# Percentile Study of χ Distribution. Application to Response Time Data

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Percentile Analysis of the χ Distribution (from k = 2 to k = 10)

**1**(p1-p50-p99),

**2**(p3-p50-p97),

**3**(p5-p50-p95),

**4**(p10-p50-p90),

**5**(p20-p50-p80),

**6**(p25-p50-p75), and

**7**(p30-p50-p70). The preceding number is just a label to identify each combination. The three numbers between brackets are the positions of the symmetric percentiles about the median (p 50). For each value of the degree of freedom from k = 2 to k = 10, a ratio was calculated as follows for each percentile combination,

#### 2.2. Response Time Experiments. A case for k = 3

**1**to

**7**in Section 2.1), the following ones covering a broader area under the curve will be determined:

**8**(p0.5-p50-p99.5),

**9**(p0.7-p50-p99.3),

**10**(p2-p50-p98),

**11**(p7-p50-p93),

**12**(p15-p50-p85), and

**13**(p40-p50-p60).

## 3. Results and Discussions

#### 3.1. Multidimensional, Anisotropic Ideal Gas (χ of k Degrees of Freedom)

#### 3.2. Reaction Times and Ideal Gas. Appearance of the Golden Ratio

**8**(p0.5-p50-p99.5), r = 1.5690,

**9**(p0.7-p50-p99.3), r = 1.4421,

**10**(p2-p50-p98), r = 1.3305,

**11**(p7-p50-p93), r = 1.1902,

**12**(p15-p50-p85), and r = 1.0424,

**13**(p40-p50-p60).

**8**(p0.5-p50-p99.5), we can see that a ratio very close to the Golden Ratio is obtained. In Table 2 we present the values of ${x}_{1}$ (where CDF(${x}_{1})=0.005)$ and ${x}_{2}$ (where CDF(${x}_{2})=0.995)$, the value of r and the relative error with respect to the Golden Ratio $\mathsf{\Phi}=\frac{1+\sqrt{5}}{2}$ [7].

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Probability (panel

**a**) and cumulative (panel

**b**) density functions of the χ distribution for k = 2 to k = 10. In both panels the value of B is 0.5 (see Equations (1) and (2)).

**Figure 2.**Percentile ratio, $r$, versus the percentile combination number. The legend for the combination number is:

**1**(p1-p50-p99),

**2**(p3-p50-p97),

**3**(p5-p50-p95),

**4**(p10-p50-p90),

**5**(p20-p50-p80),

**6**(p25-p50-p75), and

**7**(p30-p50-p70).

**Figure 3.**MB-like distribution (χ for k = 3 and $B=0.123$) obtained from response time data with a coefficient of determination of ${R}^{2}=0.90$. The position of the median is indicated by a blue vertical line and the label “p50”. A total of 7200 reaction times were used. The appearance of the Golden Ratio can be observed.

**Table 1.**Ratios (r) for different combinations of percentile. As demonstrated above, these ratios are independent of the values of B chosen in Equations (1) and (2).

Combinations | k = 2 | k = 3 | k = 4 | k = 5 | k = 6 | k = 7 | k = 8 | k = 9 | k = 10 |
---|---|---|---|---|---|---|---|---|---|

1 (p1-p50-p99) | 1.7935 | 1.5259 | 1.4075 | 1.3404 | 1.2966 | 1.2655 | 1.2421 | 1.2237 | 1.2088 |

2 (p3-p50-p97) | 1.5805 | 1.3930 | 1.3090 | 1.2607 | 1.2287 | 1.2058 | 1.1884 | 1.1747 | 1.1635 |

3 (p5-p50-p95) | 1.4821 | 1.3305 | 1.2619 | 1.2220 | 1.1955 | 1.1764 | 1.1618 | 1.1502 | 1.1408 |

4 (p10-p50-p90) | 1.3483 | 1.2435 | 1.1951 | 1.1666 | 1.1474 | 1.1335 | 1.1228 | 1.1143 | 1.1073 |

5 (p20-p50-p80) | 1.2108 | 1.1506 | 1.1221 | 1.1051 | 1.0935 | 1.0850 | 1.0784 | 1.0731 | 1.0688 |

6 (p25-p50-p75) | 1.1643 | 1.1183 | 1.0963 | 1.0831 | 1.0740 | 1.0673 | 1.0622 | 1.0580 | 1.0547 |

7 (p30-p50-p70) | 1.1248 | 1.0904 | 1.0738 | 1.0638 | 1.0570 | 1.0519 | 1.0480 | 1.0448 | 1.0422 |

**Table 2.**Values of ${x}_{1}$, ${x}_{2}$, ${x}_{\mathrm{median}}$, the ratio r, and the relative error with respect to $\mathsf{\Phi}$ (k = 3).

k | ${\mathit{x}}_{1}$ | $\mathbf{CDF}({\mathit{x}}_{1})$ | ${\mathit{x}}_{2}$ | $\mathbf{CDF}({\mathit{x}}_{2})$ | ${\mathit{x}}_{\mathbf{median}}$ | r | $\mathsf{\Phi}$ | Relative Error |
---|---|---|---|---|---|---|---|---|

3 | 0.26789 | 0.005 | 3.5831 | 0.995 | 1.5382 | 1.6098 | 1.6180 | 0.5% |

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**MDPI and ACS Style**

Castro-Palacio, J.C.; Fernández-de-Córdoba, P.; Isidro, J.M.; Navarro-Pardo, E.; Selvas Aguilar, R.
Percentile Study of χ Distribution. Application to Response Time Data. *Mathematics* **2020**, *8*, 514.
https://doi.org/10.3390/math8040514

**AMA Style**

Castro-Palacio JC, Fernández-de-Córdoba P, Isidro JM, Navarro-Pardo E, Selvas Aguilar R.
Percentile Study of χ Distribution. Application to Response Time Data. *Mathematics*. 2020; 8(4):514.
https://doi.org/10.3390/math8040514

**Chicago/Turabian Style**

Castro-Palacio, Juan Carlos, Pedro Fernández-de-Córdoba, J. M. Isidro, Esperanza Navarro-Pardo, and Romeo Selvas Aguilar.
2020. "Percentile Study of χ Distribution. Application to Response Time Data" *Mathematics* 8, no. 4: 514.
https://doi.org/10.3390/math8040514