Perceptual and Conceptual Factors in Distortions in

Copyright 1989 by the American Psychological Association, Inc. 0096-3445/89/S00.75

Journal of Experimental Psychology: General 1989, Vol. 118, No. 4, 387-398

Perceptual and Conceptual Factors in Distortions in Memory for Graphs and Maps Barbara Tversky and Diane J. Schiano Stanford University We propose that representations of visual stimuli are a consequence of both perceptual and conceptual factors that may be revealed in systematic errors in memory. Three experiments demonstrated increased (horizontal or vertical) symmetry in perception and memory of nearly symmetric curves in graphs and rivers in maps. Next, a conceptual factor, an accompanying description biasing toward symmetry or asymmetry, also distorted memory in the expected direction for the symmetric descriptions. In the two final experiments, we investigated conceptual factors in selection of a frame of reference. Subjects remembered lines in graphs, but not in maps, as closer to the imaginary 45° line. Combined with earlier research, this suggests that the reference frame for map lines is the canonical axes and for graph lines, the imaginary 45° line.

Researchers of perception by people and by machines alike agree that a promising way to conceptualize perceptual representations is in terms of structural descriptions (e.g., Biederman, 1981; Leeuwenberg, 1982; Marr & Nishihara, 1978; Minsky, 1975; Palmer, 1975; Pinker, 1984; Rock, 1983; Ullman, 1984; Witkin & Tenenbaum, 1983; also, Carmichael, Hogan, & Walter, 1932). Although no complete formulation for generating structural descriptions has been developed, there is agreement that inputs both from stimulus per se, bottom-up information, and from general knowledge, topdown information, are incorporated into structural descriptions that allow recognition and categorization of objects and scenes. The bottom-up processes can provide uninterpreted representations of objects that have little depth information; more complete representations of appearance and interpretation depend on top-down information. The bottom-up processes and certain top-down processes are probably performed on almost any visual stimulus, but other top-down processes are probably selectively applied. Top-down processes are more flexible and may utilize sequences of elementary operations. Termed visual routines by Ullman (1984), these are designed to ascertain particular information, such as indexing specific shape properties, tracing figure boundaries, and marking locations of elements with respect to reference points. Visual routines provide an alternative way to conceptualize what have been termed perceptual organizing principles by traditional students of perception. Pictures not only convey visual information directly; they also convey symbolic information. Some visual routines, such

as selection of a frame of reference, may depend on conceptual as well as perceptual processes. Maps and graphs are ideal stimuli for studying the separate and joint effects of these factors. On the one hand, maps and graphs can be made very simple visually and very similar to each other. On the other hand, they can convey either simple or complex conceptual information and quite different information in maps and graphs. Last, they are familiar and available. Some of the recent work in distortions in perception and memory of maps and environments has been discussed in the analysis of perceptual processing. Although there was a flurry of early research in which accuracy of inference from different types of graphic presentations was compared, there has been little recent psychological work on systematic errors in graphs (Poulton's 1985 study is a notable exception). Some statisticians, however, have been acutely aware that graphs may be misperceived and have produced remarkable examples of misperceptions (e.g., Berlin, 1973; Cleveland, Diaconis, & McGill, 1982; Cleveland & McGill, 1985; Kruskal, 1975; Tufte, 1983; Wainer, 1980). Cleveland and McGill (1985; Cleveland, 1985), both statisticians, developed what they called a perceptual analysis of graphs. Their analysis rests on the accuracy, determined by the psychophysics (size of exponent in power law) of the different judgments (e.g., line length, line height, angle) required to compare values represented in graphs. Maps and graphs, like music notation and architectural drawing, belong to different visual symbol systems with different syntax, semantics, and pragmatics (see, e.g., Goodman, 1968). The same line that conveys the x axis in a graph could represent a road on a map. Similarly, the same curve symbolizing a function on a graph could symbolize a river on a map. If the same visual stimulus is distorted similarly in both maps and graphs, then the distortion is most likely a consequence of perceptual factors. If, however, the same visual stimulus is distorted one way in maps and a different way in graphs, then the distortion is most likely a consequence of differing conceptual factors invoked by the two graphic symbol systems. Our experiments demonstrated systematic errors of memory attributable to both perceptual and conceptual

This research was supported by National Science Foundation Grant 1ST 8403273 to Stanford University. Preparation of the article was aided by Air Force Office of Scientific Research Grant 89-0076 to Stanford University. We are grateful to Seonghee Hong, Steve Palmer, James Pomerantz, and E. C. Poulton for helpful comments on an earlier version of the article. Correspondence concerning this article should be addressed to Barbara Tversky, Department of Psychology, Building 420, Stanford University, Stanford, California 94305.

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factors. The first two experiments showed increased symmetry in memory for curves, appearing as functions in graphs and as rivers in maps, and a third experiment showed increased symmetry in perception as well. The next experiment showed that biasing verbal descriptions can enhance or counter this effect. The final two experiments showed systematic changes in slopes of lines representing graph relations but not in lines representing roads or paths on a map. A single simple principle underlies both these errors and others as well (Goldmeier, 1982; also, Bartlett, 1932; Postman, 1954; B. Tversky, in press): to quote and then paraphrase Goldmeier, "An almost perfect, good, or typical pattern is perceived and coded as a variant of the good or typical pattern" (p. 83) and is consequently remembered as better or more typical than it actually was. This may be reminiscent of the earlier assertions of Gestalt psychologists (see Riley, 1962; and Zusne, 1970, for reviews) that memories tend toward good figure (pragnanz) over time. Those claims did not stand up to experimental test: Some memory changes observed have not been toward good figure but, rather, have been away from good figure; the changes do not always progress over time. The claim here is that some distortion may occur in the perceptual processes establishing a representation or structural description and that these distortions are maintained in memory. The claim is not for all figures but only for figures slightly distorted in encoding and for particular encoding processes. The reasons for encoding by symmetry and by frame of reference are given in turn.

Perceptual Factors: Symmetry Almost without exception, perceptual theorists have pointed to the importance of symmetry in perceptual organization (e.g., Corballis, 1976; Garner, 1974; Goldmeier, 1972, 1982; Hochberg, 1978; Howard & Templeton, 1971; Julesz, 1971;Kanizsa, 1979;Marr&Nishihara, 1978; Palmer, 1982; Pomerantz & Kubovy, 1986; Rock, 1983) in distinguishing figures from ground and in identifying figures as well. Not all symmetries are easily detected (e.g., Attneave, 1982; Kanizsa, 1979; Pomerantz & Kubovy, 1986; Rock, 1973), nor is symmetry necessarily the primary force in perceptual organization (Kanizsa, 1979). In a series of elegant demonstrations, Rock (1973) showed that people are most sensitive to symmetry around the perceived vertical axis. Symmetry around a horizontal axis is also perceived fairly readily in many stimuli, but symmetry along other axes is not. Consider the most familiar things in the world around us: people, plants, animals, and the objects designed to serve them. Many are not only symmetric but also bilaterally symmetric around a vertical axis, and so symmetry is an ecologically valid clue to "figureness" for many biological and manufactured objects (Gardner, 1979; Weyl, 1955). Symmetry, especially bilateral vertical symmetry, is readily and rapidly detected (Barlow & Reeves, 1979; Carmody, Nodine, & Locher, 1977; Chipman & Mendelson, 1979; Goldmeier, 1972) but only, of course, within limits of visual acuity. Even 4-month-old infants are sensitive to (vertical) symmetry

(Bornstein, Ferdinandsen, & Gross, 1981), which indicates that if symmetry detection is not innate, it is acquired or matures at a very early age. Yet, in the real world, people are not often presented with perfectly symmetric objects, either because the objects are viewed at an angle, however so slight, or because the objects themselves are nearly but not perfectly symmetric. Human faces provide an interesting example. They are in actuality quite asymmetric (see, e.g., Sackeim, Gur, & Saucy, 1978) but are nevertheless regarded and encoded as symmetric stimuli. Because people often view even symmetric objects off center, it makes sense to code nearly symmetric objects as symmetric. Just as in shape constancy, in which people "see" tilted quarters as round and tilted books as rectangular, people also seem to "see" slightly off-center views of faces or flowers as symmetric. Thus a figure that is nearly symmetric around a vertical or horizontal axis is likely to be perceived and represented as more symmetric than it is. Empirical evidence for this claim comes from studies by Freyd and Tversky (1984) demonstrating a symmetry bias in similarity judgments and in reaction time. In one study, subjects rated a more symmetric variant of a standard as more similar to the standard than a less symmetric variant that was physically equally different. In another study, subjects were faster to select the same figure when the other alternative was more asymmetric than the standard than when the other alternative was more symmetric than the standard. Both of these effects occurred for nearly symmetric standards, but the symmetry bias did not appear for less symmetric standards. Does the symmetry bias persist in memory? In other words, will nearly symmetric figures be remembered as more symmetric than they actually were? In the first experiment, we tested that question in maps and graphs. Students studied either maps or graphs, including some with slightly asymmetric curves, that portrayed rivers in the maps and functions in the graphs. In order to induce a natural comprehension attitude toward the maps and graphs, students were told that they would either be asked questions about the content of the maps or graphs or be asked to draw the curves or lines. The data of interest were the drawings of the curves, and the prediction was that these would be more symmetric than the original stimuli. The perceptual analysis predicts errors toward symmetry in maps and graphs alike. Another possible hypothesis, deriving from the likelihood of finding symmetric curves in graphs versus maps, would predict errors toward symmetry in graphs but not in maps. Many physical or analytic processes yield normal or other symmetric curves, and so many of the curves that we observe in graphs in science or statistics texts are in fact symmetric. Thus because graph curves are frequently symmetric, the medium of graphs may raise an expectation for symmetric curves. Because rivers on maps come in all sorts of shapes and there is no symmetrizing physical process for rivers, no expectation of symmetry should exist for maps. If we found errors toward symmetry for graphs but not for maps, then the expectancy hypothesis would be supported; if there were errors toward symmetry for both maps and graphs, then the perceptual analysis would be supported.

DISTORTIONS IN MEMORY

Experiment 1 Method Subjects A total of 160 subjects, Stanford University students fulfilling a course requirement, participated in this study. Half of the subjects were randomly assigned to the graph condition and the remaining half to the map condition. They were run in groups of about 15-20 people in a session including this and other brief experiments.

Stimuli Separate experimental booklets were compiled for the graph and map conditions. Each booklet contained an instructional page, a practice (filler) stimulus, and then four critical stimuli and four additional filler items, in random order. After each stimulus was a blank colored sheet of paper followed by an answer page. For the graph condition, each critical figure consisted of an inverted U-shaped curve, oriented along the horizontal or vertical axis (72 x 72 mm), and somewhat skewed to the right or to the left (an example of the critical curves is provided in Figure 1). There were two types of curves, one smaller with tails and the other larger and truncated. Thus there were eight possible curves split into two sets of four so that in each booklet, half of the critical stimuli were of each type, skewed in each direction, and oriented each way. Half of the subjects had each type of booklet. Axis labels were chosen to suggest plausible relations such as those commonly depicted in introductory social

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science texts (e.g., IQ Scores x Frequency; Family Size x Frequency). The answer pages for the critical stimuli contained blank axes, indicating that the subject should draw the curve. Filler stimuli showed sinusoidal curves, horizontally or vertically oriented. One (randomly chosen) filler stimulus was followed by blank axes for a drawing response. The rest of the fillers were followed by a question about the content portrayed in the graph, in order to ensure that the subjects were attending to and conceptually encoding the entire graph. Subjects were asked about the nature of the relation of the variables (i.e., "What kind of relationship is expressed by the function?"), for a brief description (i.e, "Describe the graph in your own words"), or for a specification of the axis labels (i.e., "Which variables are related by the function?"). Stimuli for the map condition were identical to those in the graph condition, except for labeling (see Figure 1). Instead of axis labels, street names were printed along the axes for each figure (e.g., "Main Street" and "Route 1"). Subjects were instructed to view each of the axes as streets (running north-south and east-west) and the curve as a river running within the area bounded by the streets. For the filler items, subjects were asked for the direction the river was flowing, for a brief description of the map, or for specification of the street names.

Procedure After the booklets were distributed, instructions were read aloud. Subjects were given 5 s to study each stimulus and unlimited time to answer. For the critical stimuli, subjects were instructed to draw only the curve or river and not to label the axes. The entire procedure took about 15 min. At the end of the experiment, subjects were asked to indicate what the experimental stimuli looked like to them. About 95% of the subjects in the graph condition reported viewing the stimuli as graphs; about 65% of those in the map condition reported viewing their stimuli as maps. Only the data from subjects who reported viewing the stimuli as instructed were included in the data analysis.

Results and Discussion

Figure 1. Examples of critical curves in the first symmetry study. (Both shapes appeared skewed both left and right and oriented horizontally and vertically.)

Three independent raters rated the response drawings for symmetry. The raters in this and the other symmetry studies were undergraduates or secretaries who did not know the purpose of the experiments. They were instructed to use a 5point scale to relate the symmetry of the response drawing to that of the stimulus curve (1 = drawing much more asymmetric than stimulus; 2 = slightly more asymmetric; 3 = identical symmetry; 4 = slightly more symmetric; 5 = very much more symmetric). The average of the three ratings for each stimulus was used as the raw data for this study. Scores greater than 3, then, indicate distortion toward symmetry, and scores less than 3 indicate distortion toward asymmetry. In order to evaluate the agreement among the raters, 3 x 3 tables were constructed for each pair of raters; in each table the entries were the number of cases that each judge called more symmetric than, equally symmetric to, or less symmetric than the original. Significant chi-squares indicated a high proportion of entries along the diagonal and thus high agreement by the judges. The chi-squares were all highly significant (362, 481, and 231; all ps < .001), and the proportion of agreement between judges in each pair was high {.71, .78, and

.64).

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Significant distortion in the direction of symmetry was present for both graphs (M - 3.52), #79) = 8.27, p < .001, and maps (M = 3.50, t(79) = 7.39, p < .001. Moreover, significant distortions in the direction of increased symmetry were found for all but one of the eight curves for each type of stimulus. The exception was the vertically oriented, leftskewed, small untmncated curve, which subjects repeatedly reported as having unique figural properties (it was most commonly seen as a "nose"). An analysis of variance (ANOVA) revealed no differences in magnitude of distortion between maps and graphs, F(l, 158) = 0.04, p > .05. The actual difference between maps and graphs was .02 on a scale from 1 to 5, or only 4% of the difference between rated map drawings (3.5) and equal symmetry (3). There were no effects of or interactions with orientation, F(l, 158) = 0.01, p > .05, or direction of skew, P(l, 158) = 1.26, p > .05. Thus memory for slightly asymmetric curves presented in both graphs and maps was significantly distorted toward symmetry. Despite previous work showing that vertical symmetry is easier to detect than horizontal, no effect of axis of orientation was found for either graphic symbol system. There was also no effect of direction of skew. Moreover, there were no detectable differences between graphs and maps in the magnitude of the distortion. The same measure that produced significant distortions toward symmetry for both maps and graphs failed to produce a difference between them; thus if there was such a difference, it was quite small in relation to the overall tendency toward symmetry. Because there are good reasons to expect graph curves to be symmetric but no reason to expect rivers to be symmetric, this finding is not consistent with expectancy as the origin of errors toward symmetry. Rather, symmetry errors in visual memory seem to be a consequence of perceptual processing. One early perceptual process is the search for figures in grounds. Symmetry is a powerful cue to figureness and is rapidly perceived. Once a figure is segregated, symmetry is a cue to the identity of the figure as well. Previous work demonstrated that figures close to symmetry are perceived as more symmetric than they actually are. Our Experiment 1 extended this phenomenon to memory. The tendency to remember nearly symmetric figures as more symmetric is not specific to a particular graphic medium but, rather, seems to be content free and general to other kinds of pictures, even where there is no expectation of symmetry. Thus it appears to be a consequence of perceptual rather than conceptual organization. The curves used in this experiment were, on the whole, quite close to symmetric. Freyd and Tversky (1984) found a symmetry bias in judgment of nearly symmetric figures, but not of less symmetric figures. It is possible that nearly symmetric figures are coded as symmetric, and thereby remembered as more symmetric than they were, but that less symmetric figures are less likely to be coded as symmetric and therefore less likely to be distorted in memory. In the second experiment we replicated the graph condition of Experiment 1, using less symmetric curves. Because there were no differences between maps and graphs and because the map stimuli were less convincing, the map condition was not replicated.

Experiment 2 Method Stimuli As in Experiment 1, there were four types of curves, formed from combining direction of skew and truncating or not truncating the curve. The curves were made less symmetric by increasing skew. Curves were oriented vertically or horizontally (see Rgure 2). The filler stimuli were the same as in the first experiment, and the booklets were composed the same way.

Subjects Ninety-two students at Oberlin College participated in the experiment for course credit. Procedure The procedure was identical to that of Experiment 1.

Results and Discussion As in Experiment 1, the drawn curves were scored as equally symmetric to, more symmetric than, or less symmetric than

Figure 2. Examples of critical curves in the second symmetry study. (Both shapes appeared skewed both left and right and oriented horizontally and vertically.)


the original curves by three new independent raters on a scale from I (drawing much more asymmetric) to 5 (drawing much more symmetric). These raters also rerated the graph drawings from Experiment 1 for comparison. Interrater agreement was again high, with significant chi-squares (168, 89, and 113; all ps < .001) and high proportions of agreement (.73, .63, and .65). Both the slightly asymmetric curves of Experiment 1 and the more asymmetric curves of Experiment 2 drawn from memory were rated as significantly more symmetric than the originals: The mean ratings for graphs were 3.32, t(69) = 4.67, p < .001, for Experiment 1 stimuli and 3.25, t(69) = 3.99, p < .001, for Experiment 2 stimuli. The difference between the stimuli in the two experiments was not significant; moreover, it was small in relation to the significant tendency toward symmetry in remembered curves. Again, subjects remembered curves in graphs as more symmetric than they actually were; this result replicated the results of Experiment 1 for curves that were less symmetric. There was neither less nor more distortion for the less symmetric curves than for the more symmetric curves.

Experiment 3 Two experiments have shown that subjects remembered fairly symmetric curves in graphs and maps as more symmetric than they really were. Given Freyd and Tversky's (1984) finding that subjects judged a more symmetric figure to be more similar to a fairly symmetric standard than a less symmetric figure, part or all of the tendency toward greater symmetry in memory may occur in perception. In order to test this, we asked a new group of subjects to copy the curves and then asked an old group of judges to rate the copied figures for symmetry.

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Results and Discussion The same judges who scored the graphs of Experiments 1 and 2 reported in Experiment 2 were asked to score the graphs of the copy condition. As before, chi-squares between pairs of judges were significant (114, 84, and 89; ps < .001) and the proportions of agreement were high (.68, .65, and .66). The mean score for the copy condition was 3.28, t(49) - 5.19, p < .001, which, again, was a significant deviation in the direction of increased symmetry. This score, however, was not significantly different from the score of the memory condition for the same graphs (3.32). The difference between the copy and memory conditions was only .04, or 14% of the significant difference between the copy condition and equal symmetry. Both graphs copied and graphs drawn from memory tended to be more symmetric than the original, slightly asymmetric graphs. This is in spite of the fact that the graphs drawn from memory were more in error than those copied; that is, the deviations in form were quite large for the drawn curves and small for the copied curves. This implies that the symmetry effect found in memory occurred in perception, with no additional tendency toward symmetry in memory of these slightly asymmetric stimuli. The finding of an equal trend toward symmetry in both perception and memory could be interpreted differently: as a response bias toward drawing graphs more symmetrically. This possibility is weakened by previous findings. Freyd and Tversky (1984) found symmetry bias in perception in two different tasks, both of which required perceptual matching rather than drawing. Similarly, Goldmeier (1982) reported increased symmetry in recognition as well as in reproduction tasks. In the next study, we attempted to weaken this possibility even more by inducing subjects to draw the same curves more or less symmetrically, depending on the accompanying description.

Method Stimuli The four critical graphs from Experiment 1 served as stimuli.

Subjects Fifty Stanford undergraduates served in this experiment

Procedure Subjects were given booklets with the four critical stimuli and a cover page containing the following instructions, which were read aloud: This is a very simple task. On each of the following 4 pages, you will see a graphed curve at the top of the page, and empty axes at the bottom of the page. Your task is simply to copy the top curve as accurately as you can in the axes presented below. You may be wondering why we are asking you to do this. We asked another group of people to draw these graphs from memory, and want to compare the graphs drawn from memory to the copied graphs. Any questions?

Conceptual Factors: Descriptions Experiment 4 The previous experiments have shown that slightly asymmetric stimuli, whether in maps or graphs, tend to be perceived and remembered as more symmetric than they actually are. Such stimuli are ambiguous and thus may be susceptible to conceptual factors that bias their interpretation. In a classic demonstration of this phenomenon, Carmichael et al. (1932) presented ambiguous line drawings with one of two interpretations to subjects to remember. O-O, for example, was described as "barbells" to one group of subjects and as "eyeglasses" to another group. Subsequent drawings from memory were distorted in the direction of the labels. In this experiment, subjects were presented with graph curves and descriptions that indirectly emphasized either the symmetry or the asymmetry of the curves. Subsequent drawings of the curves were expected to be distorted in the direction of description. This experiment also provided a test of the

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response bias interpretation of the first three experiments. If subjects produced drawings of varying degrees of symmetry for the same curves, then it would be difficult to argue that subjects' drawings were more symmetric only because they could not draw otherwise.

Method Subjects. A total of 80 subjects, Oberlin College students fulfilling a course requirement, participated in this study. Half of the subjects were randomly assigned to receive asymmetric descriptions and half to receive symmetric descriptions. They were run in groups of about 15-20 people in a session containing this and other brief experiments. Stimuli, The stimuli, two filler-labeled graphs followed by the critical curve (also labeled) were presented in booklets with answer pages and with colored blank pages between the stimuli. The critical stimuli were taken from those of Experiment 1; they were horizontally oriented and slightly asymmetric, skewed to the right for half of the subjects and to the left for the other half. Half of the subjects for each skew direction read a note below the critical stimulus biasing toward asymmetry: "Note that the function rises sharply (slowly; depending on direction of skew) and falls slowly (sharply)." The other subjects read a note below the critical stimulus biasing toward symmetry: "Note that the function rises smoothly and falls smoothly." The filler stimuli were graphs of linear functions taken from Experiment 5 with slopes of 27.5" and 62.5°. They were also accompanied by notes, the former reading, "Note that the function rises shallowly," and the latter, "Note that the function rises steeply." These were placed in random order in the booklets. For one of the fillers, selected at random, subjects were asked to draw the graph; for the other, they were asked a verbal question about the content of the graph. As usual, subjects were asked to draw the critical stimulus. Procedure. The procedure was the same as that of Experiments 1 and 2.

Results and Discussion Critical drawings were scored by the three judges of Experiments 2 and 3. Chi-squares evaluating interrater agreement were significant (25, 28, and 28; all ps < .001), and the proportions of agreement were high (.69, .66, and .64). Median scores were submitted to an ANOVA, which yielded a large effect of description, F(1, 76) = 7.13, p < .01. Only the curves with descriptions emphasizing symmetry were significantly distorted in memory, toward symmetry (average score = 3.84), f(39) = 6.30, p < .01. Moreover, each of the curve types accompanied by symmetric descriptions was significantly distorted toward symmetry in memory. The drawings of curves with descriptions emphasizing asymmetry were not significantly distorted from the original drawings (mean score = 3.30), t(39) = 1.95, p > .05, though there was a trend toward symmetry. The same curve was remembered differently, depending on the description accompanying the curve. Specifically, nearly symmetric curves were remembered as more symmetric when accompanied by a description emphasizing the symmetry of the curve than when accompanied by a description emphasizing the asymmetry of the curve. Those curves accompanied by an asymmetric description were not significantly distorted; however, there was an insignificant trend toward symmetry.

From the previous studies, we can expect that without a description, there would have been a significant memory distortion toward symmetry. Nevertheless, we can conclude that this conceptual factor, an accompanying biasing description, distorts memory for graphs, presumably by affecting the encoding of the graph curve at presentation. Because there was an insignificant tendency for the curves with asymmetric descriptions to be drawn more symmetrically than the originals and because the magnitude of that insignificant effect was close to the magnitude of significant effects in Experiments 2 and 3, we cannot rule out response bias as a possible contributing factor to the symmetry effect. However, response bias cannot account for all of the effect, first because subjects given asymmetric descriptions produced considerably more extreme drawings than did subjects given symmetric descriptions, and, second, because symmetry effects have been found in recognition tasks in perception (Freyd & Tversky, 1984) and in memory (Goldmeier, 1982), in which response bias is not an issue.

Conceptual Factors: Reference Frames In picture perception and comprehension, once figures are segregated, they are organized in relation to one another and in relation to a frame of reference. The term cognitive reference frame includes what others have termed cognitive reference points or anchors; it is a more general term. In both concrete and abstract contexts, ordinary stimuli are judged and perceived to be closer to cognitive reference points than vice versa (e.g., Rosch, 1975; A. Tversky, 1977). Magenta is judged to be closer to red than red to magenta, and Poland is perceived to be more similar to the USSR than the USSR to Poland. Cognitive reference points, stimuli that are salient by virtue of being larger, more colorful, more prototypical, or more familiar, draw other stimuli toward them in judgment. This is also so for figures in perceptual space, in judgment, and in memory. Dots are judged closer to reference dots (Coren & Girgus, 1980); dots, blobs, roads, and land masses are remembered as closer to reference points or frames (Nelson & Chaiklin, 1980; Taylor, 1961; B. Tversky, 1981); and ordinary buildings are remembered as closer to landmarks (Hirtle & Jonides, 1985; Sadalla, Burroughs, & Staplin, 1980). Cognitive reference frames are not fixed and immutable, as Holyoak and Mah (1982) demonstrated. They asked subjects to imagine themselves on either the East or the West Coast of the United States and then to judge relative distances between pairs of cities, such as Denver and Salt Lake City or Pittsburgh and Indianapolis. On the whole, from the Eastern perspective, the Western distances were judged smaller than from the Western perspective, and vice versa, confirming what New Yorker cartoons long ago caricatured. The finding that selection of reference points is flexible suggests that either perceptual or conceptual factors may influence their selection. In pictures, one very natural reference frame is provided by the sides of the page. In addition to literally framing the picture, the sides of a page are conceptually (and often retinally) horizontal and vertical lines at right angles. Horizontal and vertical lines have a special status in perception (e.g.,

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dark, 1973; Howard & Templeton, 1971); visual acuity is greater there, and judgments are more likely to be correct. In maps and graphs, horizontal and vertical lines have an added conceptual status: In maps they usually correspond to the canonical directions north-south and east-west, and in graphs they usually correspond to the canonical axes. Thus the horizontal and vertical are natural candidates for reference frames for both perceptual and conceptual reasons in both maps and graphs. For simple x-y graphs with linear functions, however, there is another candidate for a reference frame: the imaginary 45° line. This imaginary line is the identity line, where x = y, and deviations from it may be meaningful in interpreting certain kinds of graphs. Where predicted values are on the x axis and obtained on the y axis, data along the 45° line indicate an outstanding fit. In many cases, the imaginary 45° line may simply be a boundary between relatively large and relatively small slopes. For inflation, for example, a high slope would be cause for alarm, whereas for growth, a high slope may be cause for celebration, and low slopes for each would elicit the opposite response. Like actual lines, imaginary lines can act as a frame of reference and induce distortion, at least in the case of lines at assorted angles and imaginary vertical and horizontal lines (Bouma & Andriessen, 1968). Just as for symmetry of curves or rivers, there may be different expectations for line slopes in maps or graphs. The prototypical flat American city has a grid pattern of roads, inducing an expectation for lines running parallel to the sides of the page, usually corresponding to the canonical directions. In graphs, however, lines rarely are perfectly parallel to the axes and are more often sloped. No particular angle, however, seems favored in graphs. If frequency in the world induces expectations and expectations induce distortions, then slanted lines should be pulled to vertical and horizontal in maps but not in graphs. In previous research (B. Tversky, 1981), land masses and roads were in fact remembered as oriented more closely to the canonical directions than they actually were. Because such distortion also occurred for visual blobs not perceived as maps, this finding suggests that perceptual organizing factors are at work, but it does not rule out conceptual organization as an additional factor. Memory for graphs provides an interesting test case. If lines in simple x-y graphs are rotated in memory toward the horizontal or vertical axes, then either perceptual or conceptual factors, or both, are responsible. If, however, graph lines are remembered as closer to the 45° line, this is support for conceptual selection of reference points in the case of graphs and indirect support for conceptual selection of them in maps as well. In the following two experiments, single graph lines or dots embedded in coordinates were presented to subjects to remember. The graphs were labeled, and the graphs were meaningful. Subjects were sometimes asked to report the content of the graphs and were sometimes asked to reproduce the lines. This induced a natural comprehension set in the subjects, although the data of interest are the line slopes. For other groups of subjects, the lines and coordinates were called maps and were labeled appropriately. The maps served as a control condition and were less convincing than the graphs.

Experiment 5 Method Subjects. Eighty Stanford University students participated in this study as part of a course requirement. Half were randomly assigned to the graph condition and the remaining half to the map condition. Subjects were run in groups of about 15-20 people, in a session including this and other brief experiments. Materials. Separate experimental booklets were compiled for the graph and map conditions. Each booklet contained a practice (filler) stimulus and then eight test stimuli and six additional filler items randomly interspersed. For the graph condition, each test figure consisted of a function line within equal-sized axes (72 x 72 mm). Axis labels were chosen to suggest plausible linear relations such as those commonly depicted in introductory social science texts (e.g., Reaction Time x List Length). Figure 3 is an example of the graph test stimuli used in this experiment. The slopes of the lines used for the test stimuli included 20°, 25°, 30°, 35°, 55°, 60°, 65°, and 70° from the origin; that is, deviating 10°, 15°, 20°, and 25° above and below the 45° diagonal. As in Experiment 1, subjects responded by free-hand drawing a function line on the response figure, which consisted of bare axes. Filler stimuli were similar in nature to the test stimuli. Slopes of the function lines in the filler stimuli were 0°, 27.5°, and 62.5° from the origin, and 20°, 35°, 55°, and 70° from the upper endpoint of the y axis (along the left diagonal). The filler stimuli were followed by a question about the content portrayed in the graph. Subjects were asked either for a brief description of the nature of the relation

Figure 3. Examples of critical lines in the first slope study. (Angles varied from 20° to 70°.)

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between the two variables (e.g., "What is the relationship between birth order and participation in dangerous sports?") or for the relation of one variable to another specified one (e.g., "Recall decreases as increases"). A blank, colored page separated each response and stimulus figure. Stimuli for the map condition were identical to those in the graph condition, except for axis labels (see Figure 3). Subjects were instructed to view each of the axes as streets (running north-south and east-west), the hatch marks on the axes as cross streets, and the function line as a bike path leading from "home," the starting position. In response questions for the filler items, subjects were asked for the direction that one would take from home to one of the destinations (e.g., "In which direction is symphony hall from home?") or were asked for the location of one destination in relation to another (e.g., "The golf course is due north of "). Procedure. Subjects in each group were given the appropriate experimental booklets and instructed as to the nature of the task. Stimulus self-presentation was timed by the experimenter at 5 s; responses, however, were untuned. Subjects were asked to respond to filler questions on the same page as the question. For test stimuli, they were instructed to draw only the function line or the bike path and not to label axes or destination points. The entire procedure lasted about 15 min. At the end of the experiment, subjects were asked to indicate what the experimental stimuli looked like to them. About 95% of the subjects in the graph condition saw the stimuli as graphs; about 80% of those in the map condition viewed them as maps. Only the data from subjects who perceived the stimuli as instructed were included in the data analyses.

Results and Discussion Using a protractor, one of the experimenters scored the drawn lines. The scores were averaged over subjects for each line slope and condition and transformed to deviations toward 45°. The means are displayed in Table 1. An ANOVA showed a significant difference between the graph and map conditions in overall distortion, F(l, 78) = 12.90, p < .001, reflecting the consistent tendency of the graph, but not the map, stimuli to be distorted toward the 45° diagonal. When we collapsed across deviations for slopes above and below 45°, significant (or marginally significant) distortion toward the diagonal was found for each graph condition: for 10° deviation, «(39) = 1.68, p < .05; for 15°, i(39) = 2.84; p < .01; for 20°, /(39) = 2.98, p < .01; and for 25°, ?(39) = 5.77, p < .001. For the map condition, none of the deviations was significant. Overall, significant distortion was found in the graph condition, r(39) = 4.73, p < .001, but not in the map condition, r(39) = 0.34, p > .05. Also, the size

of effect for maps (0.1°) was less than 5% of the size of the effect for graphs (2.2°). Although the graph data suggest that the size of the distortion increases somewhat with the slope of the line, this effect was not significant for either graphs, t(39) = 1.11, p > .05, or maps, t(39) = 1.19, p > .05. In addition, although there seemed to be a slight tendency for greater distortion in the direction of the diagonal for slopes greater than 45°, this effect did not reach significance either overall, F(l, 79) = 3.35, p = .07, or for graphs, F(l, 39) = 1.12, p> .05, and maps, F(l, 39) = 2.34, p > .05, considered independently. The same line slope was remembered differently when it appeared in a graph than when it appeared in a map. Specifically, graph lines were distorted in the direction of the diagonal for every line tested, which suggests that the imaginary 45° line serves as a reference point for encoding and remembering slopes of lines in graphs. This pattern of effects is restricted to graphs, providing evidence for conceptual factors in selection of a frame of reference. The results for the map condition are somewhat more difficult to interpret than the graph results. Previous findings would predict errors in map memory in the direction of the north-south and east-west axes. However, no evidence of significant distortion was found for the maps, and the direction of errors was fairly evenly split between the axes and the diagonal. This may reflect the more obviously artificial nature of the maps used in this experiment, which may not have been as effective in inducing a map-reading set as in previous research. Alternatively, it might reflect the fact that the task was to remember a bike route toward a destination midway between two perpendicular streets that may suggest a diagonal shortcut. So, there may have been two conflicting or alternative reference lines for the maps that canceled each other. As in the case of symmetry, the expectancy hypothesis received no support. There was no systematic tendency for map lines to be pulled toward the canonical axes, as expectancies about prototypical street maps would predict. Although there may be expectancies that graph lines have slopes, there are no expectancies about a particular angle of slope. Because nearly all the lines presented were at angles other than 90° or 0°, the expectancy hypothesis predicts that these should be remembered as sloping but not distorted in one direction or another. The critical findings here are a significant difference between maps and graphs and significant distortion of graph, but not map, lines toward the 45° diagonal. This is evidence

Table 1 Average Deviation Toward 45° of Drawn (Remembered) Lines From Presented Lines in Maps and Graphs in Experiments 5 and 6 Angles for which + denotes larger remembered angles Line angle Experiment 5 Graph lines Map lines Experiment 6 Graph lines

Angles for which + denotes smaller remembered angles

20°

25°

30°

35°

55°

60°

65°

70°

+3.4 +0.2

+1.7 -0.2

+1.0 +0.4

+1.2 -2.2

+1.1 +0.7

+2.9 +0.5

+2.4 0.0

+4.1 +1.5

+1.5

+0.1

+1.0

+1.0

+2.8

+2.6

+2.0

+2.3

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for qualitative differences in memory for virtually identical visual stimuli, depending on the class of visual symbol system to which they belong, a conceptual factor. Different visual symbol systems suggest different reference frames; when adopted, these reference frames pull lines to them in memory.

Experiment 6 In the final experiment, we replicated and extended the graph results to slightly different stimuli. The graphs in this experiment contained unlinked data points rather than lines. Students again were requested to study and remember the graphs, and we tested their memory, sometimes by asking for a verbal description and sometimes by asking for a drawing of a line summarizing the data points. Although students did not see lines, they were asked to draw the line that they thought best represented the data points, rather than drawing the actual data. To the extent that the distortion toward the diagonal observed for graphs in Experiment 5 was a robust finding, similar results were expected in this experiment. Because no distortion had been observed in the maps in Experiment 5, and because a map analog was even more problematic for dots than for lines, only graphs were included in Experiment 6.

Method Subjects. Forty subjects, Obeilin College students fulfilling a course requirement, participated in this study. Subjects were run in groups of about 15-20 people, in a session including this and other brief experiments. Materials and procedure. The materials and procedure for this experiment were essentially the same as those for the graph condition of Experiment 5. The only differences were that axis labels were omitted, and instead of a function line, each graph portrayed seven data points at 10-mm intervals. An example of the stimuli for this experiment is shown in Figure 4. Subjects studied the stimuli as in Experiment 5, and when making the drawing response, they were instructed to draw the function line suggested by the data points, not the individual points that constituted it.

Results and Discussion Using a protractor, two Oberlin undergraduates who were unaware of the purpose of the study scored the lines. In the few cases in which they disagreed, one of the experimenters decided the score. The results for this experiment are shown in Table 1, which displays the mean deviation toward 45° for each line slope. As in Experiment 5, significant overall distortion was found for the graphs, f(39) = 4.68, />< .001, reflecting a consistent error toward the 45° diagonal. When we collapsed across deviations above and below the diagonal, significant distortion in the expected direction was found for each condition: for 10° deviation, /(39) = 3.26, p < .01; for 15°, r(39) = 2.42; p = .01; for 20°, /(39) = 1.76, p < .05; and for 25°, r(39) = 2.85, p < .01. The tendency for greater distortion of slopes above than below the diagonal reached significance, F(l, 39) = 4.41, p < .05, in this experiment.

Figure 4. Examples of critical dotted lines in the second slope study. (Angles varied from 20° to 70°.)

Drawn lines representing remembered data points were systematically and consistently closer to the 45° line than in the actual graphs. This finding replicated and extended the findings of Experiment 5, indicating that the effect was robust. These data again demonstrated a conceptually based organizational effect in visual memory. The diagonal, a natural reference point in graphs, pulls lines and summarized data points toward it in memory.

General Discussion Six experiments demonstrated systematic errors toward symmetry around a vertical or horizontal axis and toward frame of reference in memory for maps and graphs. The errors are attributed to the perceptual and conceptual processes invoked in the perceptual analyses required to comprehend the stimuli. These distortions, though systematic, are subtle and easy to overlook without both an analysis of perceptual organization that points to likely processes and places and appropriately designed stimuli and tasks. In two experiments, students remembered curves in graphs and rivers in maps as more symmetric than they really were. A third experiment showed that the error toward symmetry was as strong in perception as in memory. The symmetry bias is interpreted as a natural consequence of the perceptual processing performed on visual stimuli, in which figures are dis-

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criminated from grounds. Symmetry is a powerful factor in perceptual organization. It is an excellent cue to "figureness" for many biological and manufactured objects. Perhaps because symmetric figures are often viewed off-center, we are biased to perceive nearly symmetric objects as more symmetric than they are (Freyd & Tversky, 1984); now, this symmetry bias has also been demonstrated in memory, in curves appearing in both maps and graphs. Conceptual factors distort memory for visual stimuli selectively. One case is the distorting effects of symmetric versus asymmetric descriptions accompanying the graphs. In a fourth experiment, descriptions accompanying the curves biasing either toward symmetry or toward asymmetry had the expected effect in memory for symmetric descriptions. Another case is selection of a frame of reference, for which conceptual as well as perceptual factors may operate. Once a frame of reference is selected, stimuli encoded with respect to it are remembered as closer to it than they actually were. In two experiments, subjects remembered lines in graphs, but not in maps, as closer to an imaginary 45° line than they really were. Apparently, the diagonal serves a a reference line for graphs, but not for maps. For any particular picture or scene, many reference points or frames suggest themselves. Some of these are perceptual: the location of another figure, the horizontal and vertical axes of the world, and frame of a picture. Conceptual factors may suggest other reference frames. In the case of maps, the canonical directions, north-south and east-west, are a common reference frame; typically, these coincide with the sides of the page, and so the perceptual and conceptual reference frames are correlated by convention and by design. In the case of x-y graphs with simple line functions, one reference frame is provided by the canonical axes, which again coincide with the sides of the page. Another reference frame, however, is provided by the imaginary 45° diagonal, which represents the identity line where x = y. It provides a comparison frame for graph functions that is often meaningful. Slopes above the diagonal are usually high ones, indicating rapid growth or runaway inflation; slopes below the 45° line are neither so encouraging or worrisome. Where obtained values are plotted against predicted, as is often the case in model testing, a preponderance of data at the 45° line indicates support for the theory under consideration. Although these arguments for the 45° line as a natural anchor for graphs are speculative, there does not seem to be another compelling reason for the obtained distortions for graphs and not maps. Data from previous experiments on maps support the adoption of the canonical axes as a reference frame. For our nearly identical stimuli, subjects remembered graph lines as closer to the imaginary 45° line than they actually were, but they showed no systematic distortion of map lines. Thus in order to account for processing of and memory for graphs, cognitive as well as perceptual factors must be taken into account and not simply the psychophysical factors suggested by Cleveland (1985). No systematic distortion was observed for the map lines, contrary to previous findings of distortion of roads and land masses to the nearest axis. In our experiment, the maps, in order to be identical to the graphs save labels, were not

convincing, so it is difficult to draw conclusions about maps other than that they differ from graphs in memory for line slope. Distortion of lines and blobs toward the nearest axis has been found repeatedly in memory for maps (e.g., Byrne, 1979; Moar & Bower, 1983; B. Tversky, 1981), in perceptual judgment (e.g., Bouma & Andriessen, 1968; Howard & Templeton, 1971), and in perceptual illusions (e.g., Weintraub, Krantz, & Olson, 1980). This robust finding should not be dismissed on the basis of a single weak failure to replicate. Also, our maps could have suggested use of an imaginary 45° line as a reference line shortcut, inducing a second reference line that may have been strong enough to cancel the usual effects of a horizontal/vertical reference frame but not strong enough to overcome it. In the perceptual work, deviations from the usual migration toward vertical and horizontal have been found when a second reference point was introduced (Bouma & Andriessen, 1970; Taylor, 1961). In some cases, it appears that the additional reference point itself attracts the test stimulus. What Taylor observed in 1961 is still true: It is not known how reference frames interact. Simple summation of "gravitational" effects, though elegant, will not account for the phenomena.1 Conceptual effects in grouping and consequent distortion in memory have been demonstrated previously in maps. In local environments, for example, landmarks are grouped both by similar function and by actual proximity. Once grouped together, landmarks are remembered as closer together (Hirtle & Jonides, 1985). Over the years, we have asked people which they thought was farther east, Japan or the Phillipines. Our respondents invariably erroneously report the Phillipines, presumably because of their close political association with the United States. An alternative hypothesis, the expectancy hypothesis, was raised to account for the findings. Although the idea that we see and remember what we are used to seeing is a venerable one in psychology, it did not do a good job of accounting for our results. Because graph curves are often symmetric but rivers are not, expectancy would predict a greater memory bias toward symmetry for graphs than for maps, but in fact the symmetry bias was equal for both. Because roads are likely to be in a grid pattern but graph lines are likely to be sloped, expectancy would predict that memory for roads would go toward horizontal and vertical and would predict no systematic change for sloped graph lines, which is again contrary to our findings. Some of the perceptual errors in our study may remind readers of the errors investigated by Gestalt psychologists 50 or more years ago, although our theory and methods are quite different. The Gestaltists predicted that figures would tend to go toward good form (pragnanz) over time and collected evidence for their predictions. These claims generated counter-claims, and a lively controversy marked by increasingly sophisticated research ensued, captured by Riley (1962). The 1

We tested the gravitational hypothesis directly by systematically varying relative sizes of the two visual forms to be remembered, and we found no support for one prediction of a gravitational hypothesis that larger bodies exert a greater force than smaller ones.

DISTORTIONS IN MEMORY contention that memory changes are progressive received no empirical support. Although there were systematic errors of memory, they could not be universally and unambiguously attributed to bettering form. That proved difficult to define or substantiate; for example, a small gap in a circle was remembered as larger and a large gap in a circle was remembered as smaller (Walker & Veroff, 1956). The "final word" in that controversy was pronounced by Postman (1954). He created sets of visual patterns with different underlying structures and demonstrated systematic errors corresponding to those structures. His conclusion, then, was quite similar to the general claims here, that errors are a consequence of encoding, which is guided by knowledge about patterns of organization. Different graphic symbol systems, such as maps, graphs, games, architectural and engineering drawings, and music and dance notation, lend themselves to different patterns of organization. In music notation, notes are organized by time signatures into measures and measures into passages and movements. Although they employ the same board and pieces, the games of Go and Gomoku have different rules, which explain the different mental groupings of the same pieces that players of each game make (Eisenstadt & Kareev, 1975). It is precisely groupings that distinguish chess novices from chess experts and allow chess experts to remember chess games so well (Chase & Simon, 1973). For music, natural reference frames are provided by bars and measures. In maps, these are provided by other figures and the canonical axes; in simple x-y graphs, by the axes and the imaginary 45° line. No matter what their visual representation, in order to be cognitive reference frames, they must have significance within the graphic symbol system. The origins of cognitive reference frames in a graphic symbol system may be cognitively complex, but once they are selected, a very simple thing seems to happen: Cognitive reference frames attract elements to themselves in perception, judgment, and memory. References Attneave, F. (1982) Pragnanz and soap bubble systems: A theoretical exploration. In J. Beck (Ed.), Organization and representation in perception (pp. 11-29). Hillsdale, NJ: Erlbaum. Barlow, H. B., & Reeves, B. C. (1979). The versatility and absolute efficiency of detecting mirror symmetry in random dot displays. Vision Research, 19, 783-793. Bartlett, F. C. (1932). Remembering. Cambridge, England: Cambridge University Press. Berlin, J. (1973). Semiologiegraphique. The Hague: Gautier-Mouton. Biederman, I. (1981). On the semantics of a glance at a scene. In M. Kubovy & J. R. Pomerantz (Eds.), Perceptual organization (pp. 213-253). Hillsdale, NJ: Erlbaum. Bornstein, M. H., Ferdinandsen, K., & Gross, C. G. (1981). Perception of symmetry in infants. Developmental Psychology, 17, 8286. Bouma, H., & Andriessen, J. J. (1968). Perceived orientation of isolated line segments. Vision Research, 8, 493-507. Bouma, H., & Andriessen, J. J. (1970). Induced changes in the perceived orientation of line segments. Vision Research, 10, 333349. Byrne, R. W. (1979). Memory for urban geography. Quarterly Journal of Experimental Psychology, 31, 147-154.

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Received February 1, 1988 Revision received March 14, 1989 Accepted April 5, 1989