Special Thematic Section on "Tracking the Continuous Dynamics of Numerical Processing"

Dissociating Parallel and Serial Processing of Numerical Value

Kassandra R. Lee^*^a, Kenith V. Sobel^b, A. Kane York^b, Amrita M. Puri^c

[a] Schepens Eye Research Institute, Massachusetts Eye and Ear, Department of Ophthalmology, Harvard Medical School, Boston, MA, USA. [b] Department of Psychology and Counseling, University of Central Arkansas, Conway, AR, USA. [c] Department of Biology, University of Central Arkansas, Conway, AR, USA.

Journal of Numerical Cognition, 2018, Vol. 4(2), 360–379, https://doi.org/10.5964/jnc.v4i2.133

Received: 2017-06-02. Accepted: 2018-04-05. Published (VoR): 2018-09-07.

Handling Editors: Matthias Witte, University of Graz, Graz, Austria; Matthias Hartmann, University of Bern, Bern, Switzerland; Swiss Distance Learning University, Brig, Switzerland; Thomas J. Faulkenberry, Tarleton State University, Stephenville, TX, USA

*Corresponding author at: University of Nevada, Reno, The Center for Integrative Neuroscience, 1664 N. Virginia Street Reno, NV 89557, USA. E-mail: kassandral@nevada.unr.edu

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Digits serve as useful tools for studying the interaction between low-level perceptual representations and higher-level semantic information, and also the degree to which processing these stimulus attributes relies on similar or different mechanisms. Following a body of literature that debates the influence of high-level, semantic information on perceptual processing, and work by Van Opstal and colleagues (2011) investigating whether subliminally presented digit arrays affect estimates of numerical averages, here we explored the temporal dynamics of extracting numerical values associated with each digit in an array. Specifically, we examined reaction times (RTs) for estimating the average of digit arrays of varying sizes to determine whether numerical meaning is extracted in parallel, or instead may require serial processing of individual digits. In Experiment 1, participants completed a numerical mean estimation task, along with visual search tasks designed to yield RT patterns across increasing display sizes thought to be characteristic of serial and parallel processes. In Experiment 2, we controlled for brightness cues that could have contributed to performance in Experiment 1. In both experiments, comparing RT patterns for the numerical averaging tasks to those of the search tasks suggested that semantic information from multiple digits may be extracted by a parallel processing mechanism. Unlike in either search task, RTs improved with increasing display size, indicating a potential benefit of larger displays as has been reported for extracting ensemble, or summary statistical representations of lower-level visual information.

Keywords: parallel processing, semantic information, average estimation, ensemble coding, visual search

Letters and digits are symbols with specific meanings; however, they also possess characteristic and incidental low-level perceptual attributes such as shape and size. Here, we use digits to explore the degree to which processing of higher-level semantic information relies on similar or different mechanisms compared to perception of low-level sensory characteristics of environmental stimuli.

The question of how semantic meaning interacts with lower-level stimulus features to influence perception has engendered considerable controversy in recent years. Some evidence points to the ability of semantic information to affect judgments about perceptual attributes of a stimulus (e.g., Hansen, Olkkonen, Walter, & Gegenfurtner, 2006; Lupyan, Thompson-Schill, & Swingley, 2010; Puri & Wojciulik, 2008). Furthermore, although Wolfe and Horowitz (2004) proposed that visual search is only influenced by perceptual information, more recent work has shown effects of semantic information on finding particular letters or digits among others (e.g., Lupyan, 2008; Sobel, Puri, & Hogan, 2015), even if the semantic influence is not as strong as guidance by perceptual characteristics such as shape (Godwin, Hout, & Menneer, 2014). Others, however, have argued that visual perception is impenetrable to higher-level cognition (e.g., Firestone & Scholl, 2016), especially in its early stages, deemed “early vision” (Marr, 1982; Pylyshyn, 1999). Recently, the size congruity effect (SCE), which describes the interference between physical and numerical size observed in digit comparison (Henik & Tzelgov, 1982) and digit search tasks (Krause, Bekkering, Pratt, & Lindemann, 2017; Sobel & Puri, 2018; Sobel, Puri, & Faulkenberry, 2016), has led to disagreement about how interactions between semantic and physical attributes occur. One possibility is that they could be combined into a single representation at an early, perceptual stage (Schwarz & Heinze, 1998; Walsh, 2003). Alternatively, the representations could remain separate throughout perceptual processing and only interfere at the decision stage (Faulkenberry, Cruise, Lavro, & Shaki, 2016; Santens & Verguts, 2011). A recent study showed that although the SCE occurs reliably in a variety of contexts and demonstrates that low-level perceptual and higher-level semantic information do interact, it is likely that physical and numerical size interfere at a decision rather than perceptual processing stage (Sobel, Puri, Faulkenberry, & Dague, 2017).

In view of these debates, we wanted to explore the fundamental nature of semantic representations and investigate whether or not processing of arbitrary meanings such as numerical value can occur via similar mechanisms as extraction of lower-level visual information. One way to approach this issue is to ask participants to complete a numerical averaging task, which would require extraction of higher-level, semantic attributes of multiple digits. Van Opstal, de Lange, and Dehaene (2011) addressed whether the semantic meanings of digit stimuli can be incorporated into summary statistical representations, and if so, whether this process relies on mechanisms that support such “ensemble coding” in the purely visual domain. In this context, ensemble coding refers to the rapid extraction of statistical summary information from groups of similar stimuli (e.g., estimating the average size of a set of circles, or in the real world, glancing at a pile of apples and quickly realizing they are, on average, red), which results in an efficient and useful representation of our visually dynamic world (Ariely, 2001; Chong & Treisman, 2003; Haberman & Whitney, 2007; Parkes, Lund, Angelucci, Solomon, & Morgan, 2001).

Ensemble perception has been shown to occur for low-level stimulus features such as size, line orientation, spatial location, and direction of motion (Alvarez & Oliva, 2008; Alvarez & Oliva, 2009; Dakin & Watt, 1997; Oriet & Corbett, 2008; Parkes et al., 2001; Watamaniuk, Sekuler, & Williams, 1989; Williams & Sekuler, 1984), as well as for more complex attributes such as emotional expression of faces, identity, and gaze direction (Haberman & Whitney, 2007; Haberman & Whitney 2009; Haberman & Whitney 2010; Sweeny & Whitney, 2014). Ensemble representations are thought to depend on extraction of information from multiple items in a cluttered scene simultaneously, or in parallel, rather than via a serial process requiring attention to individual items (Alvarez, 2011; Alvarez & Oliva, 2009). Furthermore, accurate estimation of the ensemble does not appear to rely on processing of only a small subset of items as suggested by Myczek and Simons (2008), but instead depends on extraction of at least some information from the entire set (Haberman & Whitney, 2010). Thus, it appears that summary information about perceptual attributes can be gathered from a visual scene implicitly, rapidly, efficiently, and at different levels of processing complexity.

Van Opstal et al. (2011) specifically asked whether the semantic information associated with a “prime” display (a digit array presented briefly prior to presentation of a target digit array) can affect participants’ perception of the target digit arrays even when they are not consciously aware of the primes. They reported that subliminally presented digit arrays influenced estimates of the numerical average of target displays, and concluded that participants accurately performed ensemble coding of higher-level, semantic attributes (numerical meaning) in parallel, and without explicit awareness of individual items in the display. These findings are consistent with previous work investigating statistical representation of low-level stimulus attributes (Alvarez & Oliva, 2008; Ariely, 2001; Chong & Treisman, 2005). However, Van Opstal et al. (2011) also observed increases in reaction times (RTs) with larger display sizes (number of items presented in the display) that may nonetheless reflect a serial contribution to estimates of numerical averages. Furthermore, their displays were limited to relatively small numbers of items (3, 4, and 5), and thus participants may have been encouraged to attempt exact averaging despite relatively brief displays (< 800 ms). On the other hand, extracting information from a set of items may proceed in parallel until the number of items exceeds the subitizing range (i.e., four or fewer) and enters the counting range (Railo, Karhu, Mast, Pesonen, & Koivisto, 2016).

The ability to gather semantic information simultaneously from multiple digits would be indicative of parallel processing (Egeth, 1966) rather than serial processing. Serial and parallel cognitive mechanisms have traditionally been dissociated using a variety of approaches, including examining the temporal dynamics of searching for a target item among distractor items. It is generally observed that when a target item possesses a unique feature compared to distractor items and thus “pops out”, RTs increase relatively little with additional distractors. This indicates that information about that feature was processed across all items in the display in parallel. In contrast, when targets contain features that are similar to those of surrounding distractors, search is less efficient and RTs increase with the number of distractors, reflecting the possibility that individual items must be selectively attended in a serial fashion until the target is found (Duncan & Humphreys, 1989; Treisman & Gelade, 1980).

These highly replicated differences in RT patterns across different types of visual search have traditionally been viewed as reflecting a distinction between parallel and serial mechanisms, although whether less efficient search is due to a serial, attentionally demanding process as opposed to factors such as increased noise (and thus reduced discriminability of the target) is a subject of ongoing investigation (e.g., McElree & Carrasco, 1999; Moran, Zehetleitner, Liesefeld, Müller, & Usher, 2016; Townsend, 1990; Verghese, 2001). Nonetheless, for the purpose of examining the effects of additional digits on estimating numerical averages, these well-established RT patterns may provide a useful basis for comparison. Comparing performance across search and ensemble tasks may help us determine whether or not extracting semantic information from multiple digits in an array engages a relatively efficient mechanism akin to that proposed to underlie the relatively flat RT slopes in parallel, or “pop-out” visual search.

In the current study, we used displays that contained varying numbers of digits, and greater numbers of digits than previously tested (Van Opstal et al., 2011), and compared RT patterns for estimating numerical averages with those obtained during visual search for digits. In order to assess the effect of additional items on estimating the mean of digit arrays, we designed the search tasks according to well-established criteria known to yield performance consistent with traditional “serial” (increasing RTs with larger displays) or “parallel” (flat RT slopes) search tasks, and used identical digit arrays across the search and ensemble tasks. Critically, the instructions differed across the search and ensemble tasks. In both search tasks, we instructed participants to search for a number less-than- or greater-than-five, whereas in the ensemble, or averaging task, the instructions were to estimate whether the average numerical value of the digits in the display was less than or greater than five. An important difference between the two search tasks was that in the serial search, all of the digits were the same color, whereas in the parallel search, the digit that participants were searching for was always red and thus stood out compared to the rest of the display. Our goal was to directly compare RTs for estimation of the numerical average of digit displays with increasing numbers of items to those for a serial search task and a parallel search task (in which the target item pops out due to a unique low-level feature). By doing so, we aimed to determine whether the ability to rapidly extract a set statistic at the semantic level appears to proceed in a serial fashion, such that RTs increase with additional items, or whether instead, estimating a numerical average relies on processing of semantic information from multiple items in parallel.

We predicted that asking participants to estimate the average value of a set of digits would yield one of three types of RT patterns in relation to the search tasks (Figure 1). Increased RTs with increased display size, as in a serial search task, would suggest that generating the estimate is inefficient and may rely on serial processing of individual items. Alternatively, if RTs for the ensemble task show minimal or no increase with increasing display size, as characteristic of traditional parallel search tasks, this would be evidence that estimating numerical averages occurs by extracting semantic meaning from items in parallel. A third possibility is that if generating ensemble representations of numerical value relies on mechanisms similar to that proposed for ensemble perception of lower-level, perceptual stimulus attributes, RTs may actually decrease with increasing display size. This is in accordance with findings that additional items in a sequential digit display contribute to faster RTs and a more accurate ensemble representation (Brezis, Bronfman, & Usher, 2015), and other studies suggesting that additional stimuli provide more information about the ensemble average (Piazza, Sweeny, Wessel, Silver, & Whitney, 2013; Sweeny & Whitney, 2014). We did not have a strong prediction related to overall RTs for the ensemble compared to the search tasks, but for the purpose of illustration in Figure 1, we have indicated them to be similar to the serial search task but slower than the parallel search task, as pop-out searches tend to be highly efficient.

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Figure 1

Predictions for Experiment 1.

Note. For serial search (light gray), RTs are expected to increase with increasing display size, and for parallel search (dark gray), RTs typically remain flat or increase only slightly with increasing display size (all panels). The ensemble task (dashed black lines) may give rise to a RT pattern similar to that of either the serial search (left panel) or parallel search (middle panel) tasks, depending on whether numerical meaning is extracted from multiple items serially or in parallel. Alternatively, faster RTs with increasing display size (right panel) would suggest a contribution of ensemble coding mechanisms.

Experiment 1: Parallel vs. Serial Processing of Digit Displays

Method

Participants

For the current study, we predicted a medium effect size (f = .5) based on Cohen’s guidelines (Cohen, 1988). A preliminary power analysis was conducted using G*Power 3.1 software (Faul, Erdfelder, Lang, & Buchner, 2007) with a medium effect size of f = .5, an alpha level of .05, and a .95 confidence level. This analysis indicated that for 80% power, a minimum sample size of 10 participants is required for each set of tasks in this study.

Nineteen undergraduate students from a large public university in the Midwestern U.S. volunteered to participate for class credit. The university’s Institutional Review Board approved all experimental procedures. Data from three participants were excluded. Two had RTs that were greater than 2.5 standard deviations above the mean across participants (one for the ensemble task and one for the search tasks), and another had low accuracy (< 60%) in the search tasks. Thus, a total of three datasets were excluded across the three tasks to allow for within-subjects comparisons across tasks. The remaining 16 participants included 13 females and three males with normal or corrected-to-normal vision (M_age = 20.06, SD = 0.77).

Stimuli and Procedure

All task programs were written in Xojo Basic and conducted on a Mac computer connected to a built-in iMac display with a screen resolution of 1920 x 1080 pixels. Stimulus arrays were presented on the monitor and responses were gathered from the keyboard. Task (serial search, parallel search, ensemble) order was counterbalanced across participants. Stimuli for all tasks consisted of a fixation point surrounded by circular arrays of five, seven, or ten digits ranging from 1-9, but not including 5, presented in white text against a black background (Figure 2). At a viewing distance of about 60 cm, each digit was 0.92° degrees of visual angle wide by 1.8° tall. Digits were arranged on an imaginary circle with a radius of 5.9° centered on a fixation cross spanning 1.0° on each side. In the search tasks, the target digit was located in one of four quadrant locations: upper right, lower right, lower left, or upper left, and was always at least 30° of arc away from vertical to avoid ambiguity with respect to the vertical meridian.

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Figure 2

Example digit arrays for serial (left) and parallel search task (right), with the black background and white text color switched for aesthetic purposes.

Note. Displays consisted of 5, 7, or 10 digits; participants used the keyboard to indicate on which side of the display the less-than-five or greater-than-five digit appeared. In the serial search task all digits were the same color, whereas in the parallel search task targets were presented in red such that they popped out.

In different blocks, participants were instructed to search for the number greater than or less than five, and to report on which side of the display the number was located by pressing the ‘z’ key to indicate the left side of the display and the ‘/’ key for the right side. Block order within each task was counterbalanced. Trials began with a fixation cross presented for 500 ms, followed by the search display, which remained visible until the participant responded. Participants received feedback only when they responded incorrectly, in the form of a white screen with the word “Incorrect” in the middle presented for 750 ms followed by a fixation cross to begin the next trial. In the serial search task, the target digit was presented in the same white text as the distractors. The parallel search task was designed such that the target would pop out and thus yield RT patterns characteristic of parallel processing mechanisms. Displays and instructions for the parallel search task were identical to those for the serial search task, but the target digit was presented in red. Each task consisted of 12 practice trials and 120 experimental trials, with display sizes randomly interleaved.

The ensemble task consisted of a total of 288 trials that were identical to those in the serial search task, except that participants were instructed to estimate whether the average of the digits in each display was less than or greater than five (Figure 3). Participants responded using the ‘z’ key to indicate “less-than-five” and the ‘/’ key for “greater-than-five.” Averages of the digits within each display were always above or below 5 (never exactly 5); these averages ranged from ±1.2 to ±2.4 relative to 5. Displays across the search and ensemble tasks contained identical digits arrays. Each array contained only one digit that was on the “other side” of five from the average (e.g., the “3” in the display pictured on the left in Figure 3), which served as the target in the search tasks. This constraint, along with ensuring minimal digit repetitions within a display, yielded average differences from 5 for the means of the 5, 7, and 10 item displays that varied within a small range (±1.5, ±1.7, and ±2, respectively).

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Figure 3

Example digit arrays for ensemble estimation task.

Note. Displays consisted of 5, 7, or 10 digits. Participants were asked to estimate the average numerical value of the digits and use the keyboard to indicate whether the average was less than or greater than five.

Results

For each participant, we excluded all trials with RTs that were greater than the mean plus three standard deviations for that participant, or less than 100 ms; a total of 1.9%, 1.7%, and 1.5% of data points were removed for the ensemble, serial and parallel search tasks, respectively. A 3 (Task: Serial Search/Parallel Search/Ensemble) x 2 (Target Type: < 5/> 5) x 3 (Display Size: 5/7/10 items) repeated measures ANOVA was conducted on RTs. There was a significant main effect of task, F(2, 30) = 98.33, p < .001, such that RTs for the ensemble task were longer (M = 816.81, SD = 171.46) than for the serial (M = 716.04, SD = 117.33), and parallel (M = 400.29, SD = 33.97) search tasks, as shown in Figure 4. A significant main effect of target type, F(1, 15) = 18.55, p = .001 reflected faster RTs in the greater-than-five compared to the less-than-five condition for the ensemble (> 5: M = 776.82, SD = 147.92; < 5: M = 856.79, SD = 195.01) and serial tasks (> 5: M = 686.95, SD = 103.77; < 5: M = 745.13, SD = 130.89). There was also a main effect of display size, F(2, 30) = 11.92, p < .001, such that RTs increased with larger displays (5 items: M = 627.90, SD = 105.21; 7 items: M = 641.12, SD = 102.91; 10 items: M = 664.12, SD = 114.65). These main effects were qualified by a significant three-way interaction, F(4, 60) = 15.02, p < .001. We further examined this interaction by conducting a 2 (Target Type) x 3 (Display Size) repeated measures ANOVA for each task type followed by simple effects analyses and pairwise comparisons as appropriate.

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Figure 4

RTs for the serial search (light gray), parallel search (dark gray), and ensemble task (black) in Experiment 1 plotted as a function of display size and separated by target type.

Note. Dashed lines indicate < 5 and solid lines indicate > 5. Error bars represent standard error of the mean. RTs increased substantially with display size in the serial search task, but only slightly for parallel search (magnified in subfigure). RTs for the ensemble task decreased with larger displays in the greater-than-five condition (black solid line).

For the serial search task, a significant main effect of display size, F(2, 30) = 60.52, p < .001 reflected that as display size increased, RTs also increased (5 items: M = 640.37, SD = 98.07; 7 items: M = 707.51, SD = 106.89; 10 items: M = 800.25, SD = 147.04). A significant main effect of target type, F(1, 15) = 8.65, p = .01 was due to faster RTs for greater-than-five targets (M = 686.95, SD = 103.77) compared to less-than-five targets (M = 745.13, SD = 130.89). There was no interaction between these variables.

A 2 (Task: Serial Search/Parallel Search) x 3 (Display Size: 5/7/10 items) ANOVA compared the increases in RTs across the two search tasks. There was a main effect of task, F(1, 31) = 266.96, p < .001, such that RTs for the serial search (M = 716.04, SD = 119.97) were slower than for the parallel search (M = 400.29, SD = 33.47). There was also a main effect of display size, F(2, 62) = 100.47, p < .001, due to an overall increase in RTs with increasing display size. A significant interaction between task and display size, F(2, 62) = 81.30, p < .001 was followed up with pairwise comparisons to determine the relative effect of display size on RTs across the different search tasks.

For the serial search, there were substantial and significant differences in RTs between all display sizes (5 items: M = 640.37, SD = 98.07; 7 items: M = 707.51, SD = 106.89; 10 items: M = 800.25, SD = 147.04, all ps < .001), whereas the parallel search yielded an increase of less than 6 ms overall, with a significant difference only between the 5- (M = 397.68, SD = 34.07) and 10-item (M = 403.39, SD = 33.92) displays, t(31) = -2.79, p = .009.

For the ensemble task, there was a main effect of target type, F(1, 15) = 12.57, p = .003, due to faster RTs for the greater-than-five condition (M = 776.82, SD = 147.92) compared to the less-than-five condition (M = 856.79, SD = 195.01), just as observed for serial search (Figure 5). A significant main effect of display size, F(2, 30) = 5.52, p = .009, reflected a pattern different than that in either of the search tasks such that larger display sizes yielded faster RTs (5 items: M = 845.65, SD = 183.01; 7 items: M = 816.05, SD = 168.90; 10 items: M = 788.71, SD = 162.48). Because we also observed a significant interaction between target type and display size, F(2, 30) = 20.37, p < .001, we conducted further analyses to determine the simple effect of display size for each target type and found that RTs decreased with increasing display size for the greater-than-five condition, F(1, 15) = 35.86, p < .001 (5 items: M = 864.81, SD = 185.21; 7 items: M = 766.41, SD = 126.03; 10 items: M = 699.25, SD = 132.52; all ps < .01 for pairwise comparisons). The slight increases in RTs with increasing display size for the less-than-five condition were not significant, F(2,30) = 2.29, p = .119.

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Figure 5

Accuracy for the serial search (light gray), parallel search (dark gray), and ensemble (black) tasks in Experiment 1 plotted as a function of display size and separated by target type (dashed lines indicate < 5 and solid lines indicate > 5).

Note. Error bars represent standard error of the mean. The only significant effect of display size on accuracy reflected increased accuracy with larger displays in the greater-than-five condition (black solid line), F(2, 30) = 19.77, p < .001.

In order to determine whether any observed effects on RT were due to speed/accuracy trade-offs, accuracy was also submitted to a 3 (Task: Ensemble/Serial Search/Parallel Search) x 2 (Target Type: < 5/> 5) x 3 (Display Size: 5/7/10 items) repeated measures ANOVA. There was a significant main effect of task, F(2, 30) = 17.92, p < .001, such that accuracy was lower for the ensemble (M = .90, SD = .08) compared to the serial (M = 0.97, SD = .03) and parallel (M = .98, SD = .02) search tasks, consistent with the differences observed in RTs. A main effect of target type, F(1, 15) = 6.91, p = .019, was due to higher accuracy in the greater-than-five condition for the ensemble task (M = .92, SD = 0.06) compared to the less-than-five condition (M = .89, SD = 0.10). There was also a main effect of display size due to changes in accuracy with changing display sizes detailed below, F(2, 30) = 3.93, p = .031.

A significant task x target type x display size interaction, F(4, 60) = 7.24, p < .001, was further examined with a 2 (Target Type) x 3 (Display Size) ANOVA for each task type, which revealed a significant main effect of display size for the ensemble task, F(2, 30) = 8.95, p = .001, due to increased accuracy with more items in the display (5 items: M = 0.89, SD = 0.09; 7 items: M = 0.91, SD = 0.08; 10 items: M = 0.91, SD = 0.07). A significant main effect of target type, F(1, 15) = 7.11, p = .018, reflected higher accuracy for greater-than-five (M = 0.92, SD = 0.06) compared to the less-than-five displays (M = 0.88, SD = 0.10). A significant interaction between target type and display size, F(2, 30) = 8.39, p = .001, reflected that the main effect of display size described above was driven by increasing accuracy with larger displays in the greater-than-five condition, F(2, 30) = 19.77, p < .001. Pairwise comparisons revealed significantly lower accuracy for 5-item displays (M = 0.87, SD = 0.09) compared to 7- (M = 0.95, SD = 0.04) and 10-item (M = 0.96, SD = 0.04) displays in the greater-than-five condition (both ps < .005), but no significant difference between 7- and 10-item displays. There was no effect of display size in the less-than-five condition. All of these effects on accuracy are consistent with the RT effects, thus ruling out speed/accuracy trade-offs. There were no main effects of display size or target type on accuracy for either search task.

Discussion

As expected, in Experiment 1, increasing the number of items in the search task displays resulted in longer RTs for finding the digit that was greater than five or less than five, and much more so for the serial compared to parallel task in which the target was always red. This pattern indicates that in our serial search task, participants required additional processing time for additional items. Critically, when participants were asked to estimate the average of digits in the display (ensemble task), RTs decreased (for the > 5 displays) or stayed the same (for the < 5 displays), suggesting that individual items do not contribute serially to mean estimates; if anything, additional items may increase the efficiency of generating the set statistic as has been shown for high-level visual attributes such as gaze direction (Sweeny & Whitney, 2014). Moreover, accuracy improved with increased display size in the greater-than-five condition, ruling out the possibility that faster RTs for larger displays was due to a speed-accuracy tradeoff, and further supporting the notion that larger sets allow more efficient extraction of ensemble representations at the semantic level.

These results are consistent with the idea that a parallel processing mechanism underlies rapid estimation of the average of digit sets. However, because the digits we chose for the displays resulted in averages with mean differences from 5 that increased slightly across display sizes (from ±1.5 in the 5 item displays to ±2 in the 10 item displays), we wanted to determine whether this increase in distance from 5 contributed to the pattern of performance across display sizes. We therefore analyzed the subset of trials (~25 trials per display size per participant) for which the average distance from 5 was the same across display sizes (±1.8), and found the same pattern of results as in the main analysis (faster RTs for displays with more items, F(2, 30) = 4.18, p < .05, with no speed/accuracy trade-off). This additional analysis confirmed that the observed improvement in performance with larger displays was not due to the small difference in the means of displays across conditions.

Although the fact that RTs did not increase with larger display sizes in the ensemble task is consistent with parallel processing of numerical value during average estimation, the differences in RT slopes seen for the greater-than-five and less-than-five conditions in Experiment 1 were surprising. We considered the possibility that because displays with numerical means that were greater than five tended to include digits with a greater number of line segments and thus may have appeared brighter, participants could have taken advantage of this low-level visual information in performing the ensemble task. In addition to serving as a cue to whether the average was greater-than or less-than-five, this brightness difference could have resulted in a relative benefit (faster RTs) for larger displays particularly in the brighter, greater-than-five condition, and thus may explain why the decrease in RTs with larger displays occurred only for that condition. Therefore, we conducted a second experiment in which we used a restricted set of digits (2, 3 and 7, 8). This approach eliminated the systematic relationship between the correct response and the brightness of the displays present in Experiment 1, as displays composed of 2s and 3s contain the same number of line segments, on average, as those with 7s and 8s. In Experiment 2, participants performed either a serial search task in which they searched for a number less than (2 or 3) or greater than five (7 or 8), or an ensemble task in which they indicated whether the average of the digits was less than or greater than five.