# Virtual Reality as an Innovative Tool for Numerosity Perception
## 3 Results  
Six participants took part in a replication of the [PERSON] et al. study [35], which tested numerosity precision (i.e., Weber fractions) over a range of numerosities, and investigated whether humans perceive numerosity directly or infer it indirectly from texture density. Participants were required to choose which of two sets of red spheres was more numerous and their preference for the right set was plotted against the numerosity difference between the numerosity of the right and left sets normalized by the average numerosity of the range (Figure 3). These responses were fitted by a cumulative Gaussian function whose steepness indicates the precision in performing the judgment. The range necessary to move from 50% to 75% \"right more numerous\" responses indicate the extent of the zone of uncertainty reflecting the precision of the judgment. As visible from the representative participant shown in Figure 3, the three ranges considered in this study--low (avg 8), intermediate (avg 18), and high (avg 90)--produced different results. Whereas the initial two ranges produce similar curves (green and blue), the curve obtained for the high numerosity range (red) was steeper, indicating that judgments involving higher numerosities yield more precise decisions. To quantify this effect, for each participant, the Weber fraction (i.e., the JND/average numerosity) has been computed and plotted separately for the three numerosity ranges (Figure 4). As visible in the plots for low and intermediate numerosities, Weber fractions are around 13%, whereas for high numerosity they decrease to approximately 8%. The frequentist statistical analysis with one-way repeated measures ANOVA resulted in the presence of statistically significant differences among the three numerosity conditions for Weber fraction values (F(10,2) = 35, \(p<0.001\)). It is also confirmed by the Bayesian statistics for the model, showing extremely strong evidence in favor of the alternative hypothesis (BF\({}_{\text{M}}\) = 1665). [PERSON] corrected post hoc \(t\)-test for the various conditions indicate that the difference between low and intermediate range is not statistically significant (t(5) = 0.31, \(p>0.5\) two-tails) and actually there is moderate evidence for the null hypothesis (BF\({}_{10}\) = 0.26); instead, the difference between the lower conditions and the highest numerosity range is significant (low vs. high t(5) = 7.14, \(p<0.001\), intermediate vs. high  
Figure 4: Weber fractions of participants for the three numerosity conditions. Weber fraction values and error bars (**a**), and combined using average values and trend lines (**b**). Figure 3: Sample psychometric curves for a representative participant for the three numerical conditions. Raw data for the three numerosity ranges are plotted with different colors: green (low), blue (intermediate), and red (high). Raw judgments are fitted with a cumulative Gaussian curve (continuous lines). t(5) = 7.45, \(p\) < 0.001) and provide a strong support against the null hypothesis (BF\({}_{10}\) = 12.6 and BF\({}_{10}\) = 22.5, respectively). The Cohen's \(d\) values were calculated to estimate the pairwise effect sizes between group means. The results confirm the frequentist and Bayesian analyses. They indicate a very large effect for the comparisons between low vs. high (d = 2.44) and intermediate vs. high (d = 2.93), suggesting substantial differences in means. In contrast, the comparison between low and intermediate groups yielded a very small effect size (d = 0.09), implying minimal difference between these conditions. In summary, there is no significant difference between low and intermediate numerosities, while high numerosities are perceived significantly more accurately than low and intermediate conditions. Recent evidence suggests that another critical index of psychophysical performance is response times [45]. Previously, our group has demonstrated that response times for numerosity verbal estimation follows a descending pattern.