假设检验之卡方检验

假设检验简介

假设检验(Hypothesis Testing),或者叫做显著性检验(Significance Testing)是数理统计学中根据一定假设条件由样本推断总体的一种方法。其基本原理是先对总体的特征作出某种假设,然后通过抽样研究的统计推理,对此假设应该被拒绝还是接受作出推断。既然以假设为前提,那么在进行检验前需要提出相应的假设:

  • H0:原假设或零假设(null hypothesis),即需要去验证的假设;一般首先认定原假设是正确的,然后根据显著性水平选择是接受还是拒绝原假设。
  • H1:备择假设(alternative hypothesis),一般是原假设的否命题;当原假设被拒绝时,默认接受备择假设。

如原假设是假设总体均值$\mu = 0$,则备择假设为总体均值$\mu \neq 0$,检验的过程就是计算相应的统计量和显著性概率,来验证原假设应该被接受还是拒绝。

卡方检验简介

卡方检验是一种用途很广的计数资料的假设检验方法。它属于非参数检验的范畴,主要是比较两个及两个以上样本率( 构成比)以及两个分类变量的关联性分析。其根本思想就是在于比较理论频数和实际频数的吻合程度或拟合优度问题。

卡方检验是以$\chi^2$分布为基础的一种常用假设检验方法,它的无效假设H0是:观察频数与期望频数没有差别。

该检验的基本思想是:首先假设H0成立,基于此前提计算出$\chi^2$值,它表示观察值与理论值之间的偏离程度。根据$\chi^2$分布及自由度可以确定在H0假设成立的情况下获得当前统计量及更极端情况的概率P。如果P值很小,说明观察值与理论值偏离程度太大,应当拒绝无效假设,表示比较资料之间有显著差异;否则就不能拒绝无效假设,尚不能认为样本所代表的实际情况和理论假设有差别。

$\chi^2$值表示观察值与理论值之问的偏离程度。计算这种偏离程度的基本思路如下。

  • 设A代表某个类别的观察频数,E代表基于H0计算出的期望频数,A与E之差称为残差。
  • 残差可以表示某一个类别观察值和理论值的偏离程度,但如果将残差简单相加以表示各类别观察频数与期望频数的差别,则有一定的不足之处。因为残差有正有负,相加后会彼此抵消,总和仍然为0,为此可以将残差平方后求和。
  • 残差大小是一个相对的概念,相对于期望频数为10时,期望频数为20的残差非常大,但相对于期望频数为1 000时20的残差就很小了。考虑到这一点,人们又将残差平方除以期望频数再求和,以估计观察频数与期望频数的差别。

进行上述操作之后,就得到了常用的$\chi^2$统计量,由于它最初是由英国统计学家Karl Pearson在1900年首次提出的,因此也称之为Pearson $\chi^2$,其计算公式为:

$$\chi^2 = \sum \frac{(A-E)^2}{E}=\sum_{i=1}^{k}\frac{(A_i-E_i)^2}{E_i}=\sum_{i=1}^{k}\frac{(A_i-np_i)^2}{np_i}$$

其中,$A_i$为i水平的观察频数,$E_i$为i水平的期望频数,n为总频数,$p_i$为i水平的期望频率。i水平的期望频数$E_i$等于总频数$n \times i$水平的期望概率$p_i$,k为单元格数。当n比较大时,$\chi^2$统计量近似服从k-1(计算$E_i$时用到的参数个数)个自由度的卡方分布。

由卡方的计算公式可知:

  • 观察频数与期望频数完全一致时,$\chi^2$值为0;
  • 观察频数与期望频数越接近,两者之间的差异越小,$\chi^2$值越小;
  • 观察频数与期望频数差别越大,两者之间的差异越大,$\chi^2$值越大

换言之,大的$\chi^2$值表明观察频数远离期望频数,即表明远离假设。小的$\chi^2$值表明观察频数接近期望频数,接近假设。至于$\chi^2$在每个具体研究中究竟要大到什么程度才能拒绝H0,则要借助于卡方分布求出所对应的P值来确定。

1-1000自由度的卡方临界值查询表:

  显著性水平(a)
自由度 0.995 0.975 0.2 0.1 0.05 0.025 0.02 0.01 0.005 0.002 0.001
1 3.93E-05 0.000982 1.642 2.706 3.841 5.024 5.412 6.635 7.879 9.55 10.828
2 0.01 0.0506 3.219 4.605 5.991 7.378 7.824 9.21 10.597 12.429 13.816
3 0.0717 0.216 4.642 6.251 7.815 9.348 9.837 11.345 12.838 14.796 16.266
4 0.207 0.484 5.989 7.779 9.488 11.143 11.668 13.277 14.86 16.924 18.467
5 0.412 0.831 7.289 9.236 11.07 12.833 13.388 15.086 16.75 18.907 20.515
6 0.676 1.237 8.558 10.645 12.592 14.449 15.033 16.812 18.548 20.791 22.458
7 0.989 1.69 9.803 12.017 14.067 16.013 16.622 18.475 20.278 22.601 24.322
8 1.344 2.18 11.03 13.362 15.507 17.535 18.168 20.09 21.955 24.352 26.124
9 1.735 2.7 12.242 14.684 16.919 19.023 19.679 21.666 23.589 26.056 27.877
10 2.156 3.247 13.442 15.987 18.307 20.483 21.161 23.209 25.188 27.722 29.588
11 2.603 3.816 14.631 17.275 19.675 21.92 22.618 24.725 26.757 29.354 31.264
12 3.074 4.404 15.812 18.549 21.026 23.337 24.054 26.217 28.3 30.957 32.909
13 3.565 5.009 16.985 19.812 22.362 24.736 25.472 27.688 29.819 32.535 34.528
14 4.075 5.629 18.151 21.064 23.685 26.119 26.873 29.141 31.319 34.091 36.123
15 4.601 6.262 19.311 22.307 24.996 27.488 28.259 30.578 32.801 35.628 37.697
16 5.142 6.908 20.465 23.542 26.296 28.845 29.633 32 34.267 37.146 39.252
17 5.697 7.564 21.615 24.769 27.587 30.191 30.995 33.409 35.718 38.648 40.79
18 6.265 8.231 22.76 25.989 28.869 31.526 32.346 34.805 37.156 40.136 42.312
19 6.844 8.907 23.9 27.204 30.144 32.852 33.687 36.191 38.582 41.61 43.82
20 7.434 9.591 25.038 28.412 31.41 34.17 35.02 37.566 39.997 43.072 45.315
21 8.034 10.283 26.171 29.615 32.671 35.479 36.343 38.932 41.401 44.522 46.797
22 8.643 10.982 27.301 30.813 33.924 36.781 37.659 40.289 42.796 45.962 48.268
23 9.26 11.689 28.429 32.007 35.172 38.076 38.968 41.638 44.181 47.391 49.728
24 9.886 12.401 29.553 33.196 36.415 39.364 40.27 42.98 45.559 48.812 51.179
25 10.52 13.12 30.675 34.382 37.652 40.646 41.566 44.314 46.928 50.223 52.62
26 11.16 13.844 31.795 35.563 38.885 41.923 42.856 45.642 48.29 51.627 54.052
27 11.808 14.573 32.912 36.741 40.113 43.195 44.14 46.963 49.645 53.023 55.476
28 12.461 15.308 34.027 37.916 41.337 44.461 45.419 48.278 50.993 54.411 56.892
29 13.121 16.047 35.139 39.087 42.557 45.722 46.693 49.588 52.336 55.792 58.301
30 13.787 16.791 36.25 40.256 43.773 46.979 47.962 50.892 53.672 57.167 59.703
31 14.458 17.539 37.359 41.422 44.985 48.232 49.226 52.191 55.003 58.536 61.098
32 15.134 18.291 38.466 42.585 46.194 49.48 50.487 53.486 56.328 59.899 62.487
33 15.815 19.047 39.572 43.745 47.4 50.725 51.743 54.776 57.648 61.256 63.87
34 16.501 19.806 40.676 44.903 48.602 51.966 52.995 56.061 58.964 62.608 65.247
35 17.192 20.569 41.778 46.059 49.802 53.203 54.244 57.342 60.275 63.955 66.619
36 17.887 21.336 42.879 47.212 50.998 54.437 55.489 58.619 61.581 65.296 67.985
37 18.586 22.106 43.978 48.363 52.192 55.668 56.73 59.893 62.883 66.633 69.346
38 19.289 22.878 45.076 49.513 53.384 56.896 57.969 61.162 64.181 67.966 70.703
39 19.996 23.654 46.173 50.66 54.572 58.12 59.204 62.428 65.476 69.294 72.055
40 20.707 24.433 47.269 51.805 55.758 59.342 60.436 63.691 66.766 70.618 73.402
41 21.421 25.215 48.363 52.949 56.942 60.561 61.665 64.95 68.053 71.938 74.745
42 22.138 25.999 49.456 54.09 58.124 61.777 62.892 66.206 69.336 73.254 76.084
43 22.859 26.785 50.548 55.23 59.304 62.99 64.116 67.459 70.616 74.566 77.419
44 23.584 27.575 51.639 56.369 60.481 64.201 65.337 68.71 71.893 75.874 78.75
45 24.311 28.366 52.729 57.505 61.656 65.41 66.555 69.957 73.166 77.179 80.077
46 25.041 29.16 53.818 58.641 62.83 66.617 67.771 71.201 74.437 78.481 81.4
47 25.775 29.956 54.906 59.774 64.001 67.821 68.985 72.443 75.704 79.78 82.72
48 26.511 30.755 55.993 60.907 65.171 69.023 70.197 73.683 76.969 81.075 84.037
49 27.249 31.555 57.079 62.038 66.339 70.222 71.406 74.919 78.231 82.367 85.351
50 27.991 32.357 58.164 63.167 67.505 71.42 72.613 76.154 79.49 83.657 86.661
51 28.735 33.162 59.248 64.295 68.669 72.616 73.818 77.386 80.747 84.943 87.968
52 29.481 33.968 60.332 65.422 69.832 73.81 75.021 78.616 82.001 86.227 89.272
53 30.23 34.776 61.414 66.548 70.993 75.002 76.223 79.843 83.253 87.507 90.573
54 30.981 35.586 62.496 67.673 72.153 76.192 77.422 81.069 84.502 88.786 91.872
55 31.735 36.398 63.577 68.796 73.311 77.38 78.619 82.292 85.749 90.061 93.168
56 32.49 37.212 64.658 69.919 74.468 78.567 79.815 83.513 86.994 91.335 94.461
57 33.248 38.027 65.737 71.04 75.624 79.752 81.009 84.733 88.236 92.605 95.751
58 34.008 38.844 66.816 72.16 76.778 80.936 82.201 85.95 89.477 93.874 97.039
59 34.77 39.662 67.894 73.279 77.931 82.117 83.391 87.166 90.715 95.14 98.324
60 35.534 40.482 68.972 74.397 79.082 83.298 84.58 88.379 91.952 96.404 99.607
61 36.301 41.303 70.049 75.514 80.232 84.476 85.767 89.591 93.186 97.665 100.888
62 37.068 42.126 71.125 76.63 81.381 85.654 86.953 90.802 94.419 98.925 102.166
63 37.838 42.95 72.201 77.745 82.529 86.83 88.137 92.01 95.649 100.182 103.442
64 38.61 43.776 73.276 78.86 83.675 88.004 89.32 93.217 96.878 101.437 104.716
65 39.383 44.603 74.351 79.973 84.821 89.177 90.501 94.422 98.105 102.691 105.988
66 40.158 45.431 75.424 81.085 85.965 90.349 91.681 95.626 99.33 103.942 107.258
67 40.935 46.261 76.498 82.197 87.108 91.519 92.86 96.828 100.554 105.192 108.526
68 41.713 47.092 77.571 83.308 88.25 92.689 94.037 98.028 101.776 106.44 109.791
69 42.494 47.924 78.643 84.418 89.391 93.856 95.213 99.228 102.996 107.685 111.055
70 43.275 48.758 79.715 85.527 90.531 95.023 96.388 100.425 104.215 108.929 112.317
71 44.058 49.592 80.786 86.635 91.67 96.189 97.561 101.621 105.432 110.172 113.577
72 44.843 50.428 81.857 87.743 92.808 97.353 98.733 102.816 106.648 111.412 114.835
73 45.629 51.265 82.927 88.85 93.945 98.516 99.904 104.01 107.862 112.651 116.092
74 46.417 52.103 83.997 89.956 95.081 99.678 101.074 105.202 109.074 113.889 117.346
75 47.206 52.942 85.066 91.061 96.217 100.839 102.243 106.393 110.286 115.125 118.599
76 47.997 53.782 86.135 92.166 97.351 101.999 103.41 107.583 111.495 116.359 119.85
77 48.788 54.623 87.203 93.27 98.484 103.158 104.576 108.771 112.704 117.591 121.1
78 49.582 55.466 88.271 94.374 99.617 104.316 105.742 109.958 113.911 118.823 122.348
79 50.376 56.309 89.338 95.476 100.749 105.473 106.906 111.144 115.117 120.052 123.594
80 51.172 57.153 90.405 96.578 101.879 106.629 108.069 112.329 116.321 121.28 124.839
81 51.969 57.998 91.472 97.68 103.01 107.783 109.232 113.512 117.524 122.507 126.083
82 52.767 58.845 92.538 98.78 104.139 108.937 110.393 114.695 118.726 123.733 127.324
83 53.567 59.692 93.604 99.88 105.267 110.09 111.553 115.876 119.927 124.957 128.565
84 54.368 60.54 94.669 100.98 106.395 111.242 112.712 117.057 121.126 126.179 129.804
85 55.17 61.389 95.734 102.079 107.522 112.393 113.871 118.236 122.325 127.401 131.041
86 55.973 62.239 96.799 103.177 108.648 113.544 115.028 119.414 123.522 128.621 132.277
87 56.777 63.089 97.863 104.275 109.773 114.693 116.184 120.591 124.718 129.84 133.512
88 57.582 63.941 98.927 105.372 110.898 115.841 117.34 121.767 125.913 131.057 134.745
89 58.389 64.793 99.991 106.469 112.022 116.989 118.495 122.942 127.106 132.273 135.978
90 59.196 65.647 101.054 107.565 113.145 118.136 119.648 124.116 128.299 133.489 137.208
91 60.005 66.501 102.117 108.661 114.268 119.282 120.801 125.289 129.491 134.702 138.438
92 60.815 67.356 103.179 109.756 115.39 120.427 121.954 126.462 130.681 135.915 139.666
93 61.625 68.211 104.241 110.85 116.511 121.571 123.105 127.633 131.871 137.127 140.893
94 62.437 69.068 105.303 111.944 117.632 122.715 124.255 128.803 133.059 138.337 142.119
95 63.25 69.925 106.364 113.038 118.752 123.858 125.405 129.973 134.247 139.546 143.344
96 64.063 70.783 107.425 114.131 119.871 125 126.554 131.141 135.433 140.755 144.567
97 64.878 71.642 108.486 115.223 120.99 126.141 127.702 132.309 136.619 141.962 145.789
98 65.694 72.501 109.547 116.315 122.108 127.282 128.849 133.476 137.803 143.168 147.01
99 66.51 73.361 110.607 117.407 123.225 128.422 129.996 134.642 138.987 144.373 148.23
100 67.328 74.222 111.667 118.498 124.342 129.561 131.142 135.807 140.169 145.577 149.449
101 68.146 75.083 112.726 119.589 125.458 130.7 132.287 136.971 141.351 146.78 150.667
102 68.965 75.946 113.786 120.679 126.574 131.838 133.431 138.134 142.532 147.982 151.884
103 69.785 76.809 114.845 121.769 127.689 132.975 134.575 139.297 143.712 149.183 153.099
104 70.606 77.672 115.903 122.858 128.804 134.111 135.718 140.459 144.891 150.383 154.314
105 71.428 78.536 116.962 123.947 129.918 135.247 136.86 141.62 146.07 151.582 155.528
106 72.251 79.401 118.02 125.035 131.031 136.382 138.002 142.78 147.247 152.78 156.74
107 73.075 80.267 119.078 126.123 132.144 137.517 139.143 143.94 148.424 153.977 157.952
108 73.899 81.133 120.135 127.211 133.257 138.651 140.283 145.099 149.599 155.173 159.162
109 74.724 82 121.192 128.298 134.369 139.784 141.423 146.257 150.774 156.369 160.372
110 75.55 82.867 122.25 129.385 135.48 140.917 142.562 147.414 151.948 157.563 161.581
111 76.377 83.735 123.306 130.472 136.591 142.049 143.7 148.571 153.122 158.757 162.788
112 77.204 84.604 124.363 131.558 137.701 143.18 144.838 149.727 154.294 159.95 163.995
113 78.033 85.473 125.419 132.643 138.811 144.311 145.975 150.882 155.466 161.141 165.201
114 78.862 86.342 126.475 133.729 139.921 145.441 147.111 152.037 156.637 162.332 166.406
115 79.692 87.213 127.531 134.813 141.03 146.571 148.247 153.191 157.808 163.523 167.61
116 80.522 88.084 128.587 135.898 142.138 147.7 149.383 154.344 158.977 164.712 168.813
117 81.353 88.955 129.642 136.982 143.246 148.829 150.517 155.496 160.146 165.9 170.016
118 82.185 89.827 130.697 138.066 144.354 149.957 151.652 156.648 161.314 167.088 171.217
119 83.018 90.7 131.752 139.149 145.461 151.084 152.785 157.8 162.481 168.275 172.418
120 83.852 91.573 132.806 140.233 146.567 152.211 153.918 158.95 163.648 169.461 173.617
121 84.686 92.446 133.861 141.315 147.674 153.338 155.051 160.1 164.814 170.647 174.816
122 85.52 93.32 134.915 142.398 148.779 154.464 156.183 161.25 165.98 171.831 176.014
123 86.356 94.195 135.969 143.48 149.885 155.589 157.314 162.398 167.144 173.015 177.212
124 87.192 95.07 137.022 144.562 150.989 156.714 158.445 163.546 168.308 174.198 178.408
125 88.029 95.946 138.076 145.643 152.094 157.839 159.575 164.694 169.471 175.38 179.604
126 88.866 96.822 139.129 146.724 153.198 158.962 160.705 165.841 170.634 176.562 180.799
127 89.704 97.698 140.182 147.805 154.302 160.086 161.834 166.987 171.796 177.743 181.993
128 90.543 98.576 141.235 148.885 155.405 161.209 162.963 168.133 172.957 178.923 183.186
129 91.382 99.453 142.288 149.965 156.508 162.331 164.091 169.278 174.118 180.103 184.379
130 92.222 100.331 143.34 151.045 157.61 163.453 165.219 170.423 175.278 181.282 185.571
131 93.063 101.21 144.392 152.125 158.712 164.575 166.346 171.567 176.438 182.46 186.762
132 93.904 102.089 145.444 153.204 159.814 165.696 167.473 172.711 177.597 183.637 187.953
133 94.746 102.968 146.496 154.283 160.915 166.816 168.6 173.854 178.755 184.814 189.142
134 95.588 103.848 147.548 155.361 162.016 167.936 169.725 174.996 179.913 185.99 190.331
135 96.431 104.729 148.599 156.44 163.116 169.056 170.851 176.138 181.07 187.165 191.52
136 97.275 105.609 149.651 157.518 164.216 170.175 171.976 177.28 182.226 188.34 192.707
137 98.119 106.491 150.702 158.595 165.316 171.294 173.1 178.421 183.382 189.514 193.894
138 98.964 107.372 151.753 159.673 166.415 172.412 174.224 179.561 184.538 190.688 195.08
139 99.809 108.254 152.803 160.75 167.514 173.53 175.348 180.701 185.693 191.861 196.266
140 100.655 109.137 153.854 161.827 168.613 174.648 176.471 181.84 186.847 193.033 197.451
141 101.501 110.02 154.904 162.904 169.711 175.765 177.594 182.979 188.001 194.205 198.635
142 102.348 110.903 155.954 163.98 170.809 176.882 178.716 184.118 189.154 195.376 199.819
143 103.196 111.787 157.004 165.056 171.907 177.998 179.838 185.256 190.306 196.546 201.002
144 104.044 112.671 158.054 166.132 173.004 179.114 180.959 186.393 191.458 197.716 202.184
145 104.892 113.556 159.104 167.207 174.101 180.229 182.08 187.53 192.61 198.885 203.366
146 105.741 114.441 160.153 168.283 175.198 181.344 183.2 188.666 193.761 200.054 204.547
147 106.591 115.326 161.202 169.358 176.294 182.459 184.321 189.802 194.912 201.222 205.727
148 107.441 116.212 162.251 170.432 177.39 183.573 185.44 190.938 196.062 202.39 206.907
149 108.291 117.098 163.3 171.507 178.485 184.687 186.56 192.073 197.211 203.557 208.086
150 109.142 117.985 164.349 172.581 179.581 185.8 187.678 193.208 198.36 204.723 209.265
151 109.994 118.871 165.398 173.655 180.676 186.914 188.797 194.342 199.509 205.889 210.443
152 110.846 119.759 166.446 174.729 181.77 188.026 189.915 195.476 200.657 207.054 211.62
153 111.698 120.646 167.495 175.803 182.865 189.139 191.033 196.609 201.804 208.219 212.797
154 112.551 121.534 168.543 176.876 183.959 190.251 192.15 197.742 202.951 209.383 213.973
155 113.405 122.423 169.591 177.949 185.052 191.362 193.267 198.874 204.098 210.547 215.149
156 114.259 123.312 170.639 179.022 186.146 192.474 194.384 200.006 205.244 211.71 216.324
157 115.113 124.201 171.686 180.094 187.239 193.584 195.5 201.138 206.39 212.873 217.499
158 115.968 125.09 172.734 181.167 188.332 194.695 196.616 202.269 207.535 214.035 218.673
159 116.823 125.98 173.781 182.239 189.424 195.805 197.731 203.4 208.68 215.197 219.846
160 117.679 126.87 174.828 183.311 190.516 196.915 198.846 204.53 209.824 216.358 221.019
161 118.536 127.761 175.875 184.382 191.608 198.025 199.961 205.66 210.968 217.518 222.191
162 119.392 128.651 176.922 185.454 192.7 199.134 201.076 206.79 212.111 218.678 223.363
163 120.249 129.543 177.969 186.525 193.791 200.243 202.19 207.919 213.254 219.838 224.535
164 121.107 130.434 179.016 187.596 194.883 201.351 203.303 209.047 214.396 220.997 225.705
165 121.965 131.326 180.062 188.667 195.973 202.459 204.417 210.176 215.539 222.156 226.876
166 122.823 132.218 181.109 189.737 197.064 203.567 205.53 211.304 216.68 223.314 228.045
167 123.682 133.111 182.155 190.808 198.154 204.675 206.642 212.431 217.821 224.472 229.215
168 124.541 134.003 183.201 191.878 199.244 205.782 207.755 213.558 218.962 225.629 230.383
169 125.401 134.897 184.247 192.948 200.334 206.889 208.867 214.685 220.102 226.786 231.552
170 126.261 135.79 185.293 194.017 201.423 207.995 209.978 215.812 221.242 227.942 232.719
171 127.122 136.684 186.338 195.087 202.513 209.102 211.09 216.938 222.382 229.098 233.887
172 127.983 137.578 187.384 196.156 203.602 210.208 212.201 218.063 223.521 230.253 235.053
173 128.844 138.472 188.429 197.225 204.69 211.313 213.311 219.189 224.66 231.408 236.22
174 129.706 139.367 189.475 198.294 205.779 212.419 214.422 220.314 225.798 232.563 237.385
175 130.568 140.262 190.52 199.363 206.867 213.524 215.532 221.438 226.936 233.717 238.551
176 131.43 141.157 191.565 200.432 207.955 214.628 216.641 222.563 228.074 234.87 239.716
177 132.293 142.053 192.61 201.5 209.042 215.733 217.751 223.687 229.211 236.023 240.88
178 133.157 142.949 193.654 202.568 210.13 216.837 218.86 224.81 230.347 237.176 242.044
179 134.02 143.845 194.699 203.636 211.217 217.941 219.969 225.933 231.484 238.328 243.207
180 134.884 144.741 195.743 204.704 212.304 219.044 221.077 227.056 232.62 239.48 244.37
181 135.749 145.638 196.788 205.771 213.391 220.148 222.185 228.179 233.755 240.632 245.533
182 136.614 146.535 197.832 206.839 214.477 221.251 223.293 229.301 234.891 241.783 246.695
183 137.479 147.432 198.876 207.906 215.563 222.353 224.401 230.423 236.026 242.933 247.857
184 138.344 148.33 199.92 208.973 216.649 223.456 225.508 231.544 237.16 244.084 249.018
185 139.21 149.228 200.964 210.04 217.735 224.558 226.615 232.665 238.294 245.234 250.179
186 140.077 150.126 202.008 211.106 218.82 225.66 227.722 233.786 239.428 246.383 251.339
187 140.943 151.024 203.052 212.173 219.906 226.761 228.828 234.907 240.561 247.532 252.499
188 141.81 151.923 204.095 213.239 220.991 227.863 229.935 236.027 241.694 248.681 253.659
189 142.678 152.822 205.139 214.305 222.076 228.964 231.04 237.147 242.827 249.829 254.818
190 143.545 153.721 206.182 215.371 223.16 230.064 232.146 238.266 243.959 250.977 255.976
191 144.413 154.621 207.225 216.437 224.245 231.165 233.251 239.386 245.091 252.124 257.135
192 145.282 155.521 208.268 217.502 225.329 232.265 234.356 240.505 246.223 253.271 258.292
193 146.15 156.421 209.311 218.568 226.413 233.365 235.461 241.623 247.354 254.418 259.45
194 147.02 157.321 210.354 219.633 227.496 234.465 236.566 242.742 248.485 255.564 260.607
195 147.889 158.221 211.397 220.698 228.58 235.564 237.67 243.86 249.616 256.71 261.763
196 148.759 159.122 212.439 221.763 229.663 236.664 238.774 244.977 250.746 257.855 262.92
197 149.629 160.023 213.482 222.828 230.746 237.763 239.877 246.095 251.876 259.001 264.075
198 150.499 160.925 214.524 223.892 231.829 238.861 240.981 247.212 253.006 260.145 265.231
199 151.37 161.826 215.567 224.957 232.912 239.96 242.084 248.329 254.135 261.29 266.386
200 152.241 162.728 216.609 226.021 233.994 241.058 243.187 249.445 255.264 262.434 267.541
201 153.112 163.63 217.651 227.085 235.077 242.156 244.29 250.561 256.393 263.578 268.695
202 153.984 164.532 218.693 228.149 236.159 243.254 245.392 251.677 257.521 264.721 269.849
203 154.856 165.435 219.735 229.213 237.24 244.351 246.494 252.793 258.649 265.864 271.002
204 155.728 166.338 220.777 230.276 238.322 245.448 247.596 253.908 259.777 267.007 272.155
205 156.601 167.241 221.818 231.34 239.403 246.545 248.698 255.023 260.904 268.149 273.308
206 157.474 168.144 222.86 232.403 240.485 247.642 249.799 256.138 262.031 269.291 274.46
207 158.347 169.047 223.901 233.466 241.566 248.739 250.9 257.253 263.158 270.432 275.612
208 159.221 169.951 224.943 234.529 242.647 249.835 252.001 258.367 264.285 271.574 276.764
209 160.095 170.855 225.984 235.592 243.727 250.931 253.102 259.481 265.411 272.715 277.915
210 160.969 171.759 227.025 236.655 244.808 252.027 254.202 260.595 266.537 273.855 279.066
211 161.843 172.664 228.066 237.717 245.888 253.122 255.302 261.708 267.662 274.995 280.217
212 162.718 173.568 229.107 238.78 246.968 254.218 256.402 262.821 268.788 276.135 281.367
213 163.593 174.473 230.148 239.842 248.048 255.313 257.502 263.934 269.912 277.275 282.517
214 164.469 175.378 231.189 240.904 249.128 256.408 258.601 265.047 271.037 278.414 283.666
215 165.344 176.283 232.23 241.966 250.207 257.503 259.701 266.159 272.162 279.553 284.815
216 166.22 177.189 233.27 243.028 251.286 258.597 260.8 267.271 273.286 280.692 285.964
217 167.096 178.095 234.311 244.09 252.365 259.691 261.898 268.383 274.409 281.83 287.112
218 167.973 179.001 235.351 245.151 253.444 260.785 262.997 269.495 275.533 282.968 288.261
219 168.85 179.907 236.391 246.213 254.523 261.879 264.095 270.606 276.656 284.106 289.408
220 169.727 180.813 237.432 247.274 255.602 262.973 265.193 271.717 277.779 285.243 290.556
221 170.604 181.72 238.472 248.335 256.68 264.066 266.291 272.828 278.902 286.38 291.703
222 171.482 182.627 239.512 249.396 257.758 265.159 267.389 273.939 280.024 287.517 292.85
223 172.36 183.534 240.552 250.457 258.837 266.252 268.486 275.049 281.146 288.653 293.996
224 173.238 184.441 241.592 251.517 259.914 267.345 269.584 276.159 282.268 289.789 295.142
225 174.116 185.348 242.631 252.578 260.992 268.438 270.681 277.269 283.39 290.925 296.288
226 174.995 186.256 243.671 253.638 262.07 269.53 271.777 278.379 284.511 292.061 297.433
227 175.874 187.164 244.711 254.699 263.147 270.622 272.874 279.488 285.632 293.196 298.579
228 176.753 188.072 245.75 255.759 264.224 271.714 273.97 280.597 286.753 294.331 299.723
229 177.633 188.98 246.79 256.819 265.301 272.806 275.066 281.706 287.874 295.465 300.868
230 178.512 189.889 247.829 257.879 266.378 273.898 276.162 282.814 288.994 296.6 302.012
231 179.392 190.797 248.868 258.939 267.455 274.989 277.258 283.923 290.114 297.734 303.156
232 180.273 191.706 249.908 259.998 268.531 276.08 278.354 285.031 291.234 298.867 304.299
233 181.153 192.615 250.947 261.058 269.608 277.171 279.449 286.139 292.353 300.001 305.443
234 182.034 193.524 251.986 262.117 270.684 278.262 280.544 287.247 293.472 301.134 306.586
235 182.915 194.434 253.025 263.176 271.76 279.352 281.639 288.354 294.591 302.267 307.728
236 183.796 195.343 254.063 264.235 272.836 280.443 282.734 289.461 295.71 303.4 308.871
237 184.678 196.253 255.102 265.294 273.911 281.533 283.828 290.568 296.828 304.532 310.013
238 185.56 197.163 256.141 266.353 274.987 282.623 284.922 291.675 297.947 305.664 311.154
239 186.442 198.073 257.179 267.412 276.062 283.713 286.016 292.782 299.065 306.796 312.296
240 187.324 198.984 258.218 268.471 277.138 284.802 287.11 293.888 300.182 307.927 313.437
241 188.207 199.894 259.256 269.529 278.213 285.892 288.204 294.994 301.3 309.058 314.578
242 189.09 200.805 260.295 270.588 279.288 286.981 289.298 296.1 302.417 310.189 315.718
243 189.973 201.716 261.333 271.646 280.362 288.07 290.391 297.206 303.534 311.32 316.859
244 190.856 202.627 262.371 272.704 281.437 289.159 291.484 298.311 304.651 312.45 317.999
245 191.739 203.539 263.409 273.762 282.511 290.248 292.577 299.417 305.767 313.58 319.138
246 192.623 204.45 264.447 274.82 283.586 291.336 293.67 300.522 306.883 314.71 320.278
247 193.507 205.362 265.485 275.878 284.66 292.425 294.762 301.626 307.999 315.84 321.417
248 194.391 206.274 266.523 276.935 285.734 293.513 295.855 302.731 309.115 316.969 322.556
249 195.276 207.186 267.561 277.993 286.808 294.601 296.947 303.835 310.231 318.098 323.694
250 196.161 208.098 268.599 279.05 287.882 295.689 298.039 304.94 311.346 319.227 324.832
300 240.663 253.912 320.397 331.789 341.395 349.874 352.425 359.906 366.844 375.369 381.425
350 285.608 300.064 372.051 384.306 394.626 403.723 406.457 414.474 421.9 431.017 437.488
400 330.903 346.482 423.59 436.649 447.632 457.305 460.211 468.724 476.606 486.274 493.132
450 376.483 393.118 475.035 488.849 500.456 510.67 513.736 522.717 531.026 541.212 548.432
500 422.303 439.936 526.401 540.93 553.127 563.852 567.07 576.493 585.207 595.882 603.446
550 468.328 486.91 577.701 592.909 605.667 616.878 620.241 630.084 639.183 650.324 658.215
600 514.529 534.019 628.943 644.8 658.094 669.769 673.27 683.516 692.982 704.568 712.771
650 560.885 581.245 680.134 696.614 710.421 722.542 726.176 736.807 746.625 758.639 767.141
700 607.38 628.577 731.28 748.359 762.661 775.211 778.972 789.974 800.131 812.556 821.347
750 653.997 676.003 782.386 800.043 814.822 827.785 831.67 843.029 853.514 866.336 875.404
800 700.725 723.513 833.456 851.671 866.911 880.275 884.279 895.984 906.786 919.991 929.329
850 747.554 771.099 884.492 903.249 918.937 932.689 936.808 948.848 959.957 973.534 983.133
900 794.475 818.756 935.499 954.782 970.904 985.032 989.263 1001.63 1013.036 1026.974 1036.826
950 841.48 866.477 986.478 1006.272 1022.816 1037.311 1041.651 1054.334 1066.031 1080.32 1090.418
1000 888.564 914.257 1037.431 1057.724 1074.679 1089.531 1093.977 1106.969 1118.948 1133.579 1143.917
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