One-third hypothesis
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 [i.e., 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." 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 formula_1 ways. Because each group of size r can dissolve in "2" subgroups, the total number of ways all groups of size r can emerge and dissolve equals "3", 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 formula_2 and dissolve into subgroups with a probability of formula_3, then the equation reduces to formula_4 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. Early research and recent prediction. A perfect example of the OTH was illustrated by Wayne Youngquist’s 1968 “Wooden Shoes and the One-Third Hypothesis,” which documented the German population in Milwaukee little more than a century ago. As Germans approached one-third of city’s population they became more and more prominent. As they exceeded that level their importance began to abate. The first empirical test of Engelmann’s OTH came in the form of the 1967 Detroit riot. It did not explain the cause of the riots but was aimed at explaining their timing. Sam Butler, in  2011, explicitly cited Engelmann and the One-Third Hypothesis in his analysis of London's riots and their aetiology. Criticism. The OTH was never without its critics. Early on K. S. Srikantan correctly questioned the assumption that p and q are each equal to ½. Even if they are not, however, so long as p + q = 1, the maximum value of r will occur at pn/(1+p). The group most likely to emerge and persist will always be smaller than half of the population. In social dynamics the OTH is sometimes referred to as critical mass . The terminology, though  appropriate, has become ambiguous because “critical mass” is used in a variety of ways that do not suggest the OTH at all. Similarly, the OTH is sometimes called the two-thirds theory.