statistical explanation an explanation expressed in an explanatory argument containing premises and conclusions making claims about statistical probabilities. These arguments include deductions of less general from more general laws and differ from other such explanations only insofar as the contents of the laws imply claims about statistical probability.
Most philosophical discussion in the latter half of the twentieth century has focused on statistical explanation of events rather than laws. This type of argument was discussed by Ernest Nagel (The Structure of Science, 1961) under the rubric ‘probabilistic explanation,’ and by Hempel (Aspects of Scientific Explanation, 1965) as ‘inductive statistical’ explanation. The explanans contains a statement asserting that a given system responds in one of several ways specified by a sample space of possible outcomes on a trial or experiment of some type, and that the statistical probability of an event (represented by a set of points in the sample space) on the given kind of trial is also given for each such event. Thus, the statement might assert that the statistical probability is near 1 of the relative frequency r/n of heads in n tosses being close to the statistical probability p of heads on a single toss, where the sample space consists of the 2n possible sequences of heads and tails in n tosses. Nagel and Hempel understood such statistical probability statements to be covering laws, so that inductive-statistical explanation and deductivenomological explanation of events are two species of covering law explanation.
The explanans also contains a claim that an experiment of the kind mentioned in the statistical assumption has taken place (e.g., the coin has been tossed n times). The explanandum asserts that an event of some kind has occurred (e.g., the coin has landed heads approximately r times in the n tosses).
In many cases, the kind of experiment can be described equivalently as an n-fold repetition of some other kind of experiment (as a thousandfold repetition of the tossing of a given coin) or as the implementation of the kind of trial (thousand-fold tossing of the coin) one time. Hence, statistical explanation of events can always be construed as deriving conclusions about ‘single cases’ from assumptions about statistical probabilities even when the concern is to explain mass phenomena. Yet, many authors controversially contrast statistical explanation in quantum mechanics, which is alleged to require a singlecase propensity interpretation of statistical probability, with statistical explanation in statistical mechanics, genetics, and the social sciences, which allegedly calls for a frequency interpretation. The structure of the explanatory argument of such statistical explanation has the form of a direct inference from assumptions about statistical probabilities and the kind of experiment trial which has taken place to the outcome. One controversial aspect of direct inference is the problem of the reference class. Since the early nineteenth century, statistical probability has been understood to be relative to the way the experiment or trial is described. Authors like J. Venn, Peirce, R. A. Fisher, and Reichenbach, among many others, have been concerned with how to decide on which kind of trial to base a direct inference when the trial under investigation is correctly describable in several ways and the statistical probabilities of possible outcomes may differ relative to the different sorts of descriptions. The most comprehensive discussion of this problem of the reference class is found in the work of H. E. Kyburg (e.g., Probability and the Logic of Rational Belief, 1961). Hempel acknowledged its importance as an ‘epistemic ambiguity’ in inductive statistical explanation. Controversy also arises concerning inductive acceptance. May the conclusion of an explanatory direct inference be a judgment as to the subjective probability that the outcome event occurred? May a judgment that the outcome event occurred is inductively ‘accepted’ be made? Is some other mode of assessing the claim about the outcome appropriate? Hempel’s discussion of the ‘nonconjunctiveness of inductivestatistical’ explanation derives from Kyburg’s earlier account of direct inference where high probability is assumed to be sufficient for acceptance. Non-conjunctiveness has been avoided by abandoning the sufficiency of high probability (I. Levi, Gambling with Truth, 1967) or by denying that direct inference in inductive-statistical explanation involves inductive acceptance at all (R. C. Jeffrey, ‘Statistical Explanation vs. Statistical Inference,’ in Essays in Honor of C. G. Hempel, 1969). See also CAUSATION, EXPLANATION. I.L.