**Summary statistical representations are aggregate properties of the environment that are presumed to be perceived automatically and preattentively. We investigated two tasks presumed to involve these representations: judgments of the centroid of a set of spatially arrayed items and judgments of the mean size of the items in the array. The question we ask is: When similar information is required for both tasks, do observers use it with equal postfilter efficiency (Sun, Chubb, Wright, & Sperling, 2016)? We find that, according to instructions, observers can either efficiently utilize item size in making centroid judgments or ignore it almost completely. Compared to centroid judgments, however, observers estimating mean size incorporate the size of individual items into the average with low efficiency.**

^{2}) lines 2 pixels wide; the interior of each square matched the gray background luminance (46 cd/m

^{2}). The other stimuli (Figure 3B) were filled white squares (116 cd/m

^{2}) on a gray (46 cd/m

^{2}) background. The display was constructed using squares of eight fixed sizes (0.23°, 0.27°, 0.34°, 0.45°, 0.52°, 0.67°, 0.81°, 0.99°). Each set was created with sizes that were randomly selected without replacement from a discrete triangular distribution. The probability assigned to each of the eight possible sizes to appear was, respectively, 5.63%, 10.25%, 14.75%, 19.38%, 19.38%, 14.75%, 10.25%, or 5.63%. This discrete distribution was constrained to have only eight levels, because we wanted to be able to estimate the influence of each level on the size and centroid judgments. Given this constraint, this seemed a reasonable approximation of the Gaussian distribution used to determine item location. The dispersion of the location of the squares was determined by a Gaussian distribution with a standard deviation of 110 pixels (1.98°) centered in the middle of the screen. The sampling from this distribution was constrained so that the edges of two squares were never closer than 6 pixels (0.11°) to each other. In addition, because the standard deviation of the distribution of the centroids would normally be reduced by

*f*. An observer's attention filter is the vector of weights (one for each of the eight square widths used in our stimuli) used by the observer when performing a task with a particular target filter

_{φ}*φ*. The three tasks in this experiment are based on two target filters. In the equi-weighted centroid task, the target filter

*φ*gives equal weight to the squares of all eight widths

*w*—i.e.,

*i*from 1 to 8. In the size-weighted centroid task and the mean-size task, the target filter

*φ*gives weight to each square equal to its size:

*T*on a given trial has

*x*- and

*y*-coordinates

*i*in the display,

*i*, and

*x*- and

*y*-coordinates of its location. Typically, however, the response of the observer deviates from this target location.

*x*- and

*y*-coordinates of the observer's response on trial

*t*are given by

*σ*, and for some function

*w*) we have

*x*- and

*y-*coordinates of the

*i*th square in the stimulus on trial

*t*, and

*t*is

*σ*, and

*t*. And similarly, the likelihood function for the mean-size task is

*V*(which will eventually be thrown away) and sets

*V*contains guesses at the eight values of the function

*N*

_{iter}of times:

*C*in the neighborhood of the last sample

*P*> 1, set

*N*

_{iter}goes to infinity this process produces a sample from the posterior joint density characterizing the model parameter vectors (Hastings, 1970). For both the size and centroid analyses, the initial values of

*i*, and the initial value of

*σ*was 10. To ensure that the samples of this process used to generate estimates were stable,

*N*

_{iter}was 20,000 and the first 10,000 samples were discarded. To ensure that the samples used to generate estimates were independent, of the remaining 10,000 samples only every 40th was retained.

*efficiency*. Here we use

*postfilter efficiency*to emphasize that this value was estimated as the proportion of the stimulus squares that would need to be processed by an ideal observer using the

*observer's*estimated attention filter

*f*rather than the target filter

_{φ}*φ*. The value of postfilter efficiency ranges from 0 to 1. Because this is the estimate for an ideal observer, it is a lower bound on the proportion of squares that would have been processed by the actual observer.

*SD*= 0.09,

*t*(6) = 0.523,

*p*= 0.62, Bayes factor BF = 0.764.

^{1}The main effect of stimulus type was negligible. Because there are also no reliable interactions involving stimulus type or level of expertise, the reported results are collapsed across these factors. Also, to simplify the summary, we will consider the data from the singleton trials separately, so that for most of the summaries only results for trials with three and nine items are reported. Finally, we will focus on two preplanned contrasts for the task factor: one comparing the results in the equi-weighted and size-weighted centroid tasks, and one comparing the results of the size-weighted centroid task and the mean-size task.

*SD*= 0.02,

*t*(7) = 1.460,

*p*= 0.188, BF = 0.74. The preplanned contrast comparing the postfilter efficiency for the size-weighted centroid task with that for the mean-size task very strongly suggests that observers were able to use size more effectively when estimating the centroid of a group of squares than when estimating the mean size of the same group, Δ = 0.35,

*SD*= 0.15,

*t*(7) = 6.485,

*p*< 0.001, BF = 45.9.

*t*test provided evidence for a reduction of postfilter efficiency with increased numerosity (Figure 8) for all three tasks, Δ = −0.14,

*SD*= 0.08,

*t*(7) = −4.810,

*p*= 0.002, BF = 24.06.

*t*value associated with any of these interactions was 1.42, with a

*p*value of 0.198 and a BF of 0.71.

*t*(7) = 11.98,

*p*= 0.0000, BF = 141.169. There is substantially more variability across observers in the slope estimates for the size-weighted centroid task. Despite this variability, there is a reliable numerosity effect. However, as shown in Figure 10, the slope is not distinguishable from the expected slope of 1 for numerosity 3 or 9, and this result still holds if the estimates for numerosities 3 and 9 are averaged—slope = 1.055, 95% CI [0.76, 1.33],

*t*(7) = 0.43,

*p*= 0.68, BF = 0.485.

*t*(7) = −2.790,

*p*= 0.030, BF = 3.980. Further, averaging across numerosity in both cases, the slope in the mean-size task differs reliably from that in the size-weighted centroid task, Δ = −0.78,

*SD*= 0.71,

*t*(7) = −3.113,

*p*= 0.017, BF = 4.32.

*t*(7) = 5.927,

*p*= 0.001, BF = 63.97. What makes this observation striking is that it suggests that something other than the misperception of the sizes of the items must be contributing to the error observed in the three- and nine-item conditions. We reach this conclusion because the mean-size error due to misperception of item sizes would be expected to decrease as 1 over the square root of the number of observations (items). Under the extreme assumption that all singleton error is due to size misperception, the dashed black line shows the predicted RMSE. Another possibility is that rather than being due to size misperception, the error in the mean-size task arises from “late” sources—i.e., error depending on processes that come after the mean-size estimate has been created. Two examples of late sources of error are memory errors that result from having to keep a perceived mean size in memory while making the response and reproduction errors that arise because of problems correctly reproducing the correctly remembered mean size. One characteristic of late error is that it should not depend on the number of items included in the mean. Thus, an alternative but equally extreme model based on the assumption that all size error arises from late sources predicts that the dashed line in Figure 11 should be flat. However, neither size misperception errors, late errors, nor some combination of the two predict the observed increase in the RMSE with an increasing number of items. This argument suggests that there is some other component of error in the mean-size task that produces the observed increase in RMSE with

*n*.

*t*(7) = −1.979,

*p*= 0.088, BF = 0.80—but the evidence for this is weak. There was evidence for an additive component of the standard deviation of the error, 0.062°, 95% CI [0.031°, 0.094°],

*t*(7) = 4.691,

*p*= 0.002, BF = 21.53, and even stronger evidence that the standard deviation of the size error also increased as the size of the item being estimated increased, 0.106, 95% CI [0.072, 0.140],

*t*(7) = 7.300,

*p*= 0.00016, BF = 182.9. One way to get a sense of the relative importance of the additive and multiplicative contributions to the standard deviation is to compare the contribution of the multiplicative component for an average-size item (0.485°) with that of the additive component: 0.485° × 0.106/0.062° = 1.19. This suggests that the additive and multiplicative components contribute about equally to the standard deviation of the size estimation error for the singletons, with the multiplicative component possibly being slightly stronger.

*r*= 0.86). That correlation, along with the previous comparison showing that the multiplicative component made a substantial contribution to the overall error in size judgments for singletons, gives us confidence that observers were able to perceive the size differences of the stimuli used and report sizes using the response method employed in this experiment. Another window on the accuracy with which the item sizes could be perceived in the stimulus displays is provided by a comparison of the results in the size-weighted and equi-weighted centroid tasks. This comparison was done by extending the postfilter efficiency analysis (see the description in the Analysis subsection under Methods) to allow for the perturbation of item sizes. For each observer, the analysis of the data from the size-weighted centroid task used the estimated postfilter efficiency from the equi-weighted centroid task as a fixed value determining what proportion of the items in a stimulus cloud would be retained after the simulated decimation process. In addition, in this expanded analysis the size of each stimulus item was randomly perturbed prior to computing the simulated centroid judgment. The size perturbations were drawn from a Gaussian distribution with mean 0 and a standard deviation that depended on item size. The MATLAB optimization function fmincon() was used to estimate the slope and intercept of a linear function relating the standard deviation of item perturbation to item size so that the centroid response error produced in the simulation matched that produced by the observer in the size-weighted centroid task.

*p*= 0.0005, BF = 72.78; for nine items it was 0.044°, 95% CI [0.033°, 0.055°],

*t*(7) = 9.366,

*p*= 0.0000, BF = 692.5. Because there is only weak evidence for a difference between these estimates, Δ = 0.010°, 95% CI [−0.006°, 0.026°],

*t*(7) = 1.490,

*p*= 0.180, BF = 1.314, we will consider their average, 0.049°, 95% CI [0.034°, 0.063°],

*t*(7) = 7.837,

*p*= 0.0001, BF = 265.4. The slope relating the size misperception error to item size for the three-item task was 0.050, 95% CI [0.002, 0.099],

*t*(7) = 2.445,

*p*= 0.044, BF = 2.073; for nine items it was 0.038, 95% CI [−0.010, 0.085],

*t*(7) = 1.859,

*p*= 0.105, BF = 1.102. Because there is only weak evidence for a difference between these estimates, Δ = 0.013, 95% CI [−0.069, 0.094],

*t*(7) = 0.369,

*p*= 0.723, BF = 0.356, we will consider their average, 0.044, 95% CI [0.018, 0.070],

*t*(7) = 4.045,

*p*= 0.005, BF = 11.473. What is striking here is that the estimate of the additive component of the size misperception error computed in this way is similar to that estimated previously for the singleton trials in the mean-size task—0.049° versus 0.062°, Δ = 0.014°, 95% CI [−0.025°, 0.053°],

*t*(7) = 0.850,

*p*= 0.423, BF = 0.451—but the slope of the multiplicative component is substantially smaller: 0.049 versus 0.106, Δ = 0.057, 95% CI [0.022, 0.093],

*t*(7) = 3.804,

*p*= 0.007, BF = 8.975. We interpret this as evidence that the information about the size of the stimulus items in the size-weighted centroid task is more accurate than that incorporated into the mean-size judgments.

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^{1}Bayes factor computed using the calculator at http://pcl.missouri.edu/bayesfactor (Rouder, Speckman, Sun, Morey, & Iverson, 2009).