# The Properties of a Decile-Based Statistic to Measure Symmetry and Asymmetry

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## Abstract

**:**

## 1. Introduction

## 2. Decile-Based Skewness

^{th}order statistic. For 0 < p < 1, the $p$

^{th}quantile of $F$ is defined as ${x}_{p}={F}^{-1}\left(p\right)$ and the corresponding sample quantile is defined as ${X}_{\left(k\right)}$ where $k=\lceil np\rceil $, the ceiling of (the smallest integer greater than or equal to $np$). Let ${D}_{1}$ and ${D}_{9}$ be the first and nine sample deciles (0.1 and 0.9 quantiles), respectively. We consider our statistic for measuring the skewness by

**Lemma**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**2.**

#### 2.1. Asymptotic Confidence Interval

#### 2.2. Hypothesis Testing

## 3. Asymptotic Properties of the Proposed Statistic

## 4. Comparison with Alternative Measures

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The Q–Q plots versus standard normal distribution. Normal distribution: $n=50$ (

**a**), $n=1000$ (

**b**). t distribution: $n=50$ (

**c**), $n=1000$ (

**d**). Uniform distribution: $n=50$ (

**e**), $n=1000\text{}$(

**f**).

Distribution | $\mathit{n}$ | |||||
---|---|---|---|---|---|---|

$50$ | $75$ | $100$ | $200$ | $500$ | $1000$ | |

Normal (1,5) | 0.9732 | 0.9734 | 0.9743 | 0.9744 | 0.975 | 0.9761 |

t(10) | 0.9916 | 0.9928 | 0.9934 | 0.9939 | 0.9942 | 0.9947 |

U(0,1) | 0.9485 | 0.949 | 0.9491 | 0.9502 | 0.9521 | 0.9569 |

Distribution | $\mathit{n}$ | |||||
---|---|---|---|---|---|---|

$50$ | $75$ | $100$ | $200$ | $500$ | $1000$ | |

Normal (1,5) | 0.7131 | 0.7174 | 0.7899 | 0.8436 | 0.9077 | 0.9213 |

t(10) | 0.433 | 0.6515 | 0.781 | 0.8317 | 0.9603 | 0.9945 |

U(0,1) | 0.3144 | 0.5566 | 0.6034 | 0.6219 | 0.8249 | 0.9488 |

Distribution | Measure | n | ||
---|---|---|---|---|

10 | 20 | 50 | ||

Extremely Skewed | $SK$ | 0.798 | 0.989 | 1.000 |

${\gamma}_{1}$ | 0.687 | 0.942 | 1.000 | |

$S{K}_{G3}$ | 0.817 | 0.991 | 1.000 | |

$S{K}_{G2}$ | 0.834 | 0.992 | 1.000 | |

$S{K}_{G1}$ | 0.461 | 0.831 | 0.997 | |

$S{K}_{P}$ | 0.200 | 0.151 | 0.130 | |

$S{K}_{Y}$ | 0.616 | 0.869 | 0.999 | |

$S{K}_{P2}$ | 0.616 | 0.869 | 0.999 | |

$S{K}_{B}$ | 0.260 | 0.403 | 0.711 | |

Moderately Skewed | $SK$ | 0.318 | 0.597 | 0.945 |

${\gamma}_{1}$ | 0.297 | 0.530 | 0.889 | |

$S{K}_{G3}$ | 0.321 | 0.564 | 0.911 | |

$S{K}_{G2}$ | 0.318 | 0.623 | 0.941 | |

$S{K}_{G1}$ | 0.145 | 0.397 | 0.814 | |

$S{K}_{P}$ | 0.132 | 0.108 | 0.100 | |

$S{K}_{Y}$ | 0.207 | 0.344 | 0.651 | |

$S{K}_{P2}$ | 0.207 | 0.344 | 0.651 | |

$S{K}_{B}$ | 0.123 | 0.156 | 0.224 | |

Slightly Skewed | $SK$ | 0.144 | 0.163 | 0.289 |

${\gamma}_{1}$ | 0.129 | 0.165 | 0.288 | |

$S{K}_{G3}$ | 0.135 | 0.175 | 0.284 | |

$S{K}_{G2}$ | 0.143 | 0.153 | 0.282 | |

$S{K}_{G1}$ | 0.116 | 0.136 | 0.252 | |

$S{K}_{P}$ | 0.106 | 0.103 | 0.119 | |

$S{K}_{Y}$ | 0.120 | 0.116 | 0.180 | |

$S{K}_{P2}$ | 0.120 | 0.116 | 0.180 | |

$S{K}_{B}$ | 0.117 | 0.116 | 0.135 |

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**MDPI and ACS Style**

Mahmoudi, M.R.; Nasirzadeh, R.; Baleanu, D.; Pho, K.-H.
The Properties of a Decile-Based Statistic to Measure Symmetry and Asymmetry. *Symmetry* **2020**, *12*, 296.
https://doi.org/10.3390/sym12020296

**AMA Style**

Mahmoudi MR, Nasirzadeh R, Baleanu D, Pho K-H.
The Properties of a Decile-Based Statistic to Measure Symmetry and Asymmetry. *Symmetry*. 2020; 12(2):296.
https://doi.org/10.3390/sym12020296

**Chicago/Turabian Style**

Mahmoudi, Mohammad Reza, Roya Nasirzadeh, Dumitru Baleanu, and Kim-Hung Pho.
2020. "The Properties of a Decile-Based Statistic to Measure Symmetry and Asymmetry" *Symmetry* 12, no. 2: 296.
https://doi.org/10.3390/sym12020296