be noted that the algebraic/relational mode of thinking that I discussed in the second entry in this series is not a distinct or disconnected set of methods from those of statistical approaches. Already in the very early years of the 20th Century, Alfred North Whitehead, whose reputation as an educator was only exceeded by that as a metaphysical thinker, was advocating introducing students to elementary statistical methods in the social sciences early in the educational curriculum. His reasoning was that this, “affords one of the simplest examples of the application of algebraic ideas.” This was a remarkably prescient notion on Whitehead’s part, as statistical methods had not taken on their dominating position in all the sciences, even the physical ones, in the period before 1920. But notice also that the argument here is for making algebraic/relational methods concrete for students through the application of statistical methods. What was true then remains true now.
The second thing to note is that thinking about collectives and aggregates is different from thinking about individual, and strongly self-identical “things.” Now, the dividing line here is not always given to us in advance. A rock may be treated in some situations as a “ strongly self-identical thing,” while in other contexts we may deal with exclusively as a statistical aggregate of molecules. As an obstruction on our path, the former approach is the sensible one; as an illustration of the thermodynamic properties of temperature, only the latter is valid. Note that both approaches are objectively real, but their descriptive characterizations include not only the rock, but the intentional purposes of the person(s) doing the describing. (Those intentional purposes are also objectively real.)
Aggregates have a kind of “constellation” of relatedness, but they lack the strong, “centralizing” feature that provides them with a single, comparatively simple “handle” of identity such that they can be used in a non-aggregative manner. That rock, your keys, your self, have a center which enables you to cognitively – and, often enough, physically – manipulate it in a manner that does not (at least, immediately) reduce itself to absurdity. But a statistical aggregate is no longer such a coherently unified whole. But it is a whole of a kind. This is why it is statistically meaningful to deal with the constellation in an aggregative manner.
This brings us to the first real lesson of statistical thinking: There is a difference between coherently unified and statistically aggregated wholes.
Which, in turn, brings us to our second lesson: We can learn to appreciate these differences even without mastering the mathematical techniques of statistics. An analogy here would be appreciating jazz without ever becoming a jazz musician.
An example leaps to mind here. For all the climate change denialism that is spewed with such promiscuous abandon on the Internet, nothing seems so popular as the, “but we had a cool day here!” meme. It scarcely seems possible to imagine the level of willful stupidity that would spew such fatuous nonsense, but of course, we don’t have to imagine it at all: people willingly jump up to publicly exhibit it for us. And, it is true, it has been a relatively mind summer this year (2014) in the eastern half of the United States. So when we treat that vanishingly insignificant fragment of the entire planet, with its vastly interconnected, deeply relational (recall: “relational” = “algebraic thinking”) climate system as somehow definable as a “coherently unified” rather than as a “statistically aggregated” whole, we essentially abandon any pretense of reason. From a fact about the insignificant particle of Planet Earth that has enjoyed a mild summer – so far … THIS year – people jump to the vacuous and indefensible conclusion that this inconsequential particle even addresses (much less refutes) a statement (overwhelmingly supported by hundreds of independent lines of evidence) that is addressed to the statistical aggregate which is the Earth’s climate. (After all, it is called GLOBAL warming, rather than “your back yard” warming, for a reason.)
Failure to grasp the basic distinction between coherently unified, as opposed to statistically aggregated wholes, is no small part of what enables this type of pitiful nonsense to continue. (Obviously it is not the only thing; there is no small amount of willful obtuseness and bald-faced lying amongst the denialists as well.) The denialist nonsense is itself but an example of the broader fallacy of cherry-picking data, leading to a Hasty Generalization. This latter is a version of the fallacy of the “unrepresentative sample,” that latter terminology being quite recognizably statistical in nature. But notice that grasping the distinction itself, between a coherently unified whole on the one hand, and a statistically aggregated whole on the other, is a trivial matter – no upper division mathematics courses are required to understand that much. Anyone willing to even try to think well can understand that there is such a difference; and understanding that there is a difference is the most important step toward understanding what the difference is. Because understanding that there is a difference is what allows one to refine one’s understanding of that difference, it keys one’s mind and perception to attend to distinctions that might otherwise be hastily trampled over with illegitimate generalizations. (“It was cool here in Michigan! Global warming isn’t happening!”)