# A New Method for Haptic Shape Discriminability Detection
## 2 Angle Sorting
### Estimation of Standard Deviation
For example, a trial of angle sorting (\(AS_{1}\)) is \(26^{\circ}\), \(22^{\circ}\), \(24^{\circ}\), \(28^{\circ}\)  
Figure 2: In the angle sorting paradigm, when \(\mu=a\) and \(x=b\) are determined in the normal distribution, \(\sigma\) relates the participant’s haptic angle discriminability. The smaller \(\sigma_{1}\) is, the larger the probability density value, which reflects that the participants are more likely to discriminate the smallest angle, that is, the participant has better discrimination of the angle. The larger \(\sigma_{3}\) is, the smaller the probability density value, the lower the probability of choosing the smallest angle, and the worse the participant’s haptic angle discriminability. and 30\({}^{\circ}\), as shown in Figure 1. Here, we set up \(\mu=min(AS),\sigma=1.4\) and \(min(AS)\) as the smallest angle value among the remaining angle categories, then  
\[P_{2,n}=f(x;u;\sigma=1.4)=\frac{1}{1.4\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x- x}{14}\right)^{2}} \tag{1}\]  
In Equation (1), \(P_{2,n}\) is the probability that the remaining angle category stimulus is selected at the second position, \(n\in(22,\,24,\,26,\,28,\,30)\). The smallest angle among the remaining angle categories in the second position is \(22^{\circ}\), \(\mu=22\), and as the peak value of the normal distribution is \(P_{2,22}=f(x=22;u=22;\sigma=1.4)=0.2850\), the adjacent and other angle probabilities are \(P_{2,24}=f(x=24;u=22;\sigma=1.4)=0.1027\), \(P_{2,26}=f(x=26;u=22;\sigma=1.4)=0.0048\), \(P_{2,28}=f(x=28;u=22;\sigma=1.4)=0\) and \(P_{2,30}=f(x=30;u=22;\sigma=1.4)=0\). \(P_{3,n}\) is also the probability of the remaining angle category stimulus selected in the third position, \(n\in(22,\,24,\,28,\,30)\). The smallest angle among the remaining angle categories in the third position is still \(22^{\circ}\), \(\mu=22\), and as the peak value of the normal distribution is \(P_{3,22}=f(x=22;u=22;\sigma=1.4)=0.2850\), the adjacent and other angle probabilities are \(P_{3,24}=f(x=24;u=22;\sigma=1.4)=0.1027\), \(P_{3,26}=0\), \(P_{3,28}=f(x=28;u=22;\sigma=1.4)=0\) and \(P_{3,30}=f(x=30;u=22;\sigma=1.4)=0\). Here, the stimulus of \(26^{\circ}\) has been placed in the second position and the probability of \(P_{3,26}\) is 0. In the subsequent position, since each stimulus cannot be selected again once it has been selected, the probability of the angle stimulus selected before is also 0; thus, the probability of being selected in \(P_{4,n}\), \(P_{5,n}\), and \(P_{6,n}\) is also 0 in all cases. In addition, the probability of choosing the smallest angle at each position is always the greatest; that is, the probability of the angle being selected is the peak in its probability distribution, \(f(x=\mu;\mu;\sigma=1.4)=0.2850\); for example, the smallest angle in the third position is \(22^{\circ}\), which is the same as that of the second position. The probability density of the remaining angle selected decreases with increasing distance and obeys the normal distribution. Therefore, we can obtain the probability distributions of \(P_{4,n}\), \(P_{5,n}\) and \(P_{6,n}\) in the fourth, fifth and sixth positions, respectively, as shown in Table 1.  
In Table 1, the trial angle sorting is \(26^{\circ}\), \(22^{\circ}\), \(24^{\circ}\), \(28^{\circ}\) and \(30^{\circ}\). When we evaluate the probability of being selected at the second position, the probability of each angle is different. And we assume that when participants are asked to choose the smallest angle, the probability of choosing different angles obeys a normal distribution, so \(\mu=min(22,\,24,\,26,\,28,\,30)=22\).