The representation of individual planar locations and features stored in working memory can be affected by the average representation. However, less is known about how the average representation affects the short-term storage of depth information. To evaluate the possible different roles of the ensemble average in working memory for planar and depth information, we used mathematical models to fit the data collected from one study on working memory for depth and 12 studies on working memory for planar information. The pattern of recalled depth was well captured by models assuming that there was a probability of reporting the average depth instead of the individual depth, compressing the recalled front-back distance of the stimulus ensemble compared to the perceived distance. However, when modeling the recalled planar information, we found that participants tended to report individual nontarget features when the target was not memorized, and the assumption of reporting average information improves the data fitting only in very few studies. These results provide evidence for our hypothesis that average depth information can be used as a substitution for individual depth information stored in working memory, but for planar visual features, the substitution of target with the average works under a constraint that the average of to-be-remembered features is readily accessible.

_{T}is the proportion of trials in which participants make a target response, θ indicates the to-be-memorized value of the target, and \(\hat{\theta }\) indicates the recalled value. For WMd, the target-based recalls are assumed to be normally distributed, and thus φ represents a Gaussian distribution. For planar VWM, the responses are made on a circular analogue, and therefore φ are assumed to be a Von Mises distribution (Gaussian-like distribution for circular dimensions). The φ is characterized by its mean, µ, and its standard deviation, σ. Because studies showed that the recalled depth is not only contracted but also severely overestimated (e.g. Zhang, Gao, & Qian, 2021), for WMd, we set µ as the mean of participants’ recalled depths to compensate for the overestimation. For planar VWM, we set µ as 0, because no such overestimation was found in VWM for color and orientation (Wilken & Ma, 2004). The standard deviation of the response distribution, σ, which is the variability of target-based recalls, reflects the precision of memory recalls of the target – a smaller σ indicates a more precise recall. The free parameters of this component are γ

_{T}and σ. Because the stereoacuity for the most able observers is about 0.007 degrees (Carrillo, Baldwin, & Hess, 2020), and the least adjustable unit of depth in the task is 0.01 degrees, we restricted that σ cannot be smaller than 0.01 degrees for WMd.

_{A}is the proportion of trials in which participants make an average response, and \(\bar{\theta }\) indicates the ensemble average, which was set as the mean of all the individual values. The standard deviation (variability) of responses based on the average, σ

_{A}, reflects the precision of recalls for the average depth – a smaller σ

_{A}indicates a more precise recall. The free parameters of this component are γ

_{A}and σ

_{A}. For WMd, we restricted that σ

_{A}cannot be smaller than 0.01 degrees.

*m*nontarget items in one trial are donated by {\(\theta _1^*\), \(\theta _2^*\),……, \(\theta _m^*\)}. Each nontarget has an equal probability of being mistakenly reported as an estimate of the target. Because participants could not distinguish the target item from nontarget items until the testing phase, all the items should be memorized with equal precision in principle. Hence, the nontarget response and the target response share the same parameter of variability (i.e. σ

_{m}is equal to σ, and the mean of the response distribution, µ). The only free parameter of the nontarget component is γ

_{N}, which is the proportion of trials in which participants make a nontarget response.

*r*is 3.06 degrees for depths, 180 degrees for orientations of bars and Gabors, and 360 degrees for planar locations, colors, and orientations of the triangles used in the included experiments. γ

_{G}is the proportion of trials in which participants make random guesses, which is equal to 1 minus the proportions of all other types of responses.

_{j}is the value of the

*j*th item among m items on a trial. For planar VWM, the \({\bar{\theta }_{TA}}\) is the circular mean (Fisher, 1995) of the feature values on a trial.

*n*th trial can be defined as:

_{ij}is the value of the

*j*th item among m

_{i}items on the

*i*th trial in a total of n previous trials. For planar VWM, the \({\bar{\theta }_{WA}}\) is the circular mean of all the feature values on the current trial and all previous trials. Note that replacing \({\bar{\theta }_{TA}}\) in Equation 7 with \({\bar{\theta }_{WA}}\) defines the WA model.

*This model is another variant of the T&G model, which includes a target response component, a random guessing component, and a nontarget response component. No average response component is incorporated. The model can be defined as:*

**Target and Guess and Non-target model.**_{TA}and γ

_{WA}are the proportion of trials in which participants reported the TA and the WA, with the standard deviation of σ

_{TA}and σ

_{WA}, respectively. Note that, in principle, we could add a nontarget component in the TA&WA model, but this would result in too many free parameters in the model and therefore it was not evaluated.

*p*values for multiple comparisons were corrected by an adjustment of false discovery rate (FDR correction; Benjamini & Hochberg, 1995).

*t*-test (for a model including target/nontarget and average response components; e.g. Bays, Catalao, & Husain, 2009) would be conducted accordingly.

*t*-test to compare the precision of memory representation indicated by the specific response component.

*“Swap” error*section in

*Introduction*), but the results of modeling fittings suggest that the nontarget response component cannot sufficiently explain amount of contraction bias observed in the behavioral data. Overall, adding a nontarget component did not improve the fitting for the models that already had an average component. These results showed that the average response components, not the nontarget response component, are crucial in simulating the contraction bias.

*p*values of ANOVAs). For the single display, the models including the average components all predict that the probability of average response increases with set size (

*ps*< 0.024, although not for the TA component in the TA&WA model), which mirrors the trend that the contraction bias increased with set size in the single display. For the whole display, the models including the average components all predict that the probability of average response does not vary with set size (

*ps*> 0.069), which is also consistent with the raw data. For the three models, including the nontarget response, only the T&G&N model predicts that the probability of nontarget response increases with set size in the single display. Hence, these results showed that the models with an average response component generally performed better in capturing the characteristics of raw data.

*t*(14) = 2.30,

*p*= 0.037; whole-display, 0.12 vs. 0.23,

*t*(14) = 4.17,

*p*= 0.002; and for the TA model: single-display, 0.15 vs. 0.23,

*t*(14) = 3.68,

*p*= 0.003; whole-display, 0.12 vs. 0.25,

*t*(14) = 4.81,

*p*= 0.001. These results show that individual memory representation is consistently more precise than the average representation.

*t*(15) = 2.49,

*p*= 0.025; and in E11, 20.24 vs. 12.83,

*t*(15) = 3.97,

*p*= 0.002. These results seem to suggest that unlike WMd, the average representation for 2-D visual information, if generated, is more precise than individual representation.

**Data Availability Statement:**The data that support the findings of this study are available from the corresponding author upon request.

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