# Virtual Reality as an Innovative Tool for Numerosity Perception
## 2 Materials and Methods
### Data Analysis  
Each participant estimated numerosity by choosing either \"right more numerous\" or \"left more numerous\" across 120 trials for each of the three evaluated conditions. Data recorded included the numerosity values of each pair of sphere groups, the participants' binary choices (1: right more numerous, 0: left more numerous), and the response time for each decision. The responses indicating \"right more numerous\" were plotted as a function of the numerosity difference between the right and left groups of spheres, normalized by the average numerosity of the range. These response curves were then fitted with a cumulative Gaussian function, which served to model the relationship between the numerosity difference and the likelihood of perceiving the correct group as more numerous. The median of the Gaussian function represents the point of subjective equality (PSE), which is the  
Figure 2: Sample configurations for the three numerical conditions. Participants compared the numerosity of two groups of spheres positioned to the left and right of the fixation point (blue dot). The numerosity levels were categorized as low (**a**), intermediate (**b**), or high (**c**). numerosity difference at which participants are equally likely to judge either the left or right group as more numerous. The steepness of the Gaussian curve is indicative of the precision or sensitivity of the participant's judgment, with steeper curves reflecting more precise judgments and broader curves suggesting greater uncertainty or noise in the response. The Weber's fraction, which measures the smallest noticeable difference between two stimuli relative to the magnitude of the original stimulus, can be used to quantify how precisely individuals distinguish between different quantities. To calculate the Weber's fraction, the just noticeable difference (JND) was extracted from the fitted Gaussian curve. The JND represents the smallest change in numerosity required for the participant to reliably distinguish between the two groups, reaching a 75% correct response rate. The JND was then divided by the average numerosity of the respective condition (i.e., 8 for the very low numerosity condition, 18 for the low numerosity condition, and 90 for the high numerosity condition) to compute the [PERSON]'s fraction. This fraction provides a measure of the relative sensitivity to numerosity differences, with larger values indicating poorer discrimination ability. To estimate the variability and ensure robust statistical analysis, error bars were calculated using bootstrapping, which involved resampling the original data 1000 times. This technique enabled the estimation of confidence intervals for each measure, providing a more reliable representation of the underlying variability in participants' responses [58]. To statistically assess the effects on precision and response times, Weber fractions and response times were analyzed using a one-way repeated measures ANOVA to determine whether statistically significant differences exist among the groups. This was followed by Bonferroni-corrected post hoc \(t\)-tests (two-tailed) to compare group pairs. Frequentist \(t\)-test were complemented with the Bayesian statistics, repeated measures ANOVA and post hoc \(t\)-test, with estimation of Bayes Factors (BF) [59], which quantify the evidence for or against the null hypothesis by comparing the likelihoods of the alternative hypothesis (H\({}_{1}\)) and null hypotheses (H\({}_{0}\)). By convention, Bayes Factors greater than 3 indicate moderate evidence in favor of the alternative hypothesis (greater than 10 for strong evidence), whereas values below 1/3 suggest moderate evidence against it (below 1/10 for strong evidence). To estimate the required sample size (\(N\)) for this preliminary evaluation, a large effect size (\(\geq\)2) was considered, along with a high statistical power (1 \(-\)\(\beta\) = 0.95) and a significance level (\(\alpha\) = 0.05).