# One-third hypothesis

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One-third hypothesis

The one-third hypothesis (OTH) is a sociodynamic term advanced by Hugo O. Engelmann, which 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." [1]

The OTH actually 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 $\tbinom nr$ 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:

                            $(1+x)^n = \sum_{r=0}^\infty {n \choose r} x^r. \qquad$


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 $\tbinom n r p^r q^{n-r} \!$ and dissolve into subgroups with a probability of $q^{r} \!$, then the equation reduces to $\tbinom n r p^r q^n \!$ 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

                               $\tbinom n r / 2^{n+r} \!$,


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.[2]

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.[1]

Competition and conflict are not the only examples of the one-third hypothesis. Nothing prevents cultural and intellectual movements from being viewed in this context. The timely effects of the one-third hypothesis, for instance, can be scarcely overlooked in the Harlem Renaissance (1920-1930). In 1920 the African-American population of central Harlem reached 32.43% and by 1930 climbed to 34.82% in greater Harlem.[3]

A situation will arise in 2011 that may well test the OTH anew. By then, one-third of London’s population will be under the age of 25,[4] and one-third of the total population will be composed of persons who visibly belong to an ethnic minority.[5]

## Possible verification

Whether the 2011 Tottenham riot was an instance that verifies the hypothesis remains to be seen. The resulting 2011 London riots, with no apparent connection other than youth under the age of 25, are a strong indication of subgroups nestling within larger groups. As members interact with each other more and more frequently, the intensity of their behavior increases. Some become violent, a great many fewer become brutal, and a tiny fraction become vicious. Reaction to the rioting ranged from vigilantism in some neighborhoods to voluntarism in others, in an effort to clean up London. In Birmingham a mourning father appealed for calm and his plea was heard.[6]

## 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.[7] 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. That would be applicable in a case such as Thurlow Weed’s letter to Abraham Lincoln on the eve of the American Civil War at a time when the population of the North was about twice as large as the South. “I am unwilling to see a united South and a divided North,” Weed wrote.[8]

Indeed, that approach would add to the Origins of the American Civil War article, where the author avers that “two-thirds of Southern whites owned no slaves” at all.[9] In other words the planters who did own slaves formed the core of the South.

## References

1. ^ a b Hugo O. Engelmann. (1967). "Communication to the Editor." American Sociologist, November. p. 21.
2. ^ Wayne A. Youngquist. (1968). “Wooden Shoes and the One-Third Hypothesis.” Wisconsin Sociologist, vol. 6; Spring-Summer # 1 & 2
3. ^ Beverage, Andrew. (2008). Harlem's Shifting Population. Gotham Gazette. http://www.gothamgazette.com/article//20080827/255/2635
4. ^ Phillips, Trevor. (2001). Alcohol and London’s Youth
5. ^ Phillips, Trevor. (2001). Multiculturalism – Enrichment of Society or Cause for Conflict
6. ^ Tariq Jahan appeals for calm http://www.youtube.com/watch?v=UZ1VjUSKevc
7. ^ Srikantan, K. S. (1968). "A Curious Mathematical Property." American Sociologist, May. p.p. 154-155.
8. ^ Brands, H. W. (2009). “We Must Not Be Enemies.” Lincoln’s Genius. Leesburg, VA: Weider History Group. p. 25.
9. ^ Origins of the American Civil War http://en.wikipedia.org/wiki/Origins_of_the_American_Civil_War

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