# Tables of Quantiles of the Distribution of the Empirical Chiral Index in the Case of the Uniform Law and in the Case of the Normal Law

Michel Petitjean

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Michel Petitjean. Tables of Quantiles of the Distribution of the Empirical Chiral Index in the Case of the Uniform Law and in the Case of the Normal Law. 2020. hal-02614523

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# Tables of Quantiles of the Distribution of the Empirical Chiral Index in the Case of the Uniform Law and in the Case of the Normal Law

Michel Petitjean

¹ Université de Paris, BFA, CNRS UMR 8251, INSERM ERL U1133, F-75013 Paris, France.

² E-pôle de Génoinformatique, CNRS UMR 7592, Institut Jacques Monod, 75013 Paris, France.

E-mail: petitjean.chiral@gmail.com, michel.petitjean@univ-paris-diderot.fr.

http://petitjeanmichel.free.fr/itoweb.petitjean.html.

ORCID: 0000-0002-1745-5402.

**Abstract:** The empirical distribution of the chiral index is simulated for various sample sizes for the uniform law and for the normal law. The estimated quantiles $K_{0.90}$, $K_{0.95}$, $K_{0.98}$, and $K_{0.99}$, are tabulated for use in symmetry testing in the uniform case and in the normal case.

**Keywords:** symmetry test; skewness; asymmetry coefficient; chiral index; location-free; scale-free; order statistics; midranges; range lengths; Monte-Carlo quantiles estimates; uniform; normal

**2020 MSC codes:** primary 62Q05; secondary: 62F03, 62G30

## 1 Introduction

The chiral index $\chi$ is a quantitative measure of chirality which can be used as an asymmetry coefficient of multivariate distributions [1, 2]. It takes values in the interval [0; 1]. The chiral index is null if and only if there is a mirror symmetry. It is insensitive to isometries and scaling. In the case of a d-variate distribution, its upper bound $\chi^*$ depends on the dimensionality $d$. $\chi^* = 1/2$ when $d = 1$ [1], $\chi^* \in [1 - 1/\pi; 1 - 1/2\pi]$ when $d = 2$ [3], and $\chi^* \in [1/2; 1]$ when $d \ge 3$ [4]. As a consequence of a convergence theorem [1], the empirical chiral index of a mirror symmetric distribution converges to 0, provided that the parent distribution has a finite and non null inertia. So, the chiral index can be used for testing the symmetry of the parent law. The purpose of this paper is to provide tables of the estimated quantiles $K_{0.90}$, $K_{0.95}$, $K_{0.98}$, and $K_{0.99}$, in the univariate uniform case and in the normal case.

## 2 Theory

The chiral index $\chi$ of an univariate distribution is given in equation 1 [1, 2]:

$$ \chi = (1 + r_m)/2 \qquad (1) $$

where $r_m$ is the lower bound of the correlation coefficient between two identical distributions, taken over all their joint distributions. $r_m$ cannot be positive, $\chi = 0$ if and only if the distribution is symmetric, and its upper bound is $\chi^* = 1/2$. This latter is asymptotically reached for the Bernoulli distribution with parameter tending to 0 or 1 [1].

We consider a sample of size $n$ from a parent distribution with finite and non null inertia. $X_{i:n} \le \dots \le X_{n:n}$ are the order statistics, $\bar{X}$ is their mean, and $\sigma$ is their standard deviation. $M_i = (X_{n+1-i:n} + X_{i:n})/2$ is the midrange of the $i^{th}$ interval, and $L_i = (X_{n+1-i:n} - X_{i:n})/2$ is the
half range length of the $i^{th}$ interval, $i = 1...n$. The $i^{th}$ interval and the $(n-i)^{th}$ interval are identical except in their orientation. Thus $L_i$ is negative for $i > (n/2)$. The chiral index can be expressed from the studentized variance of the midranges or of the half ranges lengths: see equations 2.9.4 and 2.9.5 in [2].

$$ \chi = \left[ \sum_{i=1}^{i=n} M_i^2 - n \cdot \bar{X}^2 \right] / (n\sigma^2) \quad (2) $$

$$ \chi = 1 - \left[ \sum_{i=1}^{i=n} L_i^2 \right] / (n\sigma^2) \quad (3) $$

For practical computations, it is much simpler to use the expression of $r_m$ in equation 1, for which the minimal correlation exists and is such that the sequence of observations sorted in increasing order is correlated with the sequence of observations sorted in decreasing order [2, 5]. Thus, the algorithm to compute $\chi$ is very simple:

(a) Sort the sequence of $n$ observations in increasing order

(b) Correlate it with the sequence sorted in decreasing order

(c) Add 1 and then divide the result by 2.

Step (a) is operating in $O(n \cdot \text{Log}(n))$ and step (b) in $O(n)$, and the global algorithm is optimal worst-case in $O(n \cdot \text{Log}(n))$. The whole computation is easy on a pocket calculator.

## 3 Simulations and results

Table 1 was generated for the continuous uniform law $U(0, 1)$. Table 2 was generated for the normal law $N(0, 1)$. For each sample size $n$ from $U(0, 1)$ or from $N(0, 1)$, we compute one chiral index, i.e., one observation from the distribution of the chiral index. So, to approach this distribution, 10000 observations of the chiral index are generated. For each set of 10000 observations of the chiral index, four quantiles are computed as follow ($\chi_{i:10000}$ denotes the $i^{th}$ order statistic):

$$ K_{0.90} = (\chi_{9000:10000} + \chi_{9001:10000})/2 \quad (4) $$

$$ K_{0.95} = (\chi_{9500:10000} + \chi_{9501:10000})/2 \quad (5) $$

$$ K_{0.98} = (\chi_{9800:10000} + \chi_{9801:10000})/2 \quad (6) $$

$$ K_{0.99} = (\chi_{9900:10000} + \chi_{9901:10000})/2 \quad (7) $$

The whole simulation is repeated 100 times, and both the mean values $\bar{K}_{0.90}$, $\bar{K}_{0.95}$, $\bar{K}_{0.98}$, and $\bar{K}_{0.99}$, and the estimated standard deviations $S_{K_{0.90}}$, $S_{K_{0.95}}$, $S_{K_{0.98}}$, and $S_{K_{0.99}}$ of the four distributions of the quantiles are calculated and reported in tables 1 and 2.

The random generator is a long period pseudo-random congruential multiplicative one, used in the routine g05saf of the NAG library [6]. All data of both tables were generated sequentially within a single executable program, without reinitializing the random generator.

Tables 1 or 2 can be used as follows. Compute the chiral index $\chi$ of your sample of size $n$.

* If you assume that your sample is issued from the continuous uniform law $U(a, b)$, $a < b$, look in Table 1 if $\chi$ exceeds the quantile value at the desired confidence level (e.g. 0.95). If yes, you may reject the symmetry assumption, and thus you may reject the uniform law assumption.

* If you assume that your sample is issued from the normal law $N(m, s)$, look in Table 2 if $\chi$ exceeds the quantile value at the desired confidence level (e.g. 0.95). If yes, you may reject the symmetry assumption, and thus you may reject the normal law assumption.
<caption>**Table 1.** Simulations for the uniform law $U(0, 1)$. $n$: sample size.</caption>

$\bar{K}_{0.90}$, $\bar{K}_{0.95}$, $\bar{K}_{0.98}$, $\bar{K}_{0.99}$: mean values of estimated quantiles over 100 experiments.

$S_{K_{0.90}}$, $S_{K_{0.95}}$, $S_{K_{0.98}}$, $S_{K_{0.99}}$: estimated standard deviations of the quantiles distributions.

<table><thead><tr><th>n</th><th>K̄<sub>0.90</sub></th><th>K̄<sub>0.95</sub></th><th>K̄<sub>0.98</sub></th><th>K̄<sub>0.99</sub></th><th>S<sub>K<sub>0.90</sub></sub></th><th>S<sub>K<sub>0.95</sub></sub></th><th>S<sub>K<sub>0.98</sub></sub></th><th>S<sub>K<sub>0.99</sub></sub></th></tr></thead><tbody><tr><td>3</td><td>0.212764</td><td>0.231432</td><td>0.242661</td><td>0.246342</td><td>0.001020</td><td>0.000774</td><td>0.000583</td><td>0.000320</td></tr><tr><td>4</td><td>0.185211</td><td>0.229221</td><td>0.268552</td><td>0.287850</td><td>0.002169</td><td>0.002272</td><td>0.002310</td><td>0.002153</td></tr><tr><td>5</td><td>0.154894</td><td>0.195716</td><td>0.242379</td><td>0.270442</td><td>0.001717</td><td>0.002531</td><td>0.003358</td><td>0.003770</td></tr><tr><td>6</td><td>0.138459</td><td>0.174426</td><td>0.214915</td><td>0.242201</td><td>0.001695</td><td>0.001989</td><td>0.002476</td><td>0.003498</td></tr><tr><td>7</td><td>0.122632</td><td>0.156176</td><td>0.195287</td><td>0.221293</td><td>0.001495</td><td>0.001935</td><td>0.002647</td><td>0.003295</td></tr><tr><td>8</td><td>0.109606</td><td>0.139940</td><td>0.176472</td><td>0.201666</td><td>0.001339</td><td>0.001867</td><td>0.002653</td><td>0.003400</td></tr><tr><td>9</td><td>0.099866</td><td>0.127905</td><td>0.161753</td><td>0.185403</td><td>0.001367</td><td>0.001822</td><td>0.002665</td><td>0.003770</td></tr><tr><td>10</td><td>0.090742</td><td>0.116454</td><td>0.148675</td><td>0.171121</td><td>0.001184</td><td>0.001687</td><td>0.002358</td><td>0.003258</td></tr><tr><td>11</td><td>0.083883</td><td>0.107905</td><td>0.137897</td><td>0.158657</td><td>0.001161</td><td>0.001501</td><td>0.002188</td><td>0.003212</td></tr><tr><td>12</td><td>0.077375</td><td>0.099899</td><td>0.128247</td><td>0.148157</td><td>0.000985</td><td>0.001459</td><td>0.002142</td><td>0.002723</td></tr><tr><td>13</td><td>0.072242</td><td>0.093095</td><td>0.119908</td><td>0.138950</td><td>0.000913</td><td>0.001188</td><td>0.002112</td><td>0.002306</td></tr><tr><td>14</td><td>0.067347</td><td>0.087017</td><td>0.112577</td><td>0.130622</td><td>0.000790</td><td>0.001316</td><td>0.001573</td><td>0.002496</td></tr><tr><td>15</td><td>0.063401</td><td>0.081884</td><td>0.105740</td><td>0.123299</td><td>0.000871</td><td>0.001163</td><td>0.001712</td><td>0.002299</td></tr><tr><td>16</td><td>0.059669</td><td>0.077378</td><td>0.100094</td><td>0.116943</td><td>0.000788</td><td>0.001031</td><td>0.001639</td><td>0.002331</td></tr><tr><td>17</td><td>0.056655</td><td>0.073521</td><td>0.095419</td><td>0.111137</td><td>0.000743</td><td>0.001169</td><td>0.001649</td><td>0.002057</td></tr><tr><td>18</td><td>0.053513</td><td>0.069685</td><td>0.090491</td><td>0.105711</td><td>0.000691</td><td>0.001029</td><td>0.001668</td><td>0.002266</td></tr><tr><td>19</td><td>0.050912</td><td>0.066281</td><td>0.086226</td><td>0.100894</td><td>0.000624</td><td>0.000967</td><td>0.001355</td><td>0.002063</td></tr><tr><td>20</td><td>0.048617</td><td>0.063213</td><td>0.082350</td><td>0.096221</td><td>0.000623</td><td>0.000875</td><td>0.001360</td><td>0.001722</td></tr><tr><td>21</td><td>0.046461</td><td>0.060632</td><td>0.078984</td><td>0.092450</td><td>0.000607</td><td>0.000980</td><td>0.001557</td><td>0.001941</td></tr><tr><td>22</td><td>0.044460</td><td>0.058067</td><td>0.075878</td><td>0.088865</td><td>0.000586</td><td>0.000829</td><td>0.001383</td><td>0.001741</td></tr><tr><td>23</td><td>0.042783</td><td>0.055897</td><td>0.073019</td><td>0.085646</td><td>0.000673</td><td>0.000865</td><td>0.001218</td><td>0.001951</td></tr><tr><td>24</td><td>0.040955</td><td>0.053483</td><td>0.070111</td><td>0.082124</td><td>0.000591</td><td>0.000734</td><td>0.001247</td><td>0.001733</td></tr><tr><td>25</td><td>0.039345</td><td>0.051406</td><td>0.067428</td><td>0.079209</td><td>0.000463</td><td>0.000868</td><td>0.001282</td><td>0.001677</td></tr><tr><td>26</td><td>0.037959</td><td>0.049700</td><td>0.065121</td><td>0.076492</td><td>0.000457</td><td>0.000749</td><td>0.001253</td><td>0.001640</td></tr><tr><td>27</td><td>0.036764</td><td>0.048079</td><td>0.063067</td><td>0.074118</td><td>0.000524</td><td>0.000689</td><td>0.001010</td><td>0.001453</td></tr><tr><td>28</td><td>0.035401</td><td>0.046299</td><td>0.060849</td><td>0.071566</td><td>0.000507</td><td>0.000681</td><td>0.001274</td><td>0.001703</td></tr><tr><td>29</td><td>0.034266</td><td>0.044886</td><td>0.058923</td><td>0.069457</td><td>0.000519</td><td>0.000673</td><td>0.001147</td><td>0.001431</td></tr><tr><td>30</td><td>0.033099</td><td>0.043416</td><td>0.057067</td><td>0.067353</td><td>0.000409</td><td>0.000569</td><td>0.000986</td><td>0.001319</td></tr><tr><td>31</td><td>0.032121</td><td>0.042051</td><td>0.055308</td><td>0.065082</td><td>0.000429</td><td>0.000658</td><td>0.000945</td><td>0.001362</td></tr><tr><td>32</td><td>0.031084</td><td>0.040791</td><td>0.053830</td><td>0.063512</td><td>0.000405</td><td>0.000621</td><td>0.001171</td><td>0.001500</td></tr><tr><td>33</td><td>0.030233</td><td>0.039744</td><td>0.052259</td><td>0.061845</td><td>0.000389</td><td>0.000501</td><td>0.000990</td><td>0.001308</td></tr><tr><td>34</td><td>0.029371</td><td>0.038641</td><td>0.050966</td><td>0.060222</td><td>0.000427</td><td>0.000659</td><td>0.001019</td><td>0.001513</td></tr><tr><td>35</td><td>0.028665</td><td>0.037600</td><td>0.049672</td><td>0.058667</td><td>0.000424</td><td>0.000611</td><td>0.000893</td><td>0.001281</td></tr><tr><td>36</td><td>0.027763</td><td>0.036530</td><td>0.048173</td><td>0.057204</td><td>0.000354</td><td>0.000584</td><td>0.000949</td><td>0.001177</td></tr><tr><td>37</td><td>0.027185</td><td>0.035736</td><td>0.047072</td><td>0.055737</td><td>0.000371</td><td>0.000599</td><td>0.000971</td><td>0.001313</td></tr><tr><td>38</td><td>0.026346</td><td>0.034660</td><td>0.045839</td><td>0.054346</td><td>0.000359</td><td>0.000478</td><td>0.000817</td><td>0.001306</td></tr><tr><td>39</td><td>0.025722</td><td>0.033817</td><td>0.044709</td><td>0.053001</td><td>0.000307</td><td>0.000518</td><td>0.000888</td><td>0.001334</td></tr><tr><td>40</td><td>0.025175</td><td>0.033091</td><td>0.043765</td><td>0.051885</td><td>0.000352</td><td>0.000512</td><td>0.000833</td><td>0.001107</td></tr><tr><td>41</td><td>0.024560</td><td>0.032330</td><td>0.042778</td><td>0.050509</td><td>0.000328</td><td>0.000461</td><td>0.000848</td><td>0.001111</td></tr><tr><td>42</td><td>0.024013</td><td>0.031578</td><td>0.041900</td><td>0.049560</td><td>0.000275</td><td>0.000440</td><td>0.000796</td><td>0.001129</td></tr><tr><td>43</td><td>0.023479</td><td>0.030914</td><td>0.040873</td><td>0.048470</td><td>0.000314</td><td>0.000473</td><td>0.000764</td><td>0.001153</td></tr><tr><td>44</td><td>0.022904</td><td>0.030150</td><td>0.039905</td><td>0.047332</td><td>0.000296</td><td>0.000401</td><td>0.000707</td><td>0.000884</td></tr><tr><td>45</td><td>0.022436</td><td>0.029536</td><td>0.039130</td><td>0.046380</td><td>0.000308</td><td>0.000474</td><td>0.000796</td><td>0.001086</td></tr></tbody></table>
<table><caption>Table 1. Continued.</caption><tr><td>46</td><td>0.021912</td><td>0.028844</td><td>0.038205</td><td>0.045312</td><td>0.000271</td><td>0.000395</td><td>0.000677</td><td>0.001032</td></tr><tr><td>47</td><td>0.021513</td><td>0.028343</td><td>0.037557</td><td>0.044496</td><td>0.000293</td><td>0.000427</td><td>0.000736</td><td>0.000993</td></tr><tr><td>48</td><td>0.021034</td><td>0.027734</td><td>0.036730</td><td>0.043550</td><td>0.000261</td><td>0.000407</td><td>0.000606</td><td>0.000907</td></tr><tr><td>49</td><td>0.020630</td><td>0.027197</td><td>0.035990</td><td>0.042803</td><td>0.000279</td><td>0.000415</td><td>0.000791</td><td>0.001129</td></tr><tr><td>50</td><td>0.020233</td><td>0.026654</td><td>0.035312</td><td>0.041995</td><td>0.000260</td><td>0.000389</td><td>0.000583</td><td>0.000858</td></tr><tr><td>51</td><td>0.019919</td><td>0.026211</td><td>0.034795</td><td>0.041392</td><td>0.000283</td><td>0.000397</td><td>0.000710</td><td>0.000930</td></tr><tr><td>52</td><td>0.019437</td><td>0.025630</td><td>0.034018</td><td>0.040360</td><td>0.000289</td><td>0.000382</td><td>0.000645</td><td>0.000909</td></tr><tr><td>53</td><td>0.019145</td><td>0.025207</td><td>0.033497</td><td>0.039760</td><td>0.000257</td><td>0.000393</td><td>0.000566</td><td>0.000738</td></tr><tr><td>54</td><td>0.018779</td><td>0.024782</td><td>0.032827</td><td>0.038877</td><td>0.000271</td><td>0.000368</td><td>0.000610</td><td>0.000911</td></tr><tr><td>55</td><td>0.018481</td><td>0.024346</td><td>0.032247</td><td>0.038246</td><td>0.000234</td><td>0.000399</td><td>0.000669</td><td>0.000914</td></tr><tr><td>56</td><td>0.018195</td><td>0.023973</td><td>0.031819</td><td>0.037913</td><td>0.000223</td><td>0.000348</td><td>0.000622</td><td>0.000879</td></tr><tr><td>57</td><td>0.017848</td><td>0.023510</td><td>0.031156</td><td>0.036994</td><td>0.000235</td><td>0.000317</td><td>0.000510</td><td>0.000796</td></tr><tr><td>58</td><td>0.017540</td><td>0.023125</td><td>0.030707</td><td>0.036498</td><td>0.000227</td><td>0.000330</td><td>0.000580</td><td>0.000791</td></tr><tr><td>59</td><td>0.017256</td><td>0.022719</td><td>0.030166</td><td>0.035924</td><td>0.000244</td><td>0.000360</td><td>0.000608</td><td>0.000911</td></tr><tr><td>60</td><td>0.016933</td><td>0.022354</td><td>0.029726</td><td>0.035475</td><td>0.000199</td><td>0.000327</td><td>0.000583</td><td>0.000872</td></tr><tr><td>61</td><td>0.016717</td><td>0.022031</td><td>0.029217</td><td>0.034741</td><td>0.000206</td><td>0.000330</td><td>0.000589</td><td>0.000812</td></tr><tr><td>62</td><td>0.016420</td><td>0.021642</td><td>0.028772</td><td>0.034233</td><td>0.000215</td><td>0.000315</td><td>0.000526</td><td>0.000739</td></tr><tr><td>63</td><td>0.016188</td><td>0.021349</td><td>0.028339</td><td>0.033746</td><td>0.000210</td><td>0.000340</td><td>0.000574</td><td>0.000838</td></tr><tr><td>64</td><td>0.015933</td><td>0.021102</td><td>0.028073</td><td>0.033321</td><td>0.000208</td><td>0.000322</td><td>0.000554</td><td>0.000765</td></tr><tr><td>65</td><td>0.015717</td><td>0.020733</td><td>0.027559</td><td>0.032849</td><td>0.000210</td><td>0.000341</td><td>0.000517</td><td>0.000701</td></tr><tr><td>66</td><td>0.015451</td><td>0.020362</td><td>0.027101</td><td>0.032306</td><td>0.000208</td><td>0.000292</td><td>0.000490</td><td>0.000679</td></tr><tr><td>67</td><td>0.015220</td><td>0.020104</td><td>0.026754</td><td>0.031902</td><td>0.000212</td><td>0.000319</td><td>0.000473</td><td>0.000708</td></tr><tr><td>68</td><td>0.014980</td><td>0.019793</td><td>0.026426</td><td>0.031379</td><td>0.000220</td><td>0.000301</td><td>0.000510</td><td>0.000687</td></tr><tr><td>69</td><td>0.014789</td><td>0.019509</td><td>0.026005</td><td>0.030905</td><td>0.000200</td><td>0.000311</td><td>0.000505</td><td>0.000675</td></tr><tr><td>70</td><td>0.014596</td><td>0.019222</td><td>0.025610</td><td>0.030422</td><td>0.000180</td><td>0.000279</td><td>0.000464</td><td>0.000701</td></tr><tr><td>71</td><td>0.014384</td><td>0.018999</td><td>0.025335</td><td>0.030181</td><td>0.000197</td><td>0.000306</td><td>0.000495</td><td>0.000731</td></tr><tr><td>72</td><td>0.014147</td><td>0.018706</td><td>0.024913</td><td>0.029688</td><td>0.000171</td><td>0.000258</td><td>0.000490</td><td>0.000679</td></tr><tr><td>73</td><td>0.013969</td><td>0.018462</td><td>0.024527</td><td>0.029260</td><td>0.000163</td><td>0.000254</td><td>0.000424</td><td>0.000756</td></tr><tr><td>74</td><td>0.013788</td><td>0.018167</td><td>0.024190</td><td>0.028830</td><td>0.000201</td><td>0.000281</td><td>0.000457</td><td>0.000682</td></tr><tr><td>75</td><td>0.013624</td><td>0.018002</td><td>0.023958</td><td>0.028546</td><td>0.000193</td><td>0.000268</td><td>0.000456</td><td>0.000623</td></tr><tr><td>76</td><td>0.013447</td><td>0.017747</td><td>0.023662</td><td>0.028168</td><td>0.000190</td><td>0.000273</td><td>0.000414</td><td>0.000566</td></tr><tr><td>77</td><td>0.013302</td><td>0.017557</td><td>0.023380</td><td>0.027778</td><td>0.000177</td><td>0.000261</td><td>0.000425</td><td>0.000659</td></tr><tr><td>78</td><td>0.013119</td><td>0.017334</td><td>0.023157</td><td>0.027641</td><td>0.000177</td><td>0.000285</td><td>0.000470</td><td>0.000709</td></tr><tr><td>79</td><td>0.012959</td><td>0.017119</td><td>0.022789</td><td>0.027187</td><td>0.000178</td><td>0.000272</td><td>0.000400</td><td>0.000622</td></tr><tr><td>80</td><td>0.012777</td><td>0.016868</td><td>0.022473</td><td>0.026816</td><td>0.000175</td><td>0.000272</td><td>0.000441</td><td>0.000686</td></tr><tr><td>81</td><td>0.012627</td><td>0.016715</td><td>0.022264</td><td>0.026520</td><td>0.000164</td><td>0.000264</td><td>0.000417</td><td>0.000662</td></tr><tr><td>82</td><td>0.012499</td><td>0.016512</td><td>0.022052</td><td>0.026281</td><td>0.000178</td><td>0.000289</td><td>0.000403</td><td>0.000529</td></tr><tr><td>83</td><td>0.012352</td><td>0.016319</td><td>0.021718</td><td>0.025897</td><td>0.000158</td><td>0.000248</td><td>0.000408</td><td>0.000606</td></tr><tr><td>84</td><td>0.012192</td><td>0.016114</td><td>0.021558</td><td>0.025664</td><td>0.000182</td><td>0.000264</td><td>0.000398</td><td>0.000553</td></tr><tr><td>85</td><td>0.012077</td><td>0.015989</td><td>0.021302</td><td>0.025331</td><td>0.000169</td><td>0.000262</td><td>0.000411</td><td>0.000573</td></tr><tr><td>86</td><td>0.011906</td><td>0.015734</td><td>0.020964</td><td>0.024974</td><td>0.000153</td><td>0.000249</td><td>0.000426</td><td>0.000618</td></tr><tr><td>87</td><td>0.011756</td><td>0.015554</td><td>0.020740</td><td>0.024612</td><td>0.000149</td><td>0.000214</td><td>0.000363</td><td>0.000521</td></tr><tr><td>88</td><td>0.011650</td><td>0.015398</td><td>0.020566</td><td>0.024557</td><td>0.000172</td><td>0.000220</td><td>0.000373</td><td>0.000492</td></tr><tr><td>89</td><td>0.011506</td><td>0.015238</td><td>0.020295</td><td>0.024188</td><td>0.000179</td><td>0.000242</td><td>0.000417</td><td>0.000572</td></tr><tr><td>90</td><td>0.011421</td><td>0.015099</td><td>0.020147</td><td>0.024046</td><td>0.000166</td><td>0.000230</td><td>0.000428</td><td>0.000545</td></tr></table>
<table><caption>Table 1. Continued.</caption><tr><td>91</td><td>0.011254</td><td>0.014856</td><td>0.019800</td><td>0.023658</td><td>0.000148</td><td>0.000235</td><td>0.000372</td><td>0.000564</td></tr><tr><td>92</td><td>0.011142</td><td>0.014747</td><td>0.019618</td><td>0.023369</td><td>0.000137</td><td>0.000223</td><td>0.000367</td><td>0.000542</td></tr><tr><td>93</td><td>0.011026</td><td>0.014585</td><td>0.019455</td><td>0.023258</td><td>0.000137</td><td>0.000218</td><td>0.000377</td><td>0.000570</td></tr><tr><td>94</td><td>0.010927</td><td>0.014433</td><td>0.019267</td><td>0.023014</td><td>0.000173</td><td>0.000265</td><td>0.000363</td><td>0.000512</td></tr><tr><td>95</td><td>0.010822</td><td>0.014315</td><td>0.019090</td><td>0.022762</td><td>0.000153</td><td>0.000213</td><td>0.000364</td><td>0.000586</td></tr><tr><td>96</td><td>0.010670</td><td>0.014148</td><td>0.018876</td><td>0.022528</td><td>0.000146</td><td>0.000221</td><td>0.000355</td><td>0.000482</td></tr><tr><td>97</td><td>0.010588</td><td>0.014029</td><td>0.018659</td><td>0.022190</td><td>0.000136</td><td>0.000200</td><td>0.000335</td><td>0.000470</td></tr><tr><td>98</td><td>0.010484</td><td>0.013846</td><td>0.018503</td><td>0.022106</td><td>0.000146</td><td>0.000224</td><td>0.000319</td><td>0.000460</td></tr><tr><td>99</td><td>0.010364</td><td>0.013718</td><td>0.018361</td><td>0.021858</td><td>0.000140</td><td>0.000224</td><td>0.000321</td><td>0.000513</td></tr><tr><td>100</td><td>0.010268</td><td>0.013574</td><td>0.018043</td><td>0.021550</td><td>0.000143</td><td>0.000230</td><td>0.000368</td><td>0.000502</td></tr><tr><td>110</td><td>0.009351</td><td>0.012355</td><td>0.016481</td><td>0.019638</td><td>0.000132</td><td>0.000204</td><td>0.000356</td><td>0.000485</td></tr><tr><td>120</td><td>0.008593</td><td>0.011360</td><td>0.015157</td><td>0.018115</td><td>0.000132</td><td>0.000185</td><td>0.000281</td><td>0.000418</td></tr><tr><td>130</td><td>0.007919</td><td>0.010494</td><td>0.014075</td><td>0.016789</td><td>0.000117</td><td>0.000193</td><td>0.000276</td><td>0.000412</td></tr><tr><td>140</td><td>0.007373</td><td>0.009770</td><td>0.013062</td><td>0.015599</td><td>0.000105</td><td>0.000143</td><td>0.000224</td><td>0.000348</td></tr><tr><td>150</td><td>0.006883</td><td>0.009105</td><td>0.012201</td><td>0.014561</td><td>0.000102</td><td>0.000154</td><td>0.000235</td><td>0.000368</td></tr><tr><td>160</td><td>0.006453</td><td>0.008527</td><td>0.011409</td><td>0.013678</td><td>0.000111</td><td>0.000142</td><td>0.000247</td><td>0.000392</td></tr><tr><td>170</td><td>0.006071</td><td>0.008048</td><td>0.010753</td><td>0.012889</td><td>0.000077</td><td>0.000126</td><td>0.000203</td><td>0.000263</td></tr><tr><td>180</td><td>0.005741</td><td>0.007606</td><td>0.010169</td><td>0.012198</td><td>0.000078</td><td>0.000124</td><td>0.000215</td><td>0.000298</td></tr><tr><td>190</td><td>0.005426</td><td>0.007185</td><td>0.009627</td><td>0.011511</td><td>0.000084</td><td>0.000122</td><td>0.000207</td><td>0.000280</td></tr><tr><td>200</td><td>0.005169</td><td>0.006854</td><td>0.009178</td><td>0.010955</td><td>0.000071</td><td>0.000102</td><td>0.000167</td><td>0.000262</td></tr><tr><td>210</td><td>0.004942</td><td>0.006542</td><td>0.008748</td><td>0.010466</td><td>0.000067</td><td>0.000112</td><td>0.000178</td><td>0.000253</td></tr><tr><td>220</td><td>0.004710</td><td>0.006233</td><td>0.008337</td><td>0.010005</td><td>0.000063</td><td>0.000091</td><td>0.000161</td><td>0.000242</td></tr><tr><td>230</td><td>0.004500</td><td>0.005972</td><td>0.008006</td><td>0.009560</td><td>0.000064</td><td>0.000099</td><td>0.000162</td><td>0.000220</td></tr><tr><td>240</td><td>0.004309</td><td>0.005715</td><td>0.007667</td><td>0.009196</td><td>0.000063</td><td>0.000096</td><td>0.000151</td><td>0.000233</td></tr><tr><td>250</td><td>0.004148</td><td>0.005500</td><td>0.007370</td><td>0.008818</td><td>0.000059</td><td>0.000086</td><td>0.000150</td><td>0.000186</td></tr><tr><td>260</td><td>0.003986</td><td>0.005283</td><td>0.007073</td><td>0.008479</td><td>0.000049</td><td>0.000076</td><td>0.000137</td><td>0.000199</td></tr><tr><td>270</td><td>0.003834</td><td>0.005083</td><td>0.006809</td><td>0.008153</td><td>0.000048</td><td>0.000074</td><td>0.000130</td><td>0.000185</td></tr><tr><td>280</td><td>0.003702</td><td>0.004906</td><td>0.006573</td><td>0.007869</td><td>0.000049</td><td>0.000074</td><td>0.000118</td><td>0.000194</td></tr><tr><td>290</td><td>0.003577</td><td>0.004748</td><td>0.006358</td><td>0.007602</td><td>0.000053</td><td>0.000081</td><td>0.000127</td><td>0.000171</td></tr><tr><td>300</td><td>0.003457</td><td>0.004583</td><td>0.006140</td><td>0.007360</td><td>0.000049</td><td>0.000072</td><td>0.000126</td><td>0.000180</td></tr><tr><td>310</td><td>0.003345</td><td>0.004435</td><td>0.005942</td><td>0.007107</td><td>0.000044</td><td>0.000064</td><td>0.000117</td><td>0.000174</td></tr><tr><td>320</td><td>0.003241</td><td>0.004302</td><td>0.005772</td><td>0.006915</td><td>0.000051</td><td>0.000067</td><td>0.000116</td><td>0.000164</td></tr><tr><td>330</td><td>0.003145</td><td>0.004173</td><td>0.005590</td><td>0.006682</td><td>0.000039</td><td>0.000067</td><td>0.000119</td><td>0.000156</td></tr><tr><td>340</td><td>0.003051</td><td>0.004047</td><td>0.005413</td><td>0.006486</td><td>0.000044</td><td>0.000065</td><td>0.000109</td><td>0.000153</td></tr><tr><td>350</td><td>0.002964</td><td>0.003929</td><td>0.005261</td><td>0.006327</td><td>0.000038</td><td>0.000064</td><td>0.000106</td><td>0.000153</td></tr><tr><td>360</td><td>0.002887</td><td>0.003826</td><td>0.005119</td><td>0.006132</td><td>0.000040</td><td>0.000065</td><td>0.000097</td><td>0.000126</td></tr><tr><td>370</td><td>0.002808</td><td>0.003725</td><td>0.005005</td><td>0.005994</td><td>0.000037</td><td>0.000054</td><td>0.000096</td><td>0.000139</td></tr><tr><td>380</td><td>0.002736</td><td>0.003627</td><td>0.004863</td><td>0.005830</td><td>0.000037</td><td>0.000060</td><td>0.000097</td><td>0.000144</td></tr><tr><td>390</td><td>0.002663</td><td>0.003537</td><td>0.004752</td><td>0.005690</td><td>0.000037</td><td>0.000056</td><td>0.000090</td><td>0.000135</td></tr><tr><td>400</td><td>0.002601</td><td>0.003447</td><td>0.004639</td><td>0.005532</td><td>0.000037</td><td>0.000062</td><td>0.000094</td><td>0.000124</td></tr><tr><td>410</td><td>0.002527</td><td>0.003357</td><td>0.004514</td><td>0.005406</td><td>0.000033</td><td>0.000051</td><td>0.000095</td><td>0.000123</td></tr><tr><td>420</td><td>0.002467</td><td>0.003278</td><td>0.004406</td><td>0.005275</td><td>0.000038</td><td>0.000055</td><td>0.000078</td><td>0.000128</td></tr><tr><td>430</td><td>0.002411</td><td>0.003199</td><td>0.004288</td><td>0.005136</td><td>0.000037</td><td>0.000051</td><td>0.000095</td><td>0.000112</td></tr><tr><td>440</td><td>0.002358</td><td>0.003129</td><td>0.004184</td><td>0.005017</td><td>0.000033</td><td>0.000052</td><td>0.000080</td><td>0.000115</td></tr><tr><td>450</td><td>0.002308</td><td>0.003067</td><td>0.004123</td><td>0.004934</td><td>0.000036</td><td>0.000051</td><td>0.000090</td><td>0.000115</td></tr></table>
<table><caption>**Table 1. Continued.**</caption><tr><td>460</td><td>0.002258</td><td>0.002997</td><td>0.004027</td><td>0.004833</td><td>0.000028</td><td>0.000049</td><td>0.000081</td><td>0.000111</td></tr><tr><td>470</td><td>0.002210</td><td>0.002933</td><td>0.003937</td><td>0.004725</td><td>0.000031</td><td>0.000050</td><td>0.000079</td><td>0.000126</td></tr><tr><td>480</td><td>0.002166</td><td>0.002875</td><td>0.003854</td><td>0.004624</td><td>0.000030</td><td>0.000050</td><td>0.000075</td><td>0.000115</td></tr><tr><td>490</td><td>0.002122</td><td>0.002810</td><td>0.003771</td><td>0.004528</td><td>0.000030</td><td>0.000041</td><td>0.000070</td><td>0.000097</td></tr><tr><td>500</td><td>0.002078</td><td>0.002763</td><td>0.003699</td><td>0.004432</td><td>0.000032</td><td>0.000043</td><td>0.000076</td><td>0.000122</td></tr><tr><td>510</td><td>0.002038</td><td>0.002710</td><td>0.003645</td><td>0.004374</td><td>0.000029</td><td>0.000043</td><td>0.000074</td><td>0.000107</td></tr><tr><td>520</td><td>0.001999</td><td>0.002652</td><td>0.003554</td><td>0.004268</td><td>0.000028</td><td>0.000045</td><td>0.000072</td><td>0.000112</td></tr><tr><td>530</td><td>0.001966</td><td>0.002605</td><td>0.003487</td><td>0.004178</td><td>0.000026</td><td>0.000039</td><td>0.000061</td><td>0.000093</td></tr><tr><td>540</td><td>0.001927</td><td>0.002561</td><td>0.003430</td><td>0.004108</td><td>0.000029</td><td>0.000043</td><td>0.000071</td><td>0.000104</td></tr><tr><td>550</td><td>0.001890</td><td>0.002506</td><td>0.003367</td><td>0.004041</td><td>0.000027</td><td>0.000041</td><td>0.000070</td><td>0.000099</td></tr><tr><td>560</td><td>0.001857</td><td>0.002466</td><td>0.003310</td><td>0.003980</td><td>0.000028</td><td>0.000046</td><td>0.000067</td><td>0.000098</td></tr><tr><td>570</td><td>0.001819</td><td>0.002414</td><td>0.003229</td><td>0.003875</td><td>0.000027</td><td>0.000039</td><td>0.000064</td><td>0.000093</td></tr><tr><td>580</td><td>0.001791</td><td>0.002382</td><td>0.003199</td><td>0.003829</td><td>0.000027</td><td>0.000034</td><td>0.000067</td><td>0.000092</td></tr><tr><td>590</td><td>0.001766</td><td>0.002344</td><td>0.003142</td><td>0.003776</td><td>0.000025</td><td>0.000039</td><td>0.000055</td><td>0.000096</td></tr><tr><td>600</td><td>0.001733</td><td>0.002297</td><td>0.003087</td><td>0.003701</td><td>0.000023</td><td>0.000040</td><td>0.000064</td><td>0.000093</td></tr><tr><td>610</td><td>0.001705</td><td>0.002264</td><td>0.003035</td><td>0.003642</td><td>0.000023</td><td>0.000036</td><td>0.000058</td><td>0.000089</td></tr><tr><td>620</td><td>0.001677</td><td>0.002227</td><td>0.002978</td><td>0.003571</td><td>0.000022</td><td>0.000035</td><td>0.000057</td><td>0.000086</td></tr><tr><td>630</td><td>0.001649</td><td>0.002193</td><td>0.002936</td><td>0.003520</td><td>0.000022</td><td>0.000034</td><td>0.000064</td><td>0.000084</td></tr><tr><td>640</td><td>0.001620</td><td>0.002151</td><td>0.002892</td><td>0.003471</td><td>0.000024</td><td>0.000033</td><td>0.000058</td><td>0.000093</td></tr><tr><td>650</td><td>0.001601</td><td>0.002121</td><td>0.002851</td><td>0.003415</td><td>0.000022</td><td>0.000038</td><td>0.000059</td><td>0.000094</td></tr><tr><td>660</td><td>0.001571</td><td>0.002089</td><td>0.002805</td><td>0.003366</td><td>0.000022</td><td>0.000034</td><td>0.000053</td><td>0.000083</td></tr><tr><td>670</td><td>0.001553</td><td>0.002063</td><td>0.002765</td><td>0.003318</td><td>0.000021</td><td>0.000034</td><td>0.000051</td><td>0.000081</td></tr><tr><td>680</td><td>0.001529</td><td>0.002032</td><td>0.002726</td><td>0.003262</td><td>0.000019</td><td>0.000032</td><td>0.000056</td><td>0.000084</td></tr><tr><td>690</td><td>0.001506</td><td>0.002003</td><td>0.002676</td><td>0.003205</td><td>0.000021</td><td>0.000034</td><td>0.000052</td><td>0.000072</td></tr><tr><td>700</td><td>0.001487</td><td>0.001976</td><td>0.002653</td><td>0.003180</td><td>0.000020</td><td>0.000029</td><td>0.000054</td><td>0.000080</td></tr><tr><td>710</td><td>0.001465</td><td>0.001944</td><td>0.002612</td><td>0.003137</td><td>0.000021</td><td>0.000030</td><td>0.000050</td><td>0.000079</td></tr><tr><td>720</td><td>0.001446</td><td>0.001919</td><td>0.002581</td><td>0.003088</td><td>0.000021</td><td>0.000030</td><td>0.000048</td><td>0.000068</td></tr><tr><td>730</td><td>0.001427</td><td>0.001890</td><td>0.002536</td><td>0.003044</td><td>0.000020</td><td>0.000027</td><td>0.000044</td><td>0.000069</td></tr><tr><td>740</td><td>0.001407</td><td>0.001870</td><td>0.002509</td><td>0.003005</td><td>0.000018</td><td>0.000030</td><td>0.000046</td><td>0.000070</td></tr><tr><td>750</td><td>0.001384</td><td>0.001838</td><td>0.002461</td><td>0.002958</td><td>0.000020</td><td>0.000028</td><td>0.000049</td><td>0.000072</td></tr><tr><td>760</td><td>0.001365</td><td>0.001815</td><td>0.002440</td><td>0.002926</td><td>0.000018</td><td>0.000028</td><td>0.000050</td><td>0.000069</td></tr><tr><td>770</td><td>0.001353</td><td>0.001795</td><td>0.002408</td><td>0.002886</td><td>0.000020</td><td>0.000029</td><td>0.000049</td><td>0.000068</td></tr><tr><td>780</td><td>0.001330</td><td>0.001769</td><td>0.002376</td><td>0.002847</td><td>0.000018</td><td>0.000028</td><td>0.000046</td><td>0.000075</td></tr><tr><td>790</td><td>0.001317</td><td>0.001748</td><td>0.002349</td><td>0.002815</td><td>0.000018</td><td>0.000029</td><td>0.000045</td><td>0.000065</td></tr><tr><td>800</td><td>0.001301</td><td>0.001727</td><td>0.002316</td><td>0.002770</td><td>0.000017</td><td>0.000028</td><td>0.000049</td><td>0.000072</td></tr><tr><td>810</td><td>0.001286</td><td>0.001709</td><td>0.002297</td><td>0.002752</td><td>0.000016</td><td>0.000023</td><td>0.000039</td><td>0.000055</td></tr><tr><td>820</td><td>0.001270</td><td>0.001685</td><td>0.002264</td><td>0.002718</td><td>0.000016</td><td>0.000024</td><td>0.000043</td><td>0.000071</td></tr><tr><td>830</td><td>0.001252</td><td>0.001661</td><td>0.002229</td><td>0.002674</td><td>0.000017</td><td>0.000028</td><td>0.000050</td><td>0.000064</td></tr><tr><td>840</td><td>0.001241</td><td>0.001649</td><td>0.002206</td><td>0.002651</td><td>0.000017</td><td>0.000026</td><td>0.000041</td><td>0.000064</td></tr><tr><td>850</td><td>0.001225</td><td>0.001627</td><td>0.002184</td><td>0.002621</td><td>0.000017</td><td>0.000026</td><td>0.000043</td><td>0.000062</td></tr><tr><td>860</td><td>0.001210</td><td>0.001606</td><td>0.002152</td><td>0.002581</td><td>0.000019</td><td>0.000028</td><td>0.000043</td><td>0.000064</td></tr><tr><td>870</td><td>0.001196</td><td>0.001583</td><td>0.002128</td><td>0.002547</td><td>0.000016</td><td>0.000022</td><td>0.000041</td><td>0.000062</td></tr><tr><td>880</td><td>0.001182</td><td>0.001568</td><td>0.002105</td><td>0.002521</td><td>0.000015</td><td>0.000024</td><td>0.000040</td><td>0.000058</td></tr><tr><td>890</td><td>0.001170</td><td>0.001552</td><td>0.002087</td><td>0.002500</td><td>0.000016</td><td>0.000027</td><td>0.000049</td><td>0.000064</td></tr><tr><td>900</td><td>0.001155</td><td>0.001536</td><td>0.002067</td><td>0.002481</td><td>0.000015</td><td>0.000024</td><td>0.000039</td><td>0.000060</td></tr></table>
<table><caption>**Table 1. Continued.**</caption><tr><td>910</td><td>0.001143</td><td>0.001519</td><td>0.002041</td><td>0.002441</td><td>0.000014</td><td>0.000026</td><td>0.000043</td><td>0.000064</td></tr><tr><td>920</td><td>0.001132</td><td>0.001504</td><td>0.002018</td><td>0.002417</td><td>0.000017</td><td>0.000025</td><td>0.000043</td><td>0.000056</td></tr><tr><td>930</td><td>0.001119</td><td>0.001488</td><td>0.001996</td><td>0.002393</td><td>0.000014</td><td>0.000022</td><td>0.000041</td><td>0.000068</td></tr><tr><td>940</td><td>0.001109</td><td>0.001473</td><td>0.001972</td><td>0.002361</td><td>0.000015</td><td>0.000021</td><td>0.000038</td><td>0.000054</td></tr><tr><td>950</td><td>0.001093</td><td>0.001453</td><td>0.001952</td><td>0.002342</td><td>0.000015</td><td>0.000023</td><td>0.000042</td><td>0.000061</td></tr><tr><td>960</td><td>0.001080</td><td>0.001434</td><td>0.001928</td><td>0.002314</td><td>0.000015</td><td>0.000024</td><td>0.000040</td><td>0.000052</td></tr><tr><td>970</td><td>0.001071</td><td>0.001420</td><td>0.001907</td><td>0.002288</td><td>0.000017</td><td>0.000024</td><td>0.000041</td><td>0.000057</td></tr><tr><td>980</td><td>0.001059</td><td>0.001406</td><td>0.001895</td><td>0.002277</td><td>0.000014</td><td>0.000022</td><td>0.000040</td><td>0.000053</td></tr><tr><td>990</td><td>0.001050</td><td>0.001395</td><td>0.001870</td><td>0.002244</td><td>0.000014</td><td>0.000022</td><td>0.000033</td><td>0.000052</td></tr><tr><td>1000</td><td>0.001041</td><td>0.001384</td><td>0.001861</td><td>0.002228</td><td>0.000015</td><td>0.000025</td><td>0.000042</td><td>0.000055</td></tr><tr><td>10000</td><td>0.000104</td><td>0.000138</td><td>0.000186</td><td>0.000223</td><td>0.000001</td><td>0.000002</td><td>0.000004</td><td>0.000006</td></tr></table>
<caption>**Table 2.** Simulations for the normal law $N(0, 1)$. $n$: sample size.
$\bar{K}_{0.90}$, $\bar{K}_{0.95}$, $\bar{K}_{0.98}$, $\bar{K}_{0.99}$: mean values of estimated quantiles over 100 experiments.</caption>

$S_{K_{0.90}}$, $S_{K_{0.95}}$, $S_{K_{0.98}}$, $S_{K_{0.99}}$: estimated standard deviations of the quantiles distributions.

<table><thead><tr><th>n</th><th>K̄<sub>0.90</sub></th><th>K̄<sub>0.95</sub></th><th>K̄<sub>0.98</sub></th><th>K̄<sub>0.99</sub></th><th>S<sub>K<sub>0.90</sub></sub></th><th>S<sub>K<sub>0.95</sub></sub></th><th>S<sub>K<sub>0.98</sub></sub></th><th>S<sub>K<sub>0.99</sub></sub></th></tr></thead><tbody><tr><td>3</td><td>0.206148</td><td>0.227731</td><td>0.240969</td><td>0.245439</td><td>0.001423</td><td>0.001022</td><td>0.000689</td><td>0.000456</td></tr><tr><td>4</td><td>0.176247</td><td>0.220096</td><td>0.260605</td><td>0.281725</td><td>0.002109</td><td>0.002444</td><td>0.002400</td><td>0.002368</td></tr><tr><td>5</td><td>0.153131</td><td>0.193065</td><td>0.238453</td><td>0.266296</td><td>0.001812</td><td>0.002268</td><td>0.003269</td><td>0.003323</td></tr><tr><td>6</td><td>0.136157</td><td>0.173063</td><td>0.215660</td><td>0.243214</td><td>0.001734</td><td>0.002212</td><td>0.003252</td><td>0.003874</td></tr><tr><td>7</td><td>0.123281</td><td>0.157455</td><td>0.197204</td><td>0.223840</td><td>0.001479</td><td>0.002202</td><td>0.002835</td><td>0.003373</td></tr><tr><td>8</td><td>0.111374</td><td>0.142651</td><td>0.180713</td><td>0.206476</td><td>0.001447</td><td>0.001952</td><td>0.003128</td><td>0.003745</td></tr><tr><td>9</td><td>0.102793</td><td>0.131712</td><td>0.167606</td><td>0.192213</td><td>0.001246</td><td>0.001594</td><td>0.002683</td><td>0.003199</td></tr><tr><td>10</td><td>0.094767</td><td>0.121944</td><td>0.155632</td><td>0.179397</td><td>0.001168</td><td>0.001666</td><td>0.002249</td><td>0.003116</td></tr><tr><td>11</td><td>0.088537</td><td>0.113730</td><td>0.145479</td><td>0.167983</td><td>0.001080</td><td>0.001417</td><td>0.002225</td><td>0.003192</td></tr><tr><td>12</td><td>0.082536</td><td>0.106016</td><td>0.136342</td><td>0.157983</td><td>0.001127</td><td>0.001471</td><td>0.002004</td><td>0.002733</td></tr><tr><td>13</td><td>0.077855</td><td>0.100089</td><td>0.128930</td><td>0.149682</td><td>0.000976</td><td>0.001377</td><td>0.002225</td><td>0.003116</td></tr><tr><td>14</td><td>0.073353</td><td>0.094355</td><td>0.121794</td><td>0.141797</td><td>0.001000</td><td>0.001486</td><td>0.002356</td><td>0.003088</td></tr><tr><td>15</td><td>0.069663</td><td>0.089547</td><td>0.115307</td><td>0.134313</td><td>0.000766</td><td>0.001277</td><td>0.001811</td><td>0.002797</td></tr><tr><td>16</td><td>0.065954</td><td>0.084925</td><td>0.109532</td><td>0.127891</td><td>0.000838</td><td>0.001189</td><td>0.001799</td><td>0.002688</td></tr><tr><td>17</td><td>0.063034</td><td>0.080967</td><td>0.104286</td><td>0.121876</td><td>0.000829</td><td>0.001105</td><td>0.001908</td><td>0.002463</td></tr><tr><td>18</td><td>0.060037</td><td>0.077133</td><td>0.100098</td><td>0.116555</td><td>0.000719</td><td>0.001091</td><td>0.001589</td><td>0.002402</td></tr><tr><td>19</td><td>0.057690</td><td>0.074178</td><td>0.095784</td><td>0.111988</td><td>0.000669</td><td>0.000995</td><td>0.001640</td><td>0.002282</td></tr><tr><td>20</td><td>0.055160</td><td>0.070931</td><td>0.091782</td><td>0.107161</td><td>0.000602</td><td>0.001065</td><td>0.001546</td><td>0.002367</td></tr><tr><td>21</td><td>0.053234</td><td>0.068194</td><td>0.088279</td><td>0.103222</td><td>0.000589</td><td>0.000972</td><td>0.001720</td><td>0.002399</td></tr><tr><td>22</td><td>0.051111</td><td>0.065554</td><td>0.084898</td><td>0.099353</td><td>0.000566</td><td>0.000842</td><td>0.001475</td><td>0.001799</td></tr><tr><td>23</td><td>0.049378</td><td>0.063333</td><td>0.082050</td><td>0.096044</td><td>0.000602</td><td>0.000925</td><td>0.001492</td><td>0.001675</td></tr><tr><td>24</td><td>0.047672</td><td>0.061163</td><td>0.079168</td><td>0.092479</td><td>0.000680</td><td>0.000976</td><td>0.001557</td><td>0.002070</td></tr><tr><td>25</td><td>0.046052</td><td>0.059073</td><td>0.076322</td><td>0.089272</td><td>0.000582</td><td>0.000838</td><td>0.001330</td><td>0.001855</td></tr><tr><td>26</td><td>0.044557</td><td>0.057052</td><td>0.073518</td><td>0.086034</td><td>0.000567</td><td>0.000768</td><td>0.001238</td><td>0.001484</td></tr><tr><td>27</td><td>0.043118</td><td>0.055299</td><td>0.071604</td><td>0.083819</td><td>0.000537</td><td>0.000781</td><td>0.001316</td><td>0.001738</td></tr><tr><td>28</td><td>0.041852</td><td>0.053532</td><td>0.069190</td><td>0.080993</td><td>0.000475</td><td>0.000709</td><td>0.001165</td><td>0.001732</td></tr><tr><td>29</td><td>0.040710</td><td>0.052073</td><td>0.067286</td><td>0.078917</td><td>0.000479</td><td>0.000726</td><td>0.001115</td><td>0.001827</td></tr><tr><td>30</td><td>0.039579</td><td>0.050613</td><td>0.065551</td><td>0.076793</td><td>0.000470</td><td>0.000632</td><td>0.001084</td><td>0.001740</td></tr><tr><td>31</td><td>0.038442</td><td>0.049161</td><td>0.063557</td><td>0.074347</td><td>0.000448</td><td>0.000621</td><td>0.001051</td><td>0.001446</td></tr><tr><td>32</td><td>0.037411</td><td>0.047811</td><td>0.061803</td><td>0.072384</td><td>0.000410</td><td>0.000621</td><td>0.001012</td><td>0.001540</td></tr><tr><td>33</td><td>0.036456</td><td>0.046615</td><td>0.060265</td><td>0.070494</td><td>0.000464</td><td>0.000623</td><td>0.001030</td><td>0.001400</td></tr><tr><td>34</td><td>0.035545</td><td>0.045467</td><td>0.058717</td><td>0.068982</td><td>0.000421</td><td>0.000602</td><td>0.000886</td><td>0.001413</td></tr><tr><td>35</td><td>0.034782</td><td>0.044340</td><td>0.057245</td><td>0.066953</td><td>0.000454</td><td>0.000595</td><td>0.001068</td><td>0.001458</td></tr><tr><td>36</td><td>0.033925</td><td>0.043316</td><td>0.055951</td><td>0.065549</td><td>0.000395</td><td>0.000564</td><td>0.000877</td><td>0.001153</td></tr><tr><td>37</td><td>0.033158</td><td>0.042346</td><td>0.054555</td><td>0.063907</td><td>0.000400</td><td>0.000642</td><td>0.000892</td><td>0.001175</td></tr><tr><td>38</td><td>0.032366</td><td>0.041311</td><td>0.053444</td><td>0.062713</td><td>0.000397</td><td>0.000632</td><td>0.000945</td><td>0.001404</td></tr><tr><td>39</td><td>0.031691</td><td>0.040411</td><td>0.052273</td><td>0.061247</td><td>0.000403</td><td>0.000630</td><td>0.000988</td><td>0.001345</td></tr><tr><td>40</td><td>0.030907</td><td>0.039486</td><td>0.051016</td><td>0.059674</td><td>0.000349</td><td>0.000507</td><td>0.000737</td><td>0.001080</td></tr><tr><td>41</td><td>0.030470</td><td>0.038804</td><td>0.050175</td><td>0.058859</td><td>0.000375</td><td>0.000551</td><td>0.000989</td><td>0.001338</td></tr><tr><td>42</td><td>0.029769</td><td>0.037979</td><td>0.048965</td><td>0.057236</td><td>0.000335</td><td>0.000551</td><td>0.000792</td><td>0.001158</td></tr><tr><td>43</td><td>0.029212</td><td>0.037180</td><td>0.047996</td><td>0.056155</td><td>0.000357</td><td>0.000489</td><td>0.000840</td><td>0.001196</td></tr><tr><td>44</td><td>0.028562</td><td>0.036398</td><td>0.047058</td><td>0.055077</td><td>0.000337</td><td>0.000538</td><td>0.000892</td><td>0.001235</td></tr><tr><td>45</td><td>0.028021</td><td>0.035723</td><td>0.046013</td><td>0.053733</td><td>0.000287</td><td>0.000457</td><td>0.000768</td><td>0.001158</td></tr></tbody></table>
<table><caption>**Table 2. Continued.**</caption><tr><td>46</td><td>0.027504</td><td>0.034977</td><td>0.045128</td><td>0.052973</td><td>0.000332</td><td>0.000496</td><td>0.000797</td><td>0.001238</td></tr><tr><td>47</td><td>0.026981</td><td>0.034399</td><td>0.044468</td><td>0.052066</td><td>0.000317</td><td>0.000456</td><td>0.000664</td><td>0.000909</td></tr><tr><td>48</td><td>0.026539</td><td>0.033824</td><td>0.043593</td><td>0.051103</td><td>0.000304</td><td>0.000475</td><td>0.000769</td><td>0.001128</td></tr><tr><td>49</td><td>0.026072</td><td>0.033144</td><td>0.042658</td><td>0.050046</td><td>0.000283</td><td>0.000427</td><td>0.000674</td><td>0.000952</td></tr><tr><td>50</td><td>0.025603</td><td>0.032559</td><td>0.041924</td><td>0.049126</td><td>0.000284</td><td>0.000385</td><td>0.000723</td><td>0.001093</td></tr><tr><td>51</td><td>0.025170</td><td>0.032012</td><td>0.041354</td><td>0.048368</td><td>0.000294</td><td>0.000463</td><td>0.000723</td><td>0.001073</td></tr><tr><td>52</td><td>0.024719</td><td>0.031493</td><td>0.040598</td><td>0.047669</td><td>0.000271</td><td>0.000391</td><td>0.000664</td><td>0.001032</td></tr><tr><td>53</td><td>0.024357</td><td>0.030968</td><td>0.039957</td><td>0.046826</td><td>0.000276</td><td>0.000425</td><td>0.000722</td><td>0.001012</td></tr><tr><td>54</td><td>0.023921</td><td>0.030456</td><td>0.039183</td><td>0.045895</td><td>0.000258</td><td>0.000393</td><td>0.000729</td><td>0.001036</td></tr><tr><td>55</td><td>0.023566</td><td>0.029956</td><td>0.038575</td><td>0.045169</td><td>0.000286</td><td>0.000427</td><td>0.000683</td><td>0.000984</td></tr><tr><td>56</td><td>0.023173</td><td>0.029461</td><td>0.038006</td><td>0.044487</td><td>0.000237</td><td>0.000376</td><td>0.000699</td><td>0.000900</td></tr><tr><td>57</td><td>0.022878</td><td>0.029069</td><td>0.037395</td><td>0.043870</td><td>0.000259</td><td>0.000385</td><td>0.000655</td><td>0.000927</td></tr><tr><td>58</td><td>0.022505</td><td>0.028573</td><td>0.036883</td><td>0.043208</td><td>0.000294</td><td>0.000424</td><td>0.000634</td><td>0.000909</td></tr><tr><td>59</td><td>0.022182</td><td>0.028209</td><td>0.036341</td><td>0.042576</td><td>0.000242</td><td>0.000373</td><td>0.000608</td><td>0.000847</td></tr><tr><td>60</td><td>0.021792</td><td>0.027644</td><td>0.035612</td><td>0.041752</td><td>0.000290</td><td>0.000375</td><td>0.000614</td><td>0.000861</td></tr><tr><td>61</td><td>0.021529</td><td>0.027330</td><td>0.035223</td><td>0.041278</td><td>0.000225</td><td>0.000363</td><td>0.000579</td><td>0.000775</td></tr><tr><td>62</td><td>0.021162</td><td>0.026857</td><td>0.034567</td><td>0.040376</td><td>0.000245</td><td>0.000377</td><td>0.000591</td><td>0.000847</td></tr><tr><td>63</td><td>0.020945</td><td>0.026563</td><td>0.034225</td><td>0.040067</td><td>0.000230</td><td>0.000340</td><td>0.000558</td><td>0.000868</td></tr><tr><td>64</td><td>0.020709</td><td>0.026250</td><td>0.033690</td><td>0.039466</td><td>0.000237</td><td>0.000397</td><td>0.000602</td><td>0.000782</td></tr><tr><td>65</td><td>0.020368</td><td>0.025842</td><td>0.033178</td><td>0.038883</td><td>0.000256</td><td>0.000344</td><td>0.000515</td><td>0.000722</td></tr><tr><td>66</td><td>0.020081</td><td>0.025505</td><td>0.032858</td><td>0.038502</td><td>0.000235</td><td>0.000342</td><td>0.000491</td><td>0.000801</td></tr><tr><td>67</td><td>0.019822</td><td>0.025140</td><td>0.032367</td><td>0.037828</td><td>0.000236</td><td>0.000350</td><td>0.000497</td><td>0.000671</td></tr><tr><td>68</td><td>0.019617</td><td>0.024870</td><td>0.032023</td><td>0.037465</td><td>0.000249</td><td>0.000334</td><td>0.000581</td><td>0.000828</td></tr><tr><td>69</td><td>0.019337</td><td>0.024498</td><td>0.031553</td><td>0.037002</td><td>0.000241</td><td>0.000342</td><td>0.000628</td><td>0.000860</td></tr><tr><td>70</td><td>0.019047</td><td>0.024098</td><td>0.030976</td><td>0.036225</td><td>0.000202</td><td>0.000316</td><td>0.000484</td><td>0.000656</td></tr><tr><td>71</td><td>0.018846</td><td>0.023871</td><td>0.030774</td><td>0.035986</td><td>0.000202</td><td>0.000339</td><td>0.000499</td><td>0.000801</td></tr><tr><td>72</td><td>0.018614</td><td>0.023588</td><td>0.030298</td><td>0.035473</td><td>0.000195</td><td>0.000276</td><td>0.000474</td><td>0.000691</td></tr><tr><td>73</td><td>0.018388</td><td>0.023307</td><td>0.029923</td><td>0.035058</td><td>0.000160</td><td>0.000289</td><td>0.000468</td><td>0.000762</td></tr><tr><td>74</td><td>0.018165</td><td>0.022995</td><td>0.029629</td><td>0.034691</td><td>0.000203</td><td>0.000322</td><td>0.000469</td><td>0.000785</td></tr><tr><td>75</td><td>0.017973</td><td>0.022792</td><td>0.029273</td><td>0.034279</td><td>0.000209</td><td>0.000302</td><td>0.000517</td><td>0.000867</td></tr><tr><td>76</td><td>0.017768</td><td>0.022478</td><td>0.028907</td><td>0.033877</td><td>0.000214</td><td>0.000307</td><td>0.000491</td><td>0.000662</td></tr><tr><td>77</td><td>0.017562</td><td>0.022266</td><td>0.028636</td><td>0.033480</td><td>0.000190</td><td>0.000294</td><td>0.000464</td><td>0.000601</td></tr><tr><td>78</td><td>0.017347</td><td>0.021952</td><td>0.028287</td><td>0.033076</td><td>0.000206</td><td>0.000304</td><td>0.000483</td><td>0.000657</td></tr><tr><td>79</td><td>0.017169</td><td>0.021757</td><td>0.027986</td><td>0.032782</td><td>0.000200</td><td>0.000306</td><td>0.000506</td><td>0.000681</td></tr><tr><td>80</td><td>0.016962</td><td>0.021429</td><td>0.027523</td><td>0.032168</td><td>0.000194</td><td>0.000301</td><td>0.000481</td><td>0.000632</td></tr><tr><td>81</td><td>0.016794</td><td>0.021259</td><td>0.027344</td><td>0.032046</td><td>0.000193</td><td>0.000333</td><td>0.000521</td><td>0.000711</td></tr><tr><td>82</td><td>0.016587</td><td>0.020979</td><td>0.026945</td><td>0.031550</td><td>0.000170</td><td>0.000271</td><td>0.000465</td><td>0.000628</td></tr><tr><td>83</td><td>0.016428</td><td>0.020782</td><td>0.026712</td><td>0.031312</td><td>0.000187</td><td>0.000260</td><td>0.000449</td><td>0.000712</td></tr><tr><td>84</td><td>0.016240</td><td>0.020571</td><td>0.026359</td><td>0.030807</td><td>0.000180</td><td>0.000275</td><td>0.000483</td><td>0.000707</td></tr><tr><td>85</td><td>0.016076</td><td>0.020330</td><td>0.026066</td><td>0.030547</td><td>0.000174</td><td>0.000291</td><td>0.000514</td><td>0.000823</td></tr><tr><td>86</td><td>0.015922</td><td>0.020125</td><td>0.025828</td><td>0.030202</td><td>0.000187</td><td>0.000278</td><td>0.000435</td><td>0.000642</td></tr><tr><td>87</td><td>0.015723</td><td>0.019908</td><td>0.025545</td><td>0.029892</td><td>0.000181</td><td>0.000286</td><td>0.000469</td><td>0.000645</td></tr><tr><td>88</td><td>0.015587</td><td>0.019710</td><td>0.025353</td><td>0.029600</td><td>0.000201</td><td>0.000265</td><td>0.000469</td><td>0.000680</td></tr><tr><td>89</td><td>0.015432</td><td>0.019527</td><td>0.025107</td><td>0.029337</td><td>0.000178</td><td>0.000264</td><td>0.000394</td><td>0.000573</td></tr><tr><td>90</td><td>0.015265</td><td>0.019309</td><td>0.024773</td><td>0.029009</td><td>0.000182</td><td>0.000267</td><td>0.000409</td><td>0.000632</td></tr></table>
<table><caption>**Table 2. Continued.**</caption><tr><td>91</td><td>0.015112</td><td>0.019104</td><td>0.024485</td><td>0.028674</td><td>0.000181</td><td>0.000262</td><td>0.000414</td><td>0.000590</td></tr><tr><td>92</td><td>0.014991</td><td>0.018947</td><td>0.024253</td><td>0.028369</td><td>0.000175</td><td>0.000261</td><td>0.000445</td><td>0.000672</td></tr><tr><td>93</td><td>0.014830</td><td>0.018752</td><td>0.023998</td><td>0.028147</td><td>0.000174</td><td>0.000260</td><td>0.000465</td><td>0.000606</td></tr><tr><td>94</td><td>0.014673</td><td>0.018556</td><td>0.023807</td><td>0.027843</td><td>0.000158</td><td>0.000244</td><td>0.000417</td><td>0.000627</td></tr><tr><td>95</td><td>0.014543</td><td>0.018352</td><td>0.023520</td><td>0.027501</td><td>0.000166</td><td>0.000251</td><td>0.000402</td><td>0.000581</td></tr><tr><td>96</td><td>0.014418</td><td>0.018171</td><td>0.023371</td><td>0.027314</td><td>0.000152</td><td>0.000232</td><td>0.000405</td><td>0.000613</td></tr><tr><td>97</td><td>0.014249</td><td>0.018026</td><td>0.023127</td><td>0.027003</td><td>0.000159</td><td>0.000234</td><td>0.000426</td><td>0.000662</td></tr><tr><td>98</td><td>0.014145</td><td>0.017878</td><td>0.022875</td><td>0.026742</td><td>0.000177</td><td>0.000241</td><td>0.000414</td><td>0.000590</td></tr><tr><td>99</td><td>0.014033</td><td>0.017695</td><td>0.022721</td><td>0.026638</td><td>0.000149</td><td>0.000240</td><td>0.000372</td><td>0.000532</td></tr><tr><td>100</td><td>0.013912</td><td>0.017552</td><td>0.022499</td><td>0.026394</td><td>0.000165</td><td>0.000248</td><td>0.000405</td><td>0.000634</td></tr><tr><td>110</td><td>0.012782</td><td>0.016139</td><td>0.020678</td><td>0.024151</td><td>0.000152</td><td>0.000215</td><td>0.000344</td><td>0.000458</td></tr><tr><td>120</td><td>0.011799</td><td>0.014855</td><td>0.019032</td><td>0.022210</td><td>0.000115</td><td>0.000196</td><td>0.000331</td><td>0.000437</td></tr><tr><td>130</td><td>0.010972</td><td>0.013802</td><td>0.017673</td><td>0.020649</td><td>0.000127</td><td>0.000186</td><td>0.000275</td><td>0.000473</td></tr><tr><td>140</td><td>0.010277</td><td>0.012932</td><td>0.016503</td><td>0.019234</td><td>0.000113</td><td>0.000192</td><td>0.000281</td><td>0.000394</td></tr><tr><td>150</td><td>0.009661</td><td>0.012134</td><td>0.015499</td><td>0.018072</td><td>0.000103</td><td>0.000155</td><td>0.000266</td><td>0.000393</td></tr><tr><td>160</td><td>0.009094</td><td>0.011421</td><td>0.014616</td><td>0.017020</td><td>0.000105</td><td>0.000158</td><td>0.000239</td><td>0.000381</td></tr><tr><td>170</td><td>0.008625</td><td>0.010829</td><td>0.013816</td><td>0.016068</td><td>0.000085</td><td>0.000134</td><td>0.000252</td><td>0.000366</td></tr><tr><td>180</td><td>0.008171</td><td>0.010249</td><td>0.013074</td><td>0.015266</td><td>0.000093</td><td>0.000146</td><td>0.000226</td><td>0.000306</td></tr><tr><td>190</td><td>0.007800</td><td>0.009781</td><td>0.012456</td><td>0.014526</td><td>0.000086</td><td>0.000130</td><td>0.000207</td><td>0.000306</td></tr><tr><td>200</td><td>0.007437</td><td>0.009308</td><td>0.011870</td><td>0.013828</td><td>0.000076</td><td>0.000116</td><td>0.000194</td><td>0.000287</td></tr><tr><td>210</td><td>0.007101</td><td>0.008911</td><td>0.011343</td><td>0.013223</td><td>0.000081</td><td>0.000115</td><td>0.000186</td><td>0.000258</td></tr><tr><td>220</td><td>0.006806</td><td>0.008513</td><td>0.010829</td><td>0.012609</td><td>0.000074</td><td>0.000118</td><td>0.000192</td><td>0.000279</td></tr><tr><td>230</td><td>0.006544</td><td>0.008183</td><td>0.010406</td><td>0.012120</td><td>0.000068</td><td>0.000097</td><td>0.000166</td><td>0.000244</td></tr><tr><td>240</td><td>0.006292</td><td>0.007861</td><td>0.009999</td><td>0.011630</td><td>0.000062</td><td>0.000101</td><td>0.000163</td><td>0.000232</td></tr><tr><td>250</td><td>0.006060</td><td>0.007573</td><td>0.009640</td><td>0.011220</td><td>0.000061</td><td>0.000105</td><td>0.000167</td><td>0.000236</td></tr><tr><td>260</td><td>0.005849</td><td>0.007294</td><td>0.009282</td><td>0.010826</td><td>0.000069</td><td>0.000094</td><td>0.000158</td><td>0.000219</td></tr><tr><td>270</td><td>0.005637</td><td>0.007041</td><td>0.008924</td><td>0.010367</td><td>0.000064</td><td>0.000102</td><td>0.000148</td><td>0.000191</td></tr><tr><td>280</td><td>0.005463</td><td>0.006825</td><td>0.008668</td><td>0.010077</td><td>0.000059</td><td>0.000085</td><td>0.000140</td><td>0.000210</td></tr><tr><td>290</td><td>0.005284</td><td>0.006594</td><td>0.008368</td><td>0.009739</td><td>0.000054</td><td>0.000089</td><td>0.000137</td><td>0.000194</td></tr><tr><td>300</td><td>0.005125</td><td>0.006385</td><td>0.008094</td><td>0.009432</td><td>0.000051</td><td>0.000085</td><td>0.000129</td><td>0.000171</td></tr><tr><td>310</td><td>0.004970</td><td>0.006199</td><td>0.007860</td><td>0.009136</td><td>0.000059</td><td>0.000082</td><td>0.000121</td><td>0.000176</td></tr><tr><td>320</td><td>0.004825</td><td>0.006012</td><td>0.007631</td><td>0.008856</td><td>0.000047</td><td>0.000071</td><td>0.000127</td><td>0.000199</td></tr><tr><td>330</td><td>0.004685</td><td>0.005832</td><td>0.007392</td><td>0.008598</td><td>0.000053</td><td>0.000076</td><td>0.000118</td><td>0.000168</td></tr><tr><td>340</td><td>0.004572</td><td>0.005692</td><td>0.007204</td><td>0.008396</td><td>0.000050</td><td>0.000074</td><td>0.000116</td><td>0.000159</td></tr><tr><td>350</td><td>0.004449</td><td>0.005532</td><td>0.007010</td><td>0.008137</td><td>0.000046</td><td>0.000071</td><td>0.000114</td><td>0.000171</td></tr><tr><td>360</td><td>0.004324</td><td>0.005383</td><td>0.006808</td><td>0.007920</td><td>0.000047</td><td>0.000064</td><td>0.000113</td><td>0.000157</td></tr><tr><td>370</td><td>0.004224</td><td>0.005259</td><td>0.006654</td><td>0.007745</td><td>0.000042</td><td>0.000067</td><td>0.000097</td><td>0.000156</td></tr><tr><td>380</td><td>0.004117</td><td>0.005117</td><td>0.006471</td><td>0.007535</td><td>0.000045</td><td>0.000061</td><td>0.000091</td><td>0.000131</td></tr><tr><td>390</td><td>0.004015</td><td>0.004990</td><td>0.006310</td><td>0.007345</td><td>0.000044</td><td>0.000064</td><td>0.000105</td><td>0.000150</td></tr><tr><td>400</td><td>0.003924</td><td>0.004889</td><td>0.006189</td><td>0.007190</td><td>0.000040</td><td>0.000061</td><td>0.000096</td><td>0.000148</td></tr><tr><td>410</td><td>0.003839</td><td>0.004768</td><td>0.006034</td><td>0.007003</td><td>0.000040</td><td>0.000057</td><td>0.000093</td><td>0.000158</td></tr><tr><td>420</td><td>0.003745</td><td>0.004653</td><td>0.005875</td><td>0.006841</td><td>0.000035</td><td>0.000060</td><td>0.000102</td><td>0.000136</td></tr><tr><td>430</td><td>0.003667</td><td>0.004559</td><td>0.005763</td><td>0.006691</td><td>0.000042</td><td>0.000056</td><td>0.000086</td><td>0.000129</td></tr><tr><td>440</td><td>0.003589</td><td>0.004456</td><td>0.005625</td><td>0.006535</td><td>0.000041</td><td>0.000060</td><td>0.000096</td><td>0.000132</td></tr><tr><td>450</td><td>0.003515</td><td>0.004373</td><td>0.005522</td><td>0.006427</td><td>0.000037</td><td>0.000056</td><td>0.000097</td><td>0.000129</td></tr></table>
<table><caption>**Table 2. Continued.**</caption><tr><td>460</td><td>0.003445</td><td>0.004278</td><td>0.005416</td><td>0.006302</td><td>0.000036</td><td>0.000054</td><td>0.000097</td><td>0.000129</td></tr><tr><td>470</td><td>0.003379</td><td>0.004189</td><td>0.005281</td><td>0.006133</td><td>0.000038</td><td>0.000056</td><td>0.000081</td><td>0.000105</td></tr><tr><td>480</td><td>0.003318</td><td>0.004113</td><td>0.005189</td><td>0.006031</td><td>0.000033</td><td>0.000048</td><td>0.000087</td><td>0.000112</td></tr><tr><td>490</td><td>0.003250</td><td>0.004033</td><td>0.005100</td><td>0.005910</td><td>0.000033</td><td>0.000048</td><td>0.000075</td><td>0.000111</td></tr><tr><td>500</td><td>0.003186</td><td>0.003945</td><td>0.004973</td><td>0.005791</td><td>0.000032</td><td>0.000053</td><td>0.000077</td><td>0.000119</td></tr><tr><td>510</td><td>0.003132</td><td>0.003885</td><td>0.004904</td><td>0.005699</td><td>0.000030</td><td>0.000053</td><td>0.000082</td><td>0.000120</td></tr><tr><td>520</td><td>0.003073</td><td>0.003812</td><td>0.004811</td><td>0.005572</td><td>0.000039</td><td>0.000054</td><td>0.000077</td><td>0.000123</td></tr><tr><td>530</td><td>0.003018</td><td>0.003737</td><td>0.004717</td><td>0.005480</td><td>0.000030</td><td>0.000045</td><td>0.000075</td><td>0.000109</td></tr><tr><td>540</td><td>0.002965</td><td>0.003676</td><td>0.004638</td><td>0.005388</td><td>0.000033</td><td>0.000050</td><td>0.000071</td><td>0.000101</td></tr><tr><td>550</td><td>0.002917</td><td>0.003615</td><td>0.004550</td><td>0.005297</td><td>0.000030</td><td>0.000043</td><td>0.000067</td><td>0.000098</td></tr><tr><td>560</td><td>0.002862</td><td>0.003550</td><td>0.004478</td><td>0.005192</td><td>0.000028</td><td>0.000049</td><td>0.000068</td><td>0.000098</td></tr><tr><td>570</td><td>0.002818</td><td>0.003488</td><td>0.004407</td><td>0.005125</td><td>0.000030</td><td>0.000045</td><td>0.000071</td><td>0.000110</td></tr><tr><td>580</td><td>0.002778</td><td>0.003440</td><td>0.004334</td><td>0.005025</td><td>0.000027</td><td>0.000040</td><td>0.000073</td><td>0.000100</td></tr><tr><td>590</td><td>0.002728</td><td>0.003378</td><td>0.004254</td><td>0.004932</td><td>0.000028</td><td>0.000043</td><td>0.000069</td><td>0.000099</td></tr><tr><td>600</td><td>0.002686</td><td>0.003326</td><td>0.004195</td><td>0.004869</td><td>0.000025</td><td>0.000043</td><td>0.000066</td><td>0.000095</td></tr><tr><td>610</td><td>0.002649</td><td>0.003280</td><td>0.004128</td><td>0.004790</td><td>0.000024</td><td>0.000039</td><td>0.000060</td><td>0.000091</td></tr><tr><td>620</td><td>0.002611</td><td>0.003231</td><td>0.004059</td><td>0.004722</td><td>0.000029</td><td>0.000042</td><td>0.000061</td><td>0.000091</td></tr><tr><td>630</td><td>0.002568</td><td>0.003175</td><td>0.003993</td><td>0.004633</td><td>0.000023</td><td>0.000041</td><td>0.000060</td><td>0.000087</td></tr><tr><td>640</td><td>0.002530</td><td>0.003126</td><td>0.003937</td><td>0.004564</td><td>0.000027</td><td>0.000041</td><td>0.000066</td><td>0.000095</td></tr><tr><td>650</td><td>0.002494</td><td>0.003087</td><td>0.003893</td><td>0.004519</td><td>0.000027</td><td>0.000041</td><td>0.000064</td><td>0.000085</td></tr><tr><td>660</td><td>0.002457</td><td>0.003038</td><td>0.003828</td><td>0.004430</td><td>0.000024</td><td>0.000038</td><td>0.000061</td><td>0.000083</td></tr><tr><td>670</td><td>0.002424</td><td>0.002999</td><td>0.003773</td><td>0.004371</td><td>0.000025</td><td>0.000033</td><td>0.000054</td><td>0.000087</td></tr><tr><td>680</td><td>0.002394</td><td>0.002961</td><td>0.003726</td><td>0.004318</td><td>0.000026</td><td>0.000036</td><td>0.000055</td><td>0.000082</td></tr><tr><td>690</td><td>0.002358</td><td>0.002912</td><td>0.003665</td><td>0.004247</td><td>0.000024</td><td>0.000035</td><td>0.000055</td><td>0.000088</td></tr><tr><td>700</td><td>0.002330</td><td>0.002878</td><td>0.003626</td><td>0.004201</td><td>0.000023</td><td>0.000032</td><td>0.000056</td><td>0.000082</td></tr><tr><td>710</td><td>0.002295</td><td>0.002839</td><td>0.003565</td><td>0.004130</td><td>0.000022</td><td>0.000032</td><td>0.000049</td><td>0.000074</td></tr><tr><td>720</td><td>0.002268</td><td>0.002804</td><td>0.003528</td><td>0.004087</td><td>0.000023</td><td>0.000034</td><td>0.000051</td><td>0.000081</td></tr><tr><td>730</td><td>0.002233</td><td>0.002758</td><td>0.003479</td><td>0.004033</td><td>0.000023</td><td>0.000033</td><td>0.000057</td><td>0.000083</td></tr><tr><td>740</td><td>0.002211</td><td>0.002734</td><td>0.003436</td><td>0.003984</td><td>0.000022</td><td>0.000034</td><td>0.000052</td><td>0.000079</td></tr><tr><td>750</td><td>0.002182</td><td>0.002697</td><td>0.003398</td><td>0.003936</td><td>0.000022</td><td>0.000035</td><td>0.000052</td><td>0.000088</td></tr><tr><td>760</td><td>0.002156</td><td>0.002661</td><td>0.003350</td><td>0.003884</td><td>0.000021</td><td>0.000028</td><td>0.000054</td><td>0.000075</td></tr><tr><td>770</td><td>0.002127</td><td>0.002624</td><td>0.003294</td><td>0.003811</td><td>0.000020</td><td>0.000027</td><td>0.000048</td><td>0.000072</td></tr><tr><td>780</td><td>0.002099</td><td>0.002596</td><td>0.003266</td><td>0.003780</td><td>0.000022</td><td>0.000031</td><td>0.000052</td><td>0.000071</td></tr><tr><td>790</td><td>0.002075</td><td>0.002563</td><td>0.003220</td><td>0.003730</td><td>0.000021</td><td>0.000033</td><td>0.000054</td><td>0.000077</td></tr><tr><td>800</td><td>0.002051</td><td>0.002535</td><td>0.003192</td><td>0.003695</td><td>0.000020</td><td>0.000032</td><td>0.000054</td><td>0.000075</td></tr><tr><td>810</td><td>0.002032</td><td>0.002508</td><td>0.003154</td><td>0.003655</td><td>0.000020</td><td>0.000032</td><td>0.000049</td><td>0.000072</td></tr><tr><td>820</td><td>0.002006</td><td>0.002476</td><td>0.003108</td><td>0.003589</td><td>0.000018</td><td>0.000028</td><td>0.000049</td><td>0.000069</td></tr><tr><td>830</td><td>0.001983</td><td>0.002452</td><td>0.003078</td><td>0.003567</td><td>0.000021</td><td>0.000033</td><td>0.000049</td><td>0.000078</td></tr><tr><td>840</td><td>0.001962</td><td>0.002418</td><td>0.003038</td><td>0.003520</td><td>0.000019</td><td>0.000028</td><td>0.000054</td><td>0.000083</td></tr><tr><td>850</td><td>0.001938</td><td>0.002387</td><td>0.003007</td><td>0.003472</td><td>0.000020</td><td>0.000027</td><td>0.000044</td><td>0.000057</td></tr><tr><td>860</td><td>0.001918</td><td>0.002367</td><td>0.002974</td><td>0.003446</td><td>0.000020</td><td>0.000026</td><td>0.000043</td><td>0.000069</td></tr><tr><td>870</td><td>0.001896</td><td>0.002340</td><td>0.002944</td><td>0.003412</td><td>0.000018</td><td>0.000029</td><td>0.000046</td><td>0.000069</td></tr><tr><td>880</td><td>0.001878</td><td>0.002316</td><td>0.002905</td><td>0.003368</td><td>0.000018</td><td>0.000027</td><td>0.000046</td><td>0.000064</td></tr><tr><td>890</td><td>0.001855</td><td>0.002288</td><td>0.002873</td><td>0.003327</td><td>0.000018</td><td>0.000026</td><td>0.000048</td><td>0.000064</td></tr><tr><td>900</td><td>0.001835</td><td>0.002263</td><td>0.002841</td><td>0.003298</td><td>0.000021</td><td>0.000028</td><td>0.000049</td><td>0.000067</td></tr></table>
<table><caption>**Table 2. Continued.**</caption><tr><td>910</td><td>0.001817</td><td>0.002241</td><td>0.002813</td><td>0.003257</td><td>0.000018</td><td>0.000027</td><td>0.000045</td><td>0.000069</td></tr><tr><td>920</td><td>0.001801</td><td>0.002220</td><td>0.002781</td><td>0.003222</td><td>0.000017</td><td>0.000027</td><td>0.000048</td><td>0.000066</td></tr><tr><td>930</td><td>0.001777</td><td>0.002195</td><td>0.002757</td><td>0.003196</td><td>0.000019</td><td>0.000026</td><td>0.000043</td><td>0.000059</td></tr><tr><td>940</td><td>0.001761</td><td>0.002174</td><td>0.002732</td><td>0.003162</td><td>0.000018</td><td>0.000026</td><td>0.000046</td><td>0.000066</td></tr><tr><td>950</td><td>0.001744</td><td>0.002149</td><td>0.002696</td><td>0.003115</td><td>0.000019</td><td>0.000025</td><td>0.000042</td><td>0.000062</td></tr><tr><td>960</td><td>0.001725</td><td>0.002128</td><td>0.002671</td><td>0.003100</td><td>0.000018</td><td>0.000027</td><td>0.000049</td><td>0.000062</td></tr><tr><td>970</td><td>0.001712</td><td>0.002110</td><td>0.002648</td><td>0.003072</td><td>0.000017</td><td>0.000028</td><td>0.000043</td><td>0.000062</td></tr><tr><td>980</td><td>0.001694</td><td>0.002089</td><td>0.002618</td><td>0.003036</td><td>0.000018</td><td>0.000025</td><td>0.000042</td><td>0.000063</td></tr><tr><td>990</td><td>0.001678</td><td>0.002070</td><td>0.002596</td><td>0.003001</td><td>0.000015</td><td>0.000024</td><td>0.000036</td><td>0.000057</td></tr><tr><td>1000</td><td>0.001664</td><td>0.002051</td><td>0.002572</td><td>0.002977</td><td>0.000015</td><td>0.000024</td><td>0.000050</td><td>0.000068</td></tr><tr><td>10000</td><td>0.000184</td><td>0.000224</td><td>0.000276</td><td>0.000317</td><td>0.000002</td><td>0.000003</td><td>0.000005</td><td>0.000006</td></tr></table>
# 4 Conclusions

Tables 1 and 2 have been generated respectively for the uniform $U(0, 1)$ and for the normal $N(0, 1)$ parent distributions. The sampling distribution of the chiral index statistic issued from an uniform law $U(a, b)$, $a < b$, does not depend on the interval $[a; b]$. The sampling distribution of the chiral index statistic issued from a normal law $N(m, s)$ depends neither from the expectation $m$ nor on the standard deviation $s$. It means that the chiral index allows to perform location free and scale free symmetry tests.

It is expected that the tables will be used for symmetry testing. Nevertheless, asymptotic expressions of the sampling distribution of the chiral index remain desirable. The chiral index is easy to compute, even with a pocket calculator. So, beginners in programming should be able to generate more tables when desired. The following section of code in R [7] computes the chiral index of a sample:

```R
x <- scan(file="") # import observations from keyboard; type is double
chiral_index <- (1+cor(sort(x,decreasing=FALSE),sort(x,decreasing=TRUE)))/2
chiral_index # display the value of the chiral index
```

# References

[1] Petitjean, M. Chiral mixtures. *J. Math. Phys.* **2002**, *43*[8], 4147-4157;
https://doi.org/10.1063/1.1484559.

[2] Petitjean, M. Chirality and symmetry measures: A transdisciplinary review. *Entropy* **2003**, *5*[3], 271-312; https://doi.org/10.3390/e5030271.

[3] Coppersmith, D.; Petitjean, M. About the optimal density associated to the chiral index of a sample from a bivariate distribution. *Compt. Rend. Acad. Sci. Paris, Sér. I* **2005**, *340*[8], 599-604; https://doi.org/10.1016/j.crma.2005.03.011.

[4] Petitjean, M. About the upper bound of the chiral index of multivariate distributions. *AIP Conf. Proc.* **2008**, *1073*, 61-66; https://doi.org/10.1063/1.3039023.

[5] Petitjean, M. About second kind continuous chirality measures. 1. Planar sets. *J. Math. Chem.* **1997**, *22*[2-4], 185-201 (see appendix 1); https://doi.org/10.1023/A:1019132116175.

[6] Numerical Algorithm Group, **2017. G05 (rand) Random Number Generators**, section 2.1.1;
https://www.nag.co.uk/numeric/fl/nagdoc_fl26/html/g05/g05intro.html.

[7] R Core Team. *R: A language and environment for statistical computing.* **2020**, R Foundation for Statistical Computing, Vienna, Austria; https://www.R-project.org/.