STUDY OF THE BISECTION OPERATION

STUDY OF THE BISECTION OPERATION Sergio Cesare Masin University of Padua, Italy ABSTRACT Using the bisection procedure subjects divided initial sensory intervals in several intervals. The psychophysical function varied with the initial interval. Subjects also partitioned one initial interval with the instruction to equalize sensory differences or sensory ratios. The psychophysical function depended on the instruction. It is shown that these results may be reduced to a single psychological operation. Two values x and y of a sensation define the sensory interval (x, y). With x < y, this interval is bisected by the sensory value z = δ x + (1 - δ) y [1] where δ is a real weight varying between and 1. In bisection scaling with subjects instructed to set z x = y z, the experimenter sets δ at.5. However, empirical tests show that δ.5 (Gage, 1934; Marks, 1974, p. 252; Masin, 2). The following study explored how δ influenced the psychophysical function. Method STUDY 1 Subjects. Twelve university students took part in the study to fulfill a course requirement. Stimuli. Stimuli were black squares with a.5-mm white outline in the middle of a frontal parallel 33-25-cm black monitor screen. Viewing distance was about 11 cm. Three initial intervals of perceived area were used. Each interval was produced by two squares whose side lengths were 1 and 2, 1 and 2, or 1 and 2 mm. The duration of squares and of interstimulus intervals was.7 sec. Procedure. Subjects bisected the initial interval, the intervals produced by this bisection, and the intervals produced by the first three bisections. For each bisection, two standard squares were shown successively followed by a variable square. By adjusting it smoothly, subjects selected one perceived area of the variable so that this area was midway between the perceived areas of the standards. Each bisection was repeated four times (one time for each different combination of two possible initial values of the variable and of two possible temporal orders of the standards, with the initial value of the variable being one of the values

of the standards). The mean of the resulting four side lengths of the variable is defined here as the individual bisection side length. Results and discussion The values ψ = 1 and ψ = 1 of perceived area were assigned to the standards with side lengths 1 and 2 mm, respectively. By Equation 1 with δ =.5, a value of perceived area was calculated for each mean bisection side length. In Figure 1a, small squares show ψ as a function of mean bisection side length. The psychophysical function was estimated by fitting a least-squares cubic polynomial through the points represented by these small squares. A solid curve depicts this polynomial. The estimated ψs for the side lengths 1 and 1 mm were 4.1 and 42.2, respectively. With these estimates, ψs for the mean bisection side lengths relative to the initial intervals produced by the squares with side lengths of 1 and 2 and of 1 and 2 mm were calculated by Equation 1 with δ =.5. In Figure 1a, triangles and circles represent these ψs as a function of mean bisection side length. The psychophysical functions are discrepant. This result confirms a previous finding reported by Stewart, Fagot, and Eskildsen (1967). Presumably due to a response bias, δ.42 for brightness (Fagot & Stewart, 197) and numerousness (Masin, 1983). The above calculation procedure was repeated for each initial interval with δ =.42. Figure 1b shows the results. Now, the discrepancy between the psychophysical functions is reduced. It may be concluded that this discrepancy occurred because it was wrongly assumed that δ =.5. STUDY 2 Garner (1954, pp. 79-8) found that subjects judge a single quantitative relation when they are instructed to equalize loudness differences or loudness ratios in two pairs of tones. Parker and Schneider (1974), Schneider, Parker, and Stein (1974), and Schneider, Parker, Farrell, and Kanow (1976) had subjects estimate loudness differences or loudness ratios in pairs of tones. If judged differences differ from judged ratios, the rank order of tone pairs obtained from difference estimates must differ from the rank order of tone pairs obtained from ratio estimates. Schneider et al. (1976) found that these orders were equal. Thus, subjects must have judged a single quantitative relation. Birnbaum and Elmasian (1977) and Schneider, Parker, and Upenieks (1982) confirmed this conclusion. However, from estimated line-length differences and line-length ratios, Parker, Schneider, and Kanow (1975) found that the rank order of line pairs from difference estimates differed from the rank order of line pairs from ratio estimates. Schneider and Bissett (1988) confirmed this finding for length, area and volume. These authors concluded that subjects can judge different quantitative relations when subjects can imagine sensations to be contained one in the other. For a loudness interval (x, y), Garner s (1954) finding shows that in selecting a loudness value z b such that z b - x = y - z b and a loudness value z r such that z r / x = y / z r, subjects set z b = z r. For perceived area, Schneider and Bissett s (1988) results imply that z b z r. The following study explored this possibility.

ψ 1 5 Initial side lengths (mm) 1-2 1-2 1-2 (a) ψ 1 5 5 1 Initial side lengths (mm) 1-2 1-2 1-2 δ =.5 15 2 δ =.42 (b) 5 1 15 2 Figure 1. Value ψ of perceived area calculated by Equation 1 with δ =.5 (top) or δ =.42 (bottom) as a function of mean bisection side length, for each of three initial intervals. Different symbols indicate the side lengths for these intervals. Solid curves represents cubic polynomials fitted through the points represented by small squares. Method Subjects. Thirteen additional university students took part in the study to fulfill a course requirement. Stimuli and Procedure. Stimuli and presentation of stimuli were the same as those in Study 1 except that only the initial interval produced by the squares with side lengths 1 and 2 mm was used. There were two sessions. The procedure for one session was that used in Study 1. That is, subjects were instructed to equalize differences of perceived area. In the

other session, the procedure differed from that of Study 1 only for the instructions. For each bisection, subjects were asked to select an area of the variable so that the number of times this area was contained in the larger standard was equal to the number of times the area of the smaller standard was contained in the variable. That is, subjects were instructed to equalize ratios of perceived area. The session with ratio instructions was the first for six subjects and was the second for seven subjects. Separate analyses of results for these groups of subjects showed that results were substantially independent of the order of sessions. Results Figure 2a shows ψ as a function of mean bisection side length. By the above calculation procedure with δ =.42, ψs were calculated for each mean bisection side length obtained from difference instructions (filled circles) and from ratio instructions (empty circles). Error bars represent one standard error above and one below the corresponding mean. These results confirm the above prediction that z b z r. Discussion Functional measurement confirms the averaging operation encoded in Equation 1 (Carterette & Anderson, 1979; Weiss, 1975). With δ set at.42 to compensate for a presumable response bias, the results of Study 1 also confirm this averaging operation. The different effects of difference and ratio instructions found in Study 2 agree with the results of Parker et al. (1975) and of Schneider and Bissett (1988). For perceived area, the following interpretation of the bisection operation shows that these effects may depend only on the averaging operation. Bisections with difference instructions involve one average and bisections with ratio instructions involve two successive averages. For bisections with ratio instructions, subjects initially select a variable area equal to a weighted average of the areas of the standards. Since they can imagine one area contained in another area (Piaget, 1961), subjects evaluate that the number of times the smaller standard is contained in the initially selected area is greater than the number of times the initially selected area is contained in the larger standard. Since this evaluation implies that the initially selected area should be closer to the area of the smaller standard than it currently is, subjects select a final variable area equal to a weighted average of the initially selected area and of the area of the smaller standard.

1 ψ 5 δ δ =.42 (difference) =.42 (ratio) (a) 1 5 1 15 2 ψ 5 (b) 5 δ =.42 (difference) δ =.6 (ratio) 1 15 2 Figure 2. Value ψ of perceived area as a function of mean bisection side length. Values of perceived area were calculated by Equation 1 with δ =.42 (top) and with δ =.42 or.6 (bottom). These results were obtained from bisections with difference instructions (filled circles) or with ratio instructions (open circles). Two successive averages imply that δ for ratio instructions is larger than δ for difference instructions. Figure 2b shows ψ for difference instructions calculated with δ =.42 (filled circles), and ψ for ratio instructions calculated with δ =.6 (open circles), as a function of mean bisection side length. The solid curve depicts a least-squares cubic polynomial fitting all data points. Essentially, now the psychophysical functions coincide. Thus, we may conclude that the distinct operations presumably involved in equalizing sensory differences or sensory ratios may be reduced to the single operation of averaging.

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