Groman and Worsham (
1970) presented observers with two cards slanted differently in depth (one to each eye) to determine the nature of the integrated perception of slant. The perceived slant was very close to the arithmetic mean of the slant to each eye. Averaging of inclination was also investigated in Miller and Sheldon (
1969) along with intuitive averaging of line length. Their six-item ensembles (simultaneous presentation) contained lengths suitable to produce means of 20 to 185 cm viewed from approximately 5.2 m, resulting in means of about 2° to 20° visual angle. In contrast to Weiss and Anderson (
1969), instructions to the observers made reference to the method of computing an arithmetic mean. Three different comparison lines (moduli) were available during judgment (30 cm, 91 cm, 152 cm: between observers), and group Stevens exponents were 1.14, 1.00, and 1.04, respectively (as expected, the moduli at the extremes of the range produced exponents that deviated from unity; see Engen & Ross,
1966). Individual observer exponents ranged from 0.84 to 1.19. For inclination, ensembles of six lines with angles between 0° and 90° were used (0° = horizontal). Mean inclinations of the ensembles were between 10° and 80°. The individual exponents were more variable than the length exponents (range = 0.68–1.30), with a group exponent of 1.0, which is consistent with the linearity of inclination judgments for single items reported by Stevens and Galanter (
1957) when taken over a range less than 90°. The scaling of physical-to-perceptual angle has a history of inconsistent results. Some of these can be traced to stimulus presentation confounds such as orientation of the presented angles or alignment with context edges, reference/canonical angles, and anchoring effects (see Beery,
1968; Fischer,
1969; Jastrow,
1892; Judd,
1899; Pratt,
1926; Smith,
1962; Weene & Held,
1966, for history and development). Stanley (
1974) found a tendency toward overestimation of the average of angle pairs containing a small angle and underestimation of averages as the size of the individual angles increased (component angles ranged from 10°–70°). This is consistent with Maclean and Stacey's (
1971) result for angles judged singly and recent work by Nundy, Lotto, Coppola, Shimpi, and Purves (
2000), who also found under- then overestimation in three psychophysical tasks. Note that such a nonveridical perception seems curiously nonnormative; however, the goal of the Nundy et al. article was to show that perception is in fact statistically valid. That is, the physical angle in the environment that likely resulted in a small perceived angle was likely larger than its projection on the retina (or, mutatis mutandis, the reverse for large perceived angles). This is (par excellence) one of Gigerenzer's “good errors.” Notably, the perceived angle represents something akin to the average value of the physical angles in the environment that gave rise to that perception (see figure 6 in Nundy et al.,
2000, for an amazing correspondence between behavior and natural scene statistics). This is an intuitive average value on an extended temporal ensemble and, although nonnormative, full of survival value.