Evidence for a Global Sampling Process in Extraction of Summary Statistics of Item Sizes in a Set

Several studies have shown that our visual system may construct a “summary statistical representation” over groups of visual objects. Although there is a general understanding that human observers can accurately represent sets of a variety of features, many questions on how summary statistics, such as an average, are computed remain unanswered. This study investigated sampling properties of visual information used by human observers to extract two types of summary statistics of item sets, average and variance. We presented three models of ideal observers to extract the summary statistics: a global sampling model without sampling noise, global sampling model with sampling noise, and limited sampling model. We compared the performance of an ideal observer of each model with that of human observers using statistical efficiency analysis. Results suggest that summary statistics of items in a set may be computed without representing individual items, which makes it possible to discard the limited sampling account. Moreover, the extraction of summary statistics may not necessarily require the representation of individual objects with focused attention when the sets of items are larger than 4.


Discrimination of average size (Task in Experiment 2)
Item sizes in a set are chosen from a lognormal distribution of diameter sizes having mean D and SD σ c . Thus, Size item (item = 1,2,…,n) is expressed as follows: Size item ~ lnN(lnD, σ c 2 ) (1) Global sampling model 1: In GSM1, intrinsic noise, σ intrinsic , is added to each item in an item set, Size item_intrinsic , and a test item, Size test_intrinsic , independently. Thus, perceived size is expressed as follows: (2) Note that the value of σ intrinsic is obtained in Experiment 1. The average size of the item set, Size ave_GSM1 , is expressed as follows: where n is the number of items in a set (i.e., set size).
P ((Size test_intrinsic − Size ave_GSM1 ) > 0) is obtained and transformed to z c . Now, d′ I can be calculated by d′ I = √2z c .
Global sampling model 2: In GSM2, sampling noise, σ sampleN , is added to each item in an item set. Thus, the sampled item size, Size item_sampleN , is expressed as follows: Size item_sampleN ~ lnN(lnSize item , σ sampleN 2 ) Note that the value of σ sampleN is obtained in Experiment 1. The average size of the item set, Size ave_GSM2 , is expressed as follows: (4) P((Size test_intrinsic −Size ave_GSM2 ) > 0) is obtained and transformed to z c . Now, d′ I can be calculated by d′ I = √2z c .
Limited sampling model (LSM): LSM randomly samples four items from an item set, except for two items sampled in the set size two condition, using the same structures as GSM2. Thus, average size, Size ave_LSM , is expressed as follows: where n = 2 or 4. P(Size ave_LSM − Size test_intrinsic ) is obtained and transformed to z c . Now, d′ I can be calculated by d′ I = √2z c .

Global sampling model 2:
In GSM2, sampling noise, σ sampleN , is added to each item in a set. Thus, the sampled size in the standard set, Size stan_sampleN , and the comparison set, Size comp_sampleN , are expressed as follows: Size stan_sampleN ~ lnN (lnSize stan , σ sampleN 2 ) and Size ccomp_sampleN ~ lnN (ln Size comp , σ sampleN 2 ), respectively.
Limited sampling model (LSM): LSM randomly sampled four items from a set, except for two items sampled in the set size two condition, using the same structures as that of the GSM2 model. Thus, variances of the standard and comparison sets are given by the following equations: Size stan _ var_LSM = [(Size stan1_sampleN − Size s_ave_LSM ) 2 +…+ (Size stan_n_sampleN − Size stan_ave_LSM ) 2 ]/n and Size comp _ var_LSM = [(Size comp1_sampleN − Size comp_ave_LSM ) 2 +…+ (Size comp_n_sampleN − Size comp_ave_LSM ) 2 ]/n, provided Size stan_ave_LSM and Size comp_ave_LSM are given by Equation (4). P((Size comp _ var_GSM2 − Size stan _ var_GSM2 ) > 0) were obtained and transformed to z c . Now, d′ I can be calculated by d′ I = √2z c .