Inferring and Explaining

simple answer is one of practicality. It would be too time consuming, too expensive, or other- wise too impractical to survey the entire popu- lation. Tus we use the sample, which can be examined and described, as a clue about the whole population, which cannot. Inferences from samples to populations are classic examples of inferences to the best expla- nation. Our data are the discovery that some sample has an interesting feature or property, and we use this as evidence that the population also has this property. We ask the explanatory question—Why does the sample have P ? And our hypothesis answers that it has P because the population as a whole has P . e 1 . Sample has property P . t 0 . Population has property P . Couldn’t It Just Be a Fluke? I hope by nowyou are almost programmedwhen you see an argument such as the previous one to begin to think of rival explanations. Sure, if the population has P , that would be a good expla- nation of why the sample has P . But what else might explain the sample having P ? I get home at 6:00 on a Tuesday evening and before I can fnish looking at the mail and fx- ing a martini, the phone has rung three times, all from charitable organizations seeking con- tributions. I conclude that this Tuesday is a big push for getting money. My sample, those three phone calls, is pretty skimpy. Afer all, I’m ofering a hypothesis about the whole country (or perhaps state or county). Isn’t the follow- ing rival explanation just as plausible, perhaps more plausible, thanmy charity full court press theory? 97 statIstICs t 1 . It’s just a coincidence that those three calls were all from charitable organizations. Or more generally, t 1 . It’s just a coincidence that the sample has property P . Modern probability theory has devoted a good deal of time and attention to developing some very sophisticated mathematical tests of how likely it is that a sample will have a given property simply as a matter of random chance. Some of you may be familiar with some of these tests for what is called statistical signifcance from other courses or computer sofware. Even those of you who hate numbers or math would be well advised, in my humble opinion, to learn a bit about all this by taking an introductory statistics course. But that is not my goal in the present context. Even those of you with the least experience and confdence withmathematics know that the size of the sample matters in important ways. A sample of three calls tells us almost noth- ing, while a sample of three thousand can tell us quite a lot. We will confne our discussion to an informal treatment of what statisticians call statistical signifcance. How accurate are our measurements within samples of a given size? A contemporary philosopher of science Ronald Giere ofers what he calls a rule of thumb for answering this question. 3 He ofers the follow- ing scale for correlating the size of the sample with the accuracy of what is being measured: