**People can quickly and accurately compute not only the mean size of a set of items but also the size variability of the items. However, it remains unknown how these statistics are estimated. Here we show that neither parallel access to all items nor random subsampling of just a few items is sufficient to explain participants' estimations of size variability. In three experiments, we had participants compare two arrays of circles with different variability in their sizes. In the first two experiments, we manipulated the congruency of the range and variance of the arrays. The arrays with congruent range and variability information were judged more accurately, indicating the use of range as a proxy for variability. Experiments 2B and 3 showed that people also are not invariant to low- or mid-level visual information in the arrays, as comparing arrays with different low-level characteristics (filled vs. outlined circles) led to systematic biases. Together, these experiments indicate that range and low- or mid-level properties are both utilized as proxies for variability discrimination, and people are flexible in adopting these strategies. These strategies are at odds with the claim of parallel extraction of ensemble statistics per se and random subsampling strategies previously proposed in the literature.**

*range*-based account), and an account where the variability is computed using low- or mid-level properties of the array like spatial frequency or texture. Together, our experiments suggest that size variability of a set of items is accessed through proxies, and this view may be extended to other types of statistical summary representations.

*X*) in the array is first extracted. The values within an array are then summed and divided by the number of items in the array (

*N*). A mean of each array (

*μ*) is then calculated by the formula:

*μ*). The squared deviations between each item in the array and the mean are then summed. The sum of squared deviations is then divided by

*N*to form a variance (

*σ*

^{2}).

*standard*array, which had a certain orientation variability among the Gabor stimuli, and it was compared to the

*test*array. The test array has a larger variability compared to the standard array. The arrays were shown for 200 ms. The participants' task was to determine which of the two arrays had a larger variability (i.e., identify the test array). Participants were quite good at this task. However, the cognitive mechanism—how participants performed this judgment—was relatively unexplored. In a later study by the same group, Solomon et al. (2011, experiment 3) examined whether the just noticeable difference (JND) to detect variability differences increased as the variability of the standard array increased. This time, the authors focused on size variability instead of orientation. They systematically manipulated the variability differences between the standard and test arrays displayed in succession. They found that as the variability of the standard array increased, a larger difference between the two arrays was needed for participants to detect the difference. This is consistent with Weber's law and thus with most other domains of quantity judgment, which provides some evidence that participants may have been directly estimating variance from the size of the items in the arrays. However, the results of Solomon et al. (2011) also suggested that a simple parallel processing account cannot explain their data. They concluded that participants utilized only a subset of items in each array, and then use them to compute the summary statistics. In particular, their efficiency analysis led to an estimate that participants used five to eight items' worth of information from each array (relative to all eight items in each array) for the variability discrimination task. Interestingly, participants' reports of variability were also more accurate than one would expect based on their reports of mean size in a previous experiment. Solomon et al. (2011) suggested that this is accounted for by some form of late decision noise in the computation of mean size, but an alternative suggestion is that participants simply use different cognitive strategies in computing the two (e.g., Yang, Tokita, & Ishiguchi, 2018).

*σ*/

^{2}*n*, where

*σ*is the variance across items and

^{2}*n*is the number of items sampled. By contrast, the variance in an estimate of the

*variance*of a set of items is 2

*σ*/

^{4}*n*. This means you have much more uncertainty about the variance of a set of items from a given number of samples,

*n*, than about the mean from that same number of samples.

*N*items. In our simulation, we used an arbitrary

*N*, and set it to

*N*= 50, though the results are invariant to

*N*as long as

*N*is greater than the number of sampled items. The average sizes of the two arrays were always the same, with 10 units. One of the arrays was a

*standard*array, which had a standard deviation of 1.0 unit. The other was a

*test*array, which had a standard deviation of 1.0, 1.2, 1.4, 1.6, or 1.8 units. These translated into a 0% to 80% difference in standard deviation between the standard and test arrays. The observer's task was to identify the array with larger variability in size (i.e., the test array).

*k*items were randomly sampled from each of the two arrays, so a total of 2

*k*items were stored in working memory. The standard deviations of the two samples were then separately calculated and compared. The one with the larger sample standard deviation was chosen as the response. The same procedure was repeated 20,000 times at each level of 2

*k*and each standard deviation difference. The responses were aggregated at each level.

*k*) would not increase the rate of selecting the array with larger variability (i.e., the test array). When there was a real difference between the two arrays, accuracy (the rate of selecting the test array) grows approximately logarithmically with the number of items stored in working memory (2

*k*).

*N*items, were shown in each trial. The task was to identify the array with a larger variability.

*k*items from each array, we simulated a case where

*k*items were

*searched*in each array, and the ranges of the arrays were estimated based on the size of the largest and smallest items in the set of

*k*items that was searched. The array with a larger estimated range was deemed to be the test array. Because, in this case, the

*k*items do not need to be held in mind, but only searched, this strategy is far less cognitively demanding than the random subsampling account proposed above. Is such a strategy effective?

*SD*), searching a set of only five items in each array and estimating the range of these items (total items searched = 2

*k*= 10) yielded an accuracy of 72.6%. When the difference between the two arrays is large (difference = 0.8

*SD*), employing the range heuristic on 10 searched items yielded an impressive 85.5% accuracy. The rates of improvement are slower than utilizing 2

*k*items in working memory for random subsampling (see Figure 2a), but in this case, only a simple visual search is needed. The entire set of display need not be held in working memory. Instead, only the largest and smallest items of each array have to be stored in working memory. That is, regardless of number of items searched, only a total of four items from both arrays need to be stored in working memory. Hence, working memory requirements for the range heuristic are very low compared to the random subsampling method, and are hence within the limits of working memory capacity.

*k*items is significantly easier than computing and remembering each of their sizes.

*standard*array, in which the circles had a 0.1-log-pixel standard deviation in log-transformed diameter. The other array was the

*test*array, in which the circles were more variable in size. Participants were instructed to select the array that was more variable (i.e., the array in which the circles were more different from each other).

*range-variance congruency*across trials. Recall that the

*test*array was always more variable compared to the

*standard*array. In the congruent trials, the test array also had a larger range compared to the standard array. In the incongruent trials, the test array had a

*smaller*range compared to the standard array. The spatial positions (left/right) of the test and standard array were randomized, so the participant's task was always a two-alternative forced choice in both range congruent and range incongruent conditions.

*d*around 0.9). A power analysis indicated that we could achieve a power of 80% with a sample size

*n*= 12. To ensure normal distribution of the sampling distributions, we decided to include at least 30 participants in each of our experiments.

*M*= 73%, 6.4%) compared to those of incongruent trials (

*M*= 66%,

*SD*= 5.3%),

*F*(1, 33) = 28.5,

*p*< 0.001, Cohen's

*d*= 0.92. The results showed that participants relied on the range of circle sizes for variability discrimination. In addition, accuracy improved as standard deviation differences increased,

*F*(4, 132) = 118.1,

*p*< 0.001. The interaction between the two factors is also significant,

*F*(4, 132) = 4.7,

*p*= 0.001, indicating differential reliance on the range heuristic across levels of standard deviation difference, likely due to the failure to discriminate when the two arrays had very similar variabilities.

*F*(4, 116) = 1.558,

*p*= 0.19. The result is consistent with previous research (e.g., Allik et al., 2013; Chong & Treisman, 2003).

*close-range*condition, the three largest and three smallest circles within an array were placed closest to the fixation (marked with red circles in Figure 5a). Other circles were randomly placed in the rest of the imaginary grid. In the

*far-range*condition, the largest and smallest circles were placed far from fixation, with the remaining items placed closest to the fixation. Hence, most range information was located far away from the fixation, and in the incongruent far-range trials, the items near fixation were not incongruent (e.g., the range of the items near fixation was congruent with the variability, since the incongruent items were far from fixation). Close- and far-range trials were interleaved within each block.

*F*(1, 42) = 6.76,

*p*< 0.05, Cohen's

*d*= 0.40, a main effect of standard deviation difference level,

*F*(4, 168) = 129.8,

*p*< 0.001, and an interaction between the two factors,

*F*(4, 168), = 3.23,

*p*< 0.05. Importantly, there is also a significant interaction between range-variance congruency and spatial arrangement,

*F*(1, 42) = 8.02,

*p*< 0.01. When range-relevant items were far from fixation (far-range conditions), variability discrimination did not differ regardless of whether range and variability information were congruent (Congruent: 73%, Incongruent: 73%),

*F*= 0.03. When range-relevant circles were close to the fixation (close-range conditions), participants were able to utilize the range of the full array as proxy for variability. As a result, worse performance was seen in the range-variance incongruent trials (

*M*= 69%,

*SD*= 17%), compared to the congruent trials (

*M*= 76%,

*SD*= 15%),

*F*(1, 42) = 19.1,

*p*< 0.001, Cohen's

*d*= 0.41. The behavioral results are consistent with our simulation above. They suggest that participants were primarily (a) using the range, but (b) sampling mostly circles close to fixation. Results are summarized in Figure 7.

*F*(1, 31) = 11.4,

*p*= 0.002, Cohen's

*d*= 0.60, a main effect of standard deviation difference level,

*F*(4, 124) = 113,

*p*< 0.001. Unlike in Experiment 2A, the interaction between the two factors is not significant,

*F*< 1. It shows that the improvements in accuracy from increasing standard deviation difference are similar regardless of whether the range and variance were congruent.

*F*(1, 31) = 33.0,

*p*< 0.001. In the far-range condition, variability discrimination did not differ regardless of whether range and variance were congruent (Congruent: 72%, Incongruent: 73%),

*F*= 0.01. In the close-range condition, the range of the full array was readily available to the participants. They seemed to utilize the range information as a proxy for variability. Participants performed worse in the range-variance incongruent trials (

*M*= 65%,

*SD*= 16%), compared to the congruent trials (

*M*= 78%,

*SD*= 15%),

*F*(1, 31) = 35.0,

*p*< 0.001, Cohen's

*d*= 0.75]. Results are summarized in Figure 8.

*t*test was conducted on the accuracy data, suggesting that the difficulty in variability discrimination for filled circles (

*M*= 72.8%,

*SD*= 5.8%, Experiment 2A) was comparable to that for outlined circles (

*M*= 72.0%,

*SD*= 5.5%, Experiment 2B),

*t*(73) = 0.61,

*p*= 0.54, Cohen's

*d*= 0.14.

*M*= 78%,

*SD*= 11%), compared to when it was outlined (

*M*= 64%,

*SD*= 11%, comparing left and right panels), suggesting that participants could not completely abstract beyond the low-level features in variability discrimination,

*F*(1, 29) = 14.8,

*p*< 0.001, Cohen's

*d*= 0.70. Other comparisons in the ANOVA were consistent with previous experiments. There is a main effect of standard deviation difference,

*F*(4, 116) = 98.8,

*p*< 0.001, indicating an increased performance as differences in variability between the standard and test arrays increased. A highly significant range-variance congruency,

*F*(1, 29) = 37.9,

*p*< 0.001, Cohen's

*d*= 1.12, suggests that participants were more accurate when range and variance conveyed consistent information (

*M*= 75%,

*SD*= 6.3%), compared to the incongruent condition (

*M*= 67%,

*SD*= 5.5%).

*F*(4, 116) = 3.0,

*p*< 0.05, suggests that the range-variance congruency advantage was different as the difference between standard and test arrays increased. Lastly, a marginally significant interaction between fill and range-variance congruency,

*F*(1, 29) = 3.4,

*p*= 0.08, suggested that the range-variance congruency advantage for outlined circles (10.2%) trended toward larger than the one for filled circle arrays (6.9%).

*M*= 69.8%,

*SD*= 4.3%; Experiment 3:

*M*= 71.3%,

*SD*= 4.6%),

*t*(60) = 1.32,

*p*= 0.19. This shows that the bias toward selecting filled test arrays in Experiment 3 was compensated by the same bias against outlined test arrays.

*SD*= 4.6%), that of Experiment 2A was 72.8% (

*SD*= 5.8%),

*t*(70) = 1.24,

*p*= 0.22, and Experiment 2B was 72.0% (

*SD*= 5.5%),

*t*(59) = 0.54,

*p*= 0.59. This shows that participants were flexible in adopting different strategies, using range, low-level visual information, or a combination of these strategies to attain a reasonably high-level performance.

*range*heuristic, participants might have employed other forms of smart subsampling heuristics other than using the range information.

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