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The one-third hypothesis (OTH) is a sociodynamic idea—advanced by Hugo O. Engelmann—that asserts that a group's prominence increases as it approaches one-third of the population and diminishes when it exceeds or falls below one-third of the population. As the one-third hypothesis was stated originally by Hugo O. Engelmann in a letter to the ''American Sociologist'' in 1967: "...we would expect that the most persistent subgroups in any group would be those which approximate one-third or, by similar reasoning, a multiple of (a power of ) one-third of the total group. Being the most persistent, these groups also should be the ones most significantly implicated in ongoing sociocultural transformation. This does not mean that these groups need to be dominant, but they play prominent roles." 〔Hugo O. Engelmann. (1967). "Communication to the Editor." ''American Sociologist'', November. p. 21.〕 The OTH involves two mathematical curves. One represents the likelihood that a subgroup of a specific size will emerge; the other is the probability that it will persist. The product of the two curves is the one-third hypothesis. == Statistical formalization == Statistically speaking, the group that is one-third of the population is the one most likely to persist and the group that is two-thirds the one most likely to dissolve into splinter groups, as if reacting to the cohesiveness of the group that is one-third. According to the binomial coefficient a group of size r occurs in a population of size n in ways. Because each group of size r can dissolve in ''2'' ''r'' subgroups, the total number of ways all groups of size r can emerge and dissolve equals ''3'' ''n'', in keeping with the summation: Said otherwise, large groups close to two-thirds of the population will be more likely than any other groups to dissolve into splinter groups. A corollary of this consideration is that much smaller groups will be the ones most likely to emerge and to persist. If groups of size r occur with a probability of and dissolve into subgroups with a probability of , then the equation reduces to and given that p and q are each equal to 1/2, Engelmann's One-Third Hypothesis can be readily deduced. It takes the form of where n is the number of people and r is the size of a group and can be verified for large numbers by using the Stirling's approximation formula. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「One-third hypothesis」の詳細全文を読む スポンサード リンク
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