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Dalzell's theorem and the analysis of proportions: A methodological note.

John G. Benjafield [*]

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 in 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 produ ce results that are closer to a predicted value than will the use of the odds as a dependent measure.

A proportion is calculated by dividing a part by the whole (e.g. winners/ (winners + losers)), while the odds are calculated by dividing the number of times an event occurs by the number of times it does not occur (e.g. winners/losers). Since either one can be derived from the other (Dawes, 1988, p. 275), which one an investigator chooses to use might not appear to matter. However, reporting data in terms of proportions leads to a different distribution of results than does reporting the same data in terms of the odds. The importance of this difference will be illustrated by means of the gold section, a proportion that has long been a topic of research in psychology and other disciplines.

The golden section hypothesis

The golden section is the proportion that obtains between two quantities when 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)). The golden section is an irrational proportion with a value of approximately .618. Notice that m/M corresponds to the odds that m will occur; while M/(M + m) is the proportion that M bears to the whole. The golden section is the only proportion for which these two values are the same.

In psychology, there are two lines of research that have focused on the golden section. One of these concerns the relationship between positive and negative judgments, while the other focuses on the aesthetic properties of the golden section. As we shall see, similar methodological problems arise in both areas of research.

Studies of positive and negative judgments

According to Boucher and Osgood's (1969) Pollyanna hypothesis, people tend to use the positive poles of bipolar opposites (e.g. good) more often than the negative poles (e.g. bad). Benjafield and Adams-Webber (1976) proposed a more precise version of the Pollyanna hypothesis that they called the golden section hypothesis, which holds that the frequency of positive (P) and negative (N) judgments has a golden section relationship (i.e. P/(P+N) [cong] .618).

To test the golden section hypothesis, Benjafield and Adams-Webber (1976) used a repertory grid technique (Kelly, 1955) that required participants to sort a number of acquaintances on the basis of a series of bipolar opposites. The bipolar opposites were representative of Osgood, Suci, and Tannenbaum's (1957) semantic differential factors, with one of the poles for each dimension being psychologically positive (e.g. fair, strong and active) and the other being psychologically negative (e.g. unfair, weak and passive). In this and in subsequent studies employing a similar method, P/(P + N) ratios were calculated for each participant by counting the number of times the positive poles are used to label acquaintances (P) and dividing by the total number of positive and negative labels (P + N) (e.g. Adams-Webber, 1978; Benjafield & Green, 1978; Kahgee, Pomeroy, & Miller, 1982; Leach, 1979; Rigdon & Epting, 1982). The results of such experiments were presented as supporting the golden section hypothesis (i.e. P/(P + N) [cong] .618). The golden section hypothesis still plays a role in research concerned with estimating the relative amounts of positive and negative judgments people make (e.g. Ronan & Kendall, 1997). Gross and Miller (1997, Table 1) identified 13 articles in which a total of 50 mean P/(P + N) ratios were presented in support of the golden section hypothesis.

Dalzell's theorem

Analogues of P/(P + N) ratios have also been computed in other disciplines. For example, Duckworth (1962) counted the number of lines in successive passages in the Aeneid. Let the number of lines in the longer of two passages be M, and the number of lines in the shorter of two passages be m. Duckworth found 1044 cases in which M/(M + m) is close to .618, and claimed that this is compelling evidence that Vergil used the golden section as a regulating proportion.

In a review of Duckworth (1962), Dalzell (1963) pointed Out that Duckworth always used M/(M + m) rather than m/M. 'In the relatively few instances when the quotient is exactly .618, then m/M = M/(M + m) and it does not matter which ratio is used. But in all other cases the more complex ratio is less sensitive to deviations from the perfect figure of .618. If Duckworth had based his calculations on the simpler and more natural ratio, he would have had to discard a large proportion of his 1044 examples' (p. 316).

Fischler (1981) recognized the significance of this observation and called it Dalzell's theorem. This theorem holds that the absolute value of the difference between M/(M + m) and .618 is always less than or equal to the absolute value of the difference between m/M and .618. If Dalzell's theorem is recast in terms of Benjafield and Adams-Webber's (1976) golden section hypothesis, then the following is obtained.

\[frac{P}{P + N}]-0.618\ [leq] \[frac{N}{P}]-.618

If Dalzell's theorem is true, then the difference between P/(P+N) ratios and .618 will be smaller than the difference between N/P ratios and .618, except for those cases in which both P/(P + N) and N/P are equal to .618. In order to appreciate the force of Dalzell's theorem, consider the example in Table 1, which gives values for

P and N ranging from 0 to 100 in intervals of 10. While the resulting P/(P + N) ratios range from 0 to 1, the corresponding N/P ratios vary from 0 to [infty]. In every case, the difference between any N/P ratio and .618 is always greater than the difference between the corresponding P/(P + N) ratio and .618.

Experimental data show a similar pattern. For example, Leach (1979, p. 116) reported P/(P + N) ratios for 32 women who completed a repertory grid task similar to the one used by Benjafield and Adams-Webber (1976). If the N/P ratios corresponding to the P/(P + N) ratios reported by Leach (1979) are calculated, it turns out that the range of the former ratios is 2.7 times as large as the range of the latter. While the mean of the N/P ratios (.617) is actually closer to the golden section than the mean for the P/(P + N) ratios (.630), the standard deviation of the N/P distribution is .207, or 2.9 times as large as the standard deviation of the P/(P + N) distribution (.079). While the 95 % confidence interval for the mean of the P/(P + N) distribution is .59 to .66, the corresponding interval for the N/P distribution is .54 to .69. Moreover, the difference between any N/P ratio and .618 is always greater than the corresponding difference between P/(P + N) and .618. Any distribution of N/P scores will be less obv iously supportive of the golden section hypothesis than will the corresponding P/(P + N) distribution, even if their measures of central tendency are similar.

Studies of the aesthetics of the golden section

In addition to studies of the relative frequency of positive and negative categories in personal judgment, the phrase 'golden section hypothesis' has also been used to refer to the possibility that the golden section has special aesthetic properties (e.g. Green, 1995; H[ddot{o}]ge, 1995; Plug, 1980). Consider a study by Schiffman and Bobko (1978) that asked participants to divide 100 mm lines 'so that the resulting two line segments form the most pleasing proportion' (p. 102). Results were presented for each participant in terms of the average m/M ratio, where m is the shortest and M the longest line segment.

Although unaware of Dalzell's (1963) work, Schiffmann and Bobko (1978) used m/M instead of M/(M + m), just as Dalzell would have recommended. While the mean m/M ratio is .59, 'there is little agreement between subjects as to the most pleasing point of line segmentation' (Schiffman & Bobko, 1978, p. 103). Their point can be made even more strongly by converting their m/M ratios to M/(M + m) ratios, and comparing the two distributions. While the mean of the M/(M + m) scores is .64, the range of the M/(M + m) scores is .29, only .40 as large as that for the m / M scores (.72). The 95% confidence interval for the mean of the M/(M + m) distribution is .61 to .68, compared with a confidence interval for the m / M distribution of .50 to .67. As also occurred with N/P and P/(P + N) ratios, in each case the M /(M + m) values are closer to the golden section than the corresponding m / M values, except for the one case in which m / M = .62.


The use of a proportion (e.g. P /(P + N)) leads to a distribution of results that appear to be more consistent with the golden section hypothesis than does the use of the odds (e.g. N/P). This fact about the golden section illustrates a more general point. Since the standard error of the odds will be greater than the standard error of the corresponding proportion, then the absolute deviations from any predicted value will tend to be greater for the odds than for the proportion. Thus, a generalized form of Dalzell's theorem would be that any observed proportion will tend to be closer to any predicted value than will the corresponding odds. Researchers who predict proportions may unwittingly take advantage of this form of Dalzell's theorem, even if they are not doing research on the golden section. Obviously, in science it is desirable that data are not presented only in the way that appears most favourable to the hypothesis being put forward. Researchers concerned with predicting proportions should be aware of this issue, and consider providing corresponding data for the odds. A sufficiently precise theoretical model should be able to give equally persuasive explanations of the data regardless of the form in which they are presented.


The comments of Chris McManus were extremely helpful, as was a Conversation with Sid Segalowitz.

(*.) Requests for reprints should be addressed to John G. Benjafield, Brock University Department of Psychology, St. Catharines, Ontario, L2S 3A1, Canada (e-mail:


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                  Comparison of P/(P + N) and N/P ratios
 P   N  [frac{P}{P + N}] [frac{N}{P}] \[frac{P}{P + N}]-.618  0 100       0.0        [infty]               0.618
 10  90       0.1        9.0                   0.518
 20  80       0.2        4.0                   0.418
 30  70       0.3        2.333                 0.318
 40  60       0.4        1.50                  0.218
 50  50       0.5        1.00                  0.118
 60  40       0.6        0.667                 0.018
 70  30       0.7        0.429                 0.082
 80  20       0.8        0.250                 0.182
 90  10       0.9        0.111                 0.282
100   0       1.0        0.0                   0.382
 P  \[frac{N}{P}]-.618  0       [infty]
 10        8.382
 20        3.382
 30        1.715
 40        0.882
 50        0.382
 60        0.048
 70        0.189
 80        0.368
 90        0.507
100        0.618
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Author:Benjafield, John G.
Publication:British Journal of Psychology
Geographic Code:4EUUK
Date:May 1, 2000
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