**Information taken in by the human visual system allows individuals to form statistical representations of sets of items. One's knowledge of natural categories includes statistical information, such as average size of category members and the upper and lower boundaries of the set. Previous research suggests that when subjects attend to a particular dimension of a set of items presented over an extended duration, they quickly learn about the central tendency of the set. However, it is unclear whether such learning can occur incidentally, when subjects are not attending to the relevant dimension of the set. The present study explored whether subjects could reproduce global statistical properties of a set presented over an extended duration when oriented to task-irrelevant properties of the set. Subjects were tested for their memory of its mean, its smallest and largest exemplars, the direction of its skew, and the relative distribution of the items. Subjects were able to accurately recall the average size circle, as well as the upper and lower boundaries of a set of 4,200 circles displayed over an extended period. This suggests that even without intending to do so, they were encoding and updating a statistical summary representation of a task-irrelevant attribute of the circles over time. Such incidental encoding of statistical properties of sets is thus a plausible mechanism for establishing a representation of typicality in category membership.**

*N*= 152; 120 women, 32 men) participated in this experiment. Subjects were randomly assigned to complete either an enumeration task (indicate whether each display contains fewer or more than 15 circles;

*N*= 50), a color task (indicate whether any two circles shared the same color;

*N*= 50), or to passively view the displays (

*N*= 52). The latter task was included to ensure that the enumeration and color tasks did not inadvertently increase (or decrease) attention to size. Within each group, half of the subjects viewed sets with low variability and half viewed sets with high variability during the familiarization phase.

*M*= 86.33 pixels,

*σ*= 26.6; skew = −0.10) or between 52 and 96 pixels in diameter (low variability condition;

*M*= 86.33 pixels,

*σ*= 9.43; skew = −1.09). The color of each circle was selected at random without replacement from 20 possible colors (color name and RGB values: red: 255,0,0; green: 0,255,0; blue: 0,0,255; yellow: 255,255,0; rose: 255,155,255; pink: 255,79,150; forest green: 0,128,0; navy: 0,0,102, peach: 255,155,117; dark purple: 108,0,108; cyan: 0,255,255; grey: 185,185,185; orange: 255,140,0; brown: 103,29,0; black: 0,0,0; violet: 137,59,195; dark grey: 77,77,77; sky blue: 0,153,153; burgundy: 165,0,33; teal: 0,200,90; and presented on a white screen: 255,255,255) with the constraint that 50% of the trial displays contained two contiguous circles of the same color.

**Figure 1**

**Figure 1**

**Figure 2**

**Figure 2**

*x*and

*y*axes of an empty bar graph. Irrespective of variability condition, the

*y*axis' scale was labeled 0 to 1,600 in increments of 100 circles and the

*x*axis was labeled with the letters A through S. Each letter corresponded to one of the possible 19 circle sizes (ranging from 16 to 146 pixels) that subjects could have seen during the familiarization phase. Subjects adjusted the height of the bars corresponding to the number of circles of each size they believe appeared in the familiarization phase by pressing the letter corresponding to the bar they wished to adjust. When subjects selected a bar to adjust, a circle of the corresponding circle size appeared under the bar graph. Before adjustment, each bar location was empty, which was indicated by a counter set to zero displayed above the corresponding letter. As the height of the bar was adjusted, the counter below each bar was adjusted to reflect the change. Concurrently, a second counter was also adjusted to reflect the number of circles (out of 4,200) that had been accounted for in subjects' frequency estimates.

*α*of 0.05, and all

*t*tests were two-tailed. Mean accuracy was computed for each subject in the enumeration and color conditions only (as there was no task in the control group), as a function of task type and variability, and analyzed in a 3 × 2 ANOVA with these variables. Accuracy was much higher in the enumeration task (

*M*= 0.89;

*SD*= 0.07) than in the color task (

*M*= 0.61;

*SD*= 0.11), leading to a significant main effect of task type,

*F*(1, 96) = 225,

*MSE*= 0.008 ,

*p <*0.001

*, η*

_{P}= 0.70. The main effect of variability and the interaction were not significant in this analysis.

*t*tests, comparing each value with its counterpart in the actual set of items shown to observers.

**Table 1**

*M*= 8.79;

*SD*= 31.1) than in the low variability condition (

*M*= −1.12;

*SD*= 26.3), leading to a significant main effect of variability,

*F*(1, 146) = 4.44,

*MSE =*854,

*p*< 0.04

*, η*

_{p}= 0.03. Although accuracy in the familiarization phase suggests that the color task was considerably more difficult than the enumeration task, this difference in difficulty had no measurable effect on the drawing task, with neither the main effect of condition nor the interaction reaching significance, both

*F*s < 1. Indeed, error in estimates was consistently smaller in the low variability condition (ranging from −0.89 to −1.45 pixels across the three task types) than in the high variability condition (ranging from 7.11 to 11.4 pixels).

**Figure 3**

**Figure 3**

*t*tests, comparing the observed error against 0. The error observed in the high variability condition estimates of mean size was reliably greater than zero,

*t*(75) = 2.46,

*p*< 0.02, 95% CI [1.65, 15.9], but the error observed in the low variability condition was not,

*t*(75) < 1,

*p*> 0.63, 95% CI [−7.22, 4.82].

*perceived midpoint*of the set, estimated from subjects' drawings of the smallest and largest circles in the set, and the agreement between the diameter of the average sized circle and the

*perceived mode*of the set, estimated from the distributions produced by subjects in the test phase. Both sets of correlations were significant (mean/midpoint,

*r*= 0.44,

*p*< 0.001; mean/mode,

*r*= 0.20,

*p*< 0.02), so we then compared the error observed in judgments of mean size to the error that subjects would make if their estimate of the average circle size was based on the actual mean, the perceived midpoint, and the perceived mode. After excluding data from any subject who adjusted the smallest circle to a diameter larger than the average circle or larger than the largest circle in the set, data from 121 subjects were analyzed in a one-way repeated measures ANOVA. The effect of reference point was significant,

*F*(2, 240) = 6.25,

*MSE =*373,

*p*< 0.003,

*η*

_{p}= 0.05, and suggested that estimates would be more accurate if they were based on the actual mean (mean error = 0.79,

*SD*= 25.9,

*t*(120) = 0.34,

*p*> 0.73, 95% CI [−3.87, 5.46]) than if they were based on subjects' own perceived midpoint (

*M*= −5.26,

*t*(120) = −3.46,

*SD*= 16.7,

*p*< 0.01, 95% CI [−8.26, −2.25]) or perceived mode (

*M*= −7.74,

*t*(120) = −3.26,

*SD*= 26.1,

*p*< 0.01, 95% CI [−12.4, −3.03]).

*M*= 132;

*SD*= 32.0) than in the low variability condition (

*M*= 64;

*SD*= 20.8), leading to a significant main effect of variability,

*F*(1,115) = 197,

*MSE =*707,

*p <*.001

*, η*

_{P}= .63. Although subjects overestimated the range in both conditions, the observed average range of 132 pixels did not reliably differ from the actual range of 130 pixels in the high variability condition,

*t*(64) < 1,

*p*> 0.58, 95% CI [−5.76, 10.1], for the difference. Moreover, the diameter of the smallest and largest circles drawn by subjects was an average of 1.41 and 3.60 pixels larger than in the actual set of items shown in this condition, and neither value was statistically different from zero, both

*t*s < 1, both

*p*s > 0.32. The observed average range of 60.2 pixels in the low variability condition, however, was reliably greater than the actual range of 44 pixels,

*t*(55) = 7.33,

*SD*= 20.8,

*p*< 0.01, 95% CI [14.7, 25.9] for the difference, with the smallest circle underestimated by an average of 5.93 pixels,

*t*(55) = −2.71,

*SD*= 16.4,

*p*< 0.01, 95% CI [−10.3, −1.54], and the largest circle overestimated by an average of 14.4 pixels,

*t*(55) = 5.18,

*SD*= 20.9,

*p*< 0.001, 95% CI [8.84, 20.0].

*D*, a measure of the extent to which observers' distributions diverge from the actual distributions they saw, are shown in Figure 4 for each condition;

*D*is expected to be close to zero if observers' reproductions of the distributions they saw are accurate (Kullback & Leibler, 1951).

*D*values were strikingly consistent across the three tasks in the high variability condition (

*D*= 0.19 to 0.21), highlighting the null effect of familiarization task difficulty. More revealing is the fact that D values for the low variability distributions (

*D*= 0.78 to 1.47) were approximately 4 to 8 times greater than for the high variability distributions, and much less consistent across the three groups, underscoring that subjects in the low variability condition produced lower fidelity distributions than those in the high variability condition.

**Figure 4**

**Figure 4**

*N*= 150). The difference between the actual mode in each distribution (92 in the high variability condition and 88 in the low variability condition) and the mode as judged by subjects was computed and this error score served as the dependent variable in a 3 (condition) × 2 (variability) ANOVA. The mode was underestimated by an average of 8.93 pixels, and the extent of this underestimation was similar across all groups with neither main effect nor the interaction approaching significance, all

*F*s < 1, all

*p*s > 0.43. The error in judging the mode reliably differed from zero,

*t*(149) = −4.19,

*SD*= 26.1,

*p*< 0.001, 95% CI [−13.1, −4.71]. Within the low variability condition, the median and modal value of the mode were the same (73), whereas in the high variability condition, the median value was 81 and the modal value was 73.

*M*= 19.7;

*SD*= 6.85) than in the high variability condition (

*M*= 32;

*SD*= 7.09), although both significantly overestimated the actual standard deviation in these conditions (low = 9.43; high = 26.6, both

*p*s < 0.001).

^{1}

*skew*= m

_{3}/m

_{2}

^{3/2}, where m

_{2}and m

_{3}represent the second and third moments of the distributions, respectively. The difference between this value and the actual skew of the distributions of circles shown to subjects was then analyzed in a 3 (condition) × 2 (variability) ANOVA. The average skew of the distributions created in the high variability condition was closer to the actual skew (mean error = 0.15;

*SD*= 0.51) than in the low variability condition (mean error = 0.95;

*SD*= 0.51), leading to a significant main effect of variability,

*F*(1, 146) = 92.7,

*MSE =*0.27 ,

*p <*0.001

*, η*

_{P}= 0.39. Neither the main effect of condition nor the interaction were significant, both

*F*s < 1, both

*p*s > 0.77. Although the estimated skew in the high variability condition was closer to the actual skew than in the low variability condition, the error in both estimates was reliably different from zero, both

*t*s > 2.5, both

*p*s < 0.02. Nevertheless, the estimated average skew in the low variability condition (

*M*= −0.14;

*SD*= 0.51) was significantly different from that expected if subjects had produced an unskewed distribution,

*t*(75) = −2.37, p < 0.03, 95% CI [−0.25, −0.02] for the difference, but it was not different from an unskewed distribution in the high variability condition,

*t*(75) < 1,

*p*> 0.40, 95% CI [−0.07, 0.17] for the difference.

^{2}

*greater*by about 9 pixels than the actual mean size in the high variability condition. Similarly precise representations of mean size can be inferred from the first half of Duffy and colleagues' (2010) experiment 1 (i.e., before the distribution from which items were drawn changed covertly) and experiment 2, in which the estimated point of zero bias occurred within 2 or 3 pixels of the mean of the entire set of lines, although subjects attended to size in those experiments and were not tested for their recollection of the mean.

*overestimate*the variability of the distribution, particularly in the low variability condition. This might reflect the fact that the distributions are skewed, but in their reconstructions, subjects tended to make them symmetrical. Thus, in reproducing the smallest and largest circles in the set and the distribution of circles, subjects may have chosen endpoints to be equidistant from the (accurately recalled) mean, effectively “unskewing” the distributions and overestimating their variability as a result. Consistent with this interpretation, the error in estimating the size of the largest circle in the low frequency condition was greater (approx. 14 pixels) than the error in estimating the smallest circle (approx. 6 pixels).

^{3}The model predicting perceived mean scores from the last display and the preceding five displays accounted for significant variability, adjusted

*R*

^{2}= 0.053,

*F*(2, 151) = 5.19,

*p*< 0.008. In contrast to what is expected if only the last display was referenced, only the mean of displays 15–19 emerged as a significant predictor,

*β*= 0.27,

*t*(151) = 3.22,

*p*< 0.003, with

*β*= 0.09,

*t*(151) = 1.07,

*p*> 0.28 for the mean of the last display. Similarly, for the perceived range, the model again accounted for significant variance, adjusted

*R*

^{2}= .37,

*F*(2, 151) = 44.5,

*p*< 0.001, and again only the range of items experienced over displays 15 to 19 emerged as a significant predictor,

*β*= 0.61,

*t*(151) = 9.43,

*p*< 0.001, with

*β*= −0.02,

*t*(151) = −0.26,

*p*> 0.79 for the range of the last display. Note that although increasing the number of displays naturally brings the mean and range closer to their actual values in the population, they will only be better predictors of observed performance if subjects' judgments also incorporate more information than is available in the last display. Thus, the outcome of these analyses suggest that it is unlikely that subjects are basing their responses on information contained in the last display only.

*toward*the mean when recalling the smallest and largest items in the low variability condition, subjects appeared to bias them

*away*from the mean. However, it is important to note that when reconstructing the distributions, subjects made them much flatter than they actually were. Hence, the overestimation of the range in the low variability condition may reflect subjects' attempts to reconcile their recollection of seeing many stimuli that were very similar to one another with an erroneous belief that they must have appeared with similar frequency.

**Concluding remarks**

*Psychological Science*, 21, 560–567. doi:10.1177/0956797610363543.

*Vision Research*, 83, 25–39. doi:10.1016/j.visres.2013.02.018.

*Psychological Science*, 12, 157–162.

*Journal of Experimental Psychology: Human Perception and Performance*, 40, 1440–1449. doi:10.1037/a0036206.

*Perception & Psychophysics*, 67 (1), 1–13. doi:10.3758/BF03195009.

*Cognitive Psychology*, 36, 28–71. doi:10.1006/cogp.1998.0681.

*Journal of Experimental Psychology: Human Perception and Performance*, 40, 1915–1925. doi:10.1037/a0037375.

*Acta Psychologica*, 138, 289–301. doi:10.1016/j.actpsy.2011.08.002.

*Visual Cognition*, 20, 211–231. doi:10.1080/13506285.2012.657261.

*Psychonomic Bulletin & Review*, 17, 224–230. doi:10.3758/PBR.17.2.224.

*The American Journal of Psychology*, 60 (1), 1–29.

*Perception & Psychophysics*, 63 (5), 918–926. doi:10.3758/BF03194447.

*Journal of Experimental Psychology: General*, 129, 220–241.

*Attention, Perception, & Psychophysics*, 75, 278–286. doi:10.3758/s13414-012-0399-4.

*The Quarterly Journal of Experimental Psychology: Section A*, 54, 1105–1124.

*Journal of Experimental Psychology: General*, 131, 287–297.

*Annals of Mathematical Sciences*, 22, 79–86.

*Acta Psychologica*, 142, 245–250. doi:10.1016/j.actpsy.2012.11.002.

*Psychological Review*, 118, 280–315. doi:10.1037/a0022494.

*Vision Research*, 79, 8–16. doi:10.1016/j.visres.2012.12.004.

*Journal of Experimental Psychology*, 52, 112–118.

*Journal of Experimental Psychology*, 58, 239–246.

*Neurocomputing*, 70, 1808–1812. doi:10.1016/j.neucom.2006.10.069.

*The American Journal of Psychology*, 78, 392–402.

*Acta Psychologica*, 146, 7–18. doi:10.1016/j.actpsy.2013.11.012.

*Journal of Experimental Psychology*, 82, 52–63. doi:10.1037/h0028028.

*Psychonomic Bulletin & Review*, 18, 484–489. doi:10.3758/s13423-011-0071-3.

*Journal of Vision*,

*13*(9): 1087, doi:10.1167/13.9.1087 [Abstract].

^{1}Distributions created by subjects in the color task were slightly more variable (

*M*= 27.9;

*SD*= 8.44) than distributions created by subjects in the enumeration (

*M*= 25.3;

*SD*= 9.29) or control (

*M*= 24.7;

*SD*= 10.0) conditions, leading to a significant main effect of task type,

*F*(2, 146) = 3.10,

*MSE =*47.6,

*p <*0.05

*, η*

_{P}= 0.041.

^{2}Despite the greater difficulty of the color task, estimates of mean size were no less precise following this task than following enumeration or passive viewing. This may be an indication that the mean was encoded accurately irrespective of how attention was allocated to the displays (cf. Chong & Treisman, 2005). However, a limitation of our design is that, to avoid creating a dual-task situation (which would make the color and enumeration tasks different from the passive viewing task in two ways), we did not assess the distribution of attention in the familiarization phase and as such, no strong conclusions can be drawn.

^{3}Note: Five displays were used because with this number, there was sufficient variability in the predictor values across subjects. When the preceding six displays were used, all but seven subjects in the high variability condition had experienced the full range of stimuli, leaving insufficient variability in this predictor for carrying out a multiple regression analysis.