Dalzell's theorem and the analysis of proportions: a methodological note.
The golden section is a well-known proportion that occurs when something (e.g. a line) is divided into two unequal parts such that the smaller (m) is to the larger (M) as the larger is to the sum of the two (i.e. m/M = M/(M + m) = .618). Dalzell's theorem holds that the absolute value of the difference between M/(M + m) and .618 will tend to be smaller than the corresponding difference between m/M and .618. This means that the use of M/(M + m) ratios leads to results that are more supportive of the golden section hypothesis than does the use of m/M ratios. Notice that M/(M + m) corresponds to the proportion of Ms that will occur; while m/M corresponds to the odds that m will occur. While these are mathematically equivalent, in practice they may lead to different interpretations of the same data. Although originally envisaged as applying to the golden section, Dalzell's theorem may have implications for any study that uses either a proportion or the odds as a dependent measure. The use of proportions may produce results that are closer to a predicted value than will the use of the odds as a dependent measure.