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중심극한정리 [2012/03/19 16:37] moonrepeat 새로 만듦 |
중심극한정리 [2021/03/10 21:42] |
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| 줄 1: | 줄 1: | ||
| - | ====== 중심극한정리 (Central Limit Theorem : CLT) ====== | ||
| - | ===== 정의 ===== | ||
| - | $X_{1}, \ ... \ ,X_{n}$이 [[평균]]이 $\mu$이고 [[분산]]이 $\sigma^{2}$인 [[분포]]로 부터의 [[확률표본]]일 때, $X$의 [[분포]]와 관계없이 [[표본평균]] $\overline{X}$의 [[분포]]는 [[표본크기]] $n$이 커짐에 따라 [[정규분포]] $\overline{X} \sim N(\mu,\sigma^{2}/n)$로 근접한다. | ||
| - | * $$ \overline{X} \sim N(\mu,\sigma^{2}/n) $$ | ||
| - | ===== 예제1 ===== | ||
| - | 아래의 자료는 $Exp(2)$를 따르는 [[확률표본]]으로 부터 [[랜덤]]하게 크기 20인 [[샘플]]을 100회 추출한 자료와 각 [[샘플군]]의 [[표본평균]]을 구한 것이다. | ||
| - | |||
| - | ^ ^ 1 ^ 2 ^ 3 ^ 4 ^ 5 ^ 6 ^ 7 ^ 8 ^ 9 ^ 10 ^ 11 ^ 12 ^ 13 ^ 14 ^ 15 ^ 16 ^ 17 ^ 18 ^ 19 ^ 20 ^ $\overline{X}$ | | ||
| - | ^ 1 | 1.826 | 0.459 | 0.574 | 0.285 | 1.781 | 0.565 | 6.415 | 0.108 | 0.835 | 0.538 | 0.067 | 1.040 | 0.643 | 0.149 | 1.404 | 0.668 | 1.711 | 0.579 | 2.091 | 0.838 ^ 1.1288 | | ||
| - | ^ 2 | 0.718 | 1.124 | 0.476 | 5.273 | 4.298 | 6.084 | 0.115 | 3.493 | 0.565 | 0.673 | 1.528 | 2.163 | 0.048 | 1.529 | 0.789 | 0.194 | 0.531 | 1.885 | 2.454 | 0.585 ^ 1.7263 | | ||
| - | ^ 3 | 1.497 | 1.910 | 1.201 | 0.436 | 1.876 | 1.533 | 2.121 | 1.448 | 5.285 | 1.421 | 5.795 | 0.587 | 1.859 | 1.583 | 1.492 | 4.896 | 3.067 | 3.969 | 0.036 | 2.152 ^ 2.2082 | | ||
| - | ^ 4 | 5.657 | 1.830 | 3.352 | 5.105 | 3.193 | 1.228 | 0.984 | 0.503 | 3.605 | 4.681 | 0.207 | 2.409 | 1.743 | 1.220 | 0.307 | 1.204 | 5.379 | 0.789 | 2.719 | 4.310 ^ 2.5213 | | ||
| - | ^ 5 | 1.532 | 0.912 | 0.486 | 0.379 | 1.766 | 0.115 | 0.086 | 3.709 | 0.213 | 2.844 | 0.992 | 1.347 | 0.241 | 0.102 | 1.561 | 1.392 | 1.784 | 0.530 | 1.182 | 2.950 ^ 1.2062 | | ||
| - | ^ 6 | 1.781 | 0.830 | 1.911 | 0.672 | 0.668 | 10.977 | 2.629 | 3.948 | 0.669 | 1.266 | 1.348 | 1.682 | 0.575 | 2.317 | 1.175 | 3.917 | 8.151 | 1.764 | 4.380 | 3.240 ^ 2.6950 | | ||
| - | ^ 7 | 4.547 | 1.080 | 2.890 | 1.852 | 0.763 | 1.928 | 4.421 | 3.993 | 5.604 | 0.044 | 1.206 | 0.022 | 0.164 | 2.453 | 2.125 | 0.407 | 3.234 | 0.207 | 1.483 | 2.792 ^ 2.0608 | | ||
| - | ^ 8 | 0.805 | 0.913 | 3.543 | 0.514 | 1.816 | 0.791 | 2.551 | 0.008 | 1.955 | 2.198 | 1.582 | 0.288 | 3.831 | 0.406 | 2.382 | 0.787 | 0.544 | 2.457 | 0.708 | 0.921 ^ 1.4500 | | ||
| - | ^ 9 | 3.688 | 0.827 | 2.711 | 1.002 | 2.887 | 1.751 | 1.029 | 0.977 | 3.054 | 1.611 | 1.852 | 3.031 | 0.180 | 0.578 | 0.959 | 4.181 | 4.620 | 1.994 | 4.133 | 0.276 ^ 2.0671 | | ||
| - | ^ 10 | 0.434 | 2.149 | 4.434 | 1.583 | 2.198 | 1.442 | 8.956 | 1.379 | 3.858 | 1.527 | 1.039 | 1.314 | 1.680 | 0.381 | 0.494 | 0.547 | 2.847 | 6.612 | 0.665 | 0.742 ^ 2.2141 | | ||
| - | ^ 11 | 0.850 | 3.892 | 5.116 | 1.165 | 0.690 | 2.461 | 0.009 | 0.445 | 1.026 | 0.003 | 0.500 | 0.424 | 3.764 | 0.550 | 1.076 | 6.645 | 0.008 | 0.191 | 4.195 | 0.147 ^ 1.6579 | | ||
| - | ^ 12 | 0.182 | 1.304 | 0.098 | 2.119 | 1.089 | 1.059 | 1.252 | 6.181 | 1.352 | 3.107 | 0.679 | 1.418 | 0.274 | 6.114 | 0.668 | 0.186 | 4.097 | 11.275 | 0.491 | 0.089 ^ 2.1517 | | ||
| - | ^ 13 | 1.021 | 1.890 | 2.557 | 0.094 | 0.979 | 0.300 | 0.881 | 0.203 | 0.282 | 1.476 | 0.699 | 2.579 | 1.139 | 6.226 | 2.014 | 3.405 | 0.845 | 3.100 | 1.336 | 0.818 ^ 1.5922 | | ||
| - | ^ 14 | 0.573 | 0.164 | 1.618 | 3.065 | 5.699 | 0.560 | 0.747 | 0.196 | 2.716 | 1.059 | 1.389 | 2.091 | 1.343 | 2.436 | 0.462 | 0.165 | 1.189 | 1.933 | 0.805 | 0.138 ^ 1.4174 | | ||
| - | ^ 15 | 0.059 | 0.736 | 3.429 | 2.624 | 1.740 | 3.556 | 0.630 | 0.716 | 5.208 | 1.487 | 1.346 | 0.172 | 0.120 | 2.465 | 0.883 | 1.767 | 1.644 | 0.130 | 2.231 | 0.418 ^ 1.5681 | | ||
| - | ^ 16 | 1.105 | 7.029 | 0.999 | 1.378 | 2.533 | 3.386 | 0.140 | 4.064 | 0.049 | 0.818 | 7.244 | 4.372 | 0.267 | 1.044 | 1.959 | 1.983 | 1.372 | 4.922 | 7.564 | 0.235 ^ 2.6232 | | ||
| - | ^ 17 | 0.079 | 5.579 | 0.296 | 5.031 | 0.193 | 4.197 | 0.262 | 2.566 | 4.255 | 3.268 | 0.042 | 0.619 | 0.844 | 1.031 | 0.979 | 1.304 | 2.291 | 1.102 | 0.005 | 1.714 ^ 1.7829 | | ||
| - | ^ 18 | 1.480 | 0.789 | 0.806 | 0.938 | 9.819 | 2.107 | 1.803 | 3.362 | 0.333 | 0.057 | 1.437 | 1.863 | 0.721 | 0.224 | 3.962 | 0.935 | 0.085 | 0.219 | 0.962 | 3.193 ^ 1.7548 | | ||
| - | ^ 19 | 1.770 | 0.462 | 1.081 | 5.194 | 1.127 | 0.589 | 1.081 | 0.416 | 3.179 | 0.235 | 0.463 | 2.010 | 6.402 | 1.667 | 3.621 | 0.295 | 2.684 | 2.352 | 2.997 | 0.691 ^ 1.9158 | | ||
| - | ^ 20 | 5.821 | 1.357 | 6.253 | 0.089 | 0.933 | 8.626 | 5.259 | 4.058 | 0.847 | 1.834 | 3.937 | 0.176 | 0.351 | 0.103 | 4.102 | 0.030 | 5.512 | 0.222 | 1.458 | 1.280 ^ 2.6124 | | ||
| - | ^ 21 | 10.232 | 0.458 | 1.119 | 2.145 | 1.683 | 2.013 | 1.320 | 3.515 | 4.400 | 2.585 | 2.712 | 0.440 | 0.225 | 0.435 | 2.366 | 2.116 | 0.303 | 1.615 | 1.195 | 2.446 ^ 2.1662 | | ||
| - | ^ 22 | 7.348 | 0.570 | 2.567 | 2.740 | 2.572 | 0.482 | 5.927 | 1.410 | 1.355 | 1.811 | 0.124 | 0.245 | 1.037 | 0.587 | 1.876 | 3.468 | 0.036 | 3.971 | 3.532 | 5.757 ^ 2.3708 | | ||
| - | ^ 23 | 0.239 | 1.576 | 0.465 | 3.990 | 5.428 | 0.827 | 3.043 | 0.023 | 0.868 | 1.733 | 1.301 | 1.958 | 6.417 | 3.324 | 0.255 | 4.525 | 4.908 | 0.746 | 7.210 | 1.930 ^ 2.5383 | | ||
| - | ^ 24 | 2.331 | 1.923 | 0.901 | 2.500 | 1.447 | 12.422 | 4.296 | 1.724 | 3.278 | 2.355 | 2.867 | 0.967 | 1.465 | 7.185 | 0.847 | 0.665 | 2.270 | 0.720 | 0.089 | 0.036 ^ 2.5144 | | ||
| - | ^ 25 | 0.370 | 0.980 | 0.464 | 0.802 | 0.898 | 0.747 | 0.820 | 3.191 | 7.604 | 2.794 | 2.408 | 1.944 | 0.556 | 4.643 | 0.105 | 4.919 | 0.256 | 4.776 | 1.745 | 1.233 ^ 2.0628 | | ||
| - | ^ 26 | 0.755 | 3.160 | 0.685 | 0.815 | 0.365 | 1.397 | 2.948 | 2.840 | 1.058 | 0.356 | 3.726 | 1.519 | 3.046 | 1.553 | 0.439 | 1.280 | 0.033 | 3.254 | 1.239 | 4.016 ^ 1.7242 | | ||
| - | ^ 27 | 2.214 | 0.614 | 1.869 | 2.534 | 5.769 | 0.720 | 1.441 | 0.381 | 0.462 | 0.572 | 1.187 | 1.517 | 0.309 | 7.689 | 0.596 | 1.023 | 1.304 | 0.546 | 1.926 | 2.929 ^ 1.7801 | | ||
| - | ^ 28 | 0.605 | 1.422 | 3.912 | 2.467 | 0.374 | 0.197 | 0.595 | 2.495 | 0.822 | 0.214 | 0.275 | 0.487 | 0.808 | 0.024 | 0.641 | 0.468 | 1.872 | 0.031 | 0.895 | 0.399 ^ 0.9502 | | ||
| - | ^ 29 | 0.688 | 0.464 | 0.438 | 0.591 | 0.448 | 0.350 | 3.037 | 2.389 | 0.412 | 2.662 | 0.841 | 2.552 | 3.720 | 2.506 | 3.885 | 0.963 | 5.636 | 1.010 | 3.908 | 0.462 ^ 1.8481 | | ||
| - | ^ 30 | 0.275 | 0.566 | 0.355 | 0.135 | 2.877 | 2.052 | 0.267 | 0.132 | 0.655 | 1.567 | 0.002 | 1.228 | 0.200 | 0.153 | 8.453 | 0.652 | 8.562 | 1.676 | 2.219 | 0.401 ^ 1.6214 | | ||
| - | ^ 31 | 0.066 | 7.474 | 2.167 | 5.027 | 2.410 | 3.255 | 2.278 | 4.683 | 1.002 | 2.687 | 3.236 | 0.672 | 5.000 | 6.620 | 0.331 | 3.143 | 0.122 | 0.097 | 1.328 | 2.074 ^ 2.6836 | | ||
| - | ^ 32 | 2.460 | 2.436 | 0.202 | 1.503 | 2.364 | 0.843 | 0.643 | 8.652 | 0.666 | 0.448 | 0.630 | 0.228 | 4.449 | 2.385 | 2.181 | 9.295 | 1.054 | 0.160 | 1.962 | 0.689 ^ 2.1625 | | ||
| - | ^ 33 | 0.255 | 1.751 | 1.423 | 0.051 | 0.107 | 2.067 | 2.331 | 0.220 | 1.295 | 3.392 | 1.869 | 0.068 | 0.824 | 1.438 | 0.233 | 1.029 | 0.457 | 1.333 | 1.825 | 3.656 ^ 1.2812 | | ||
| - | ^ 34 | 0.789 | 3.135 | 2.076 | 0.561 | 1.856 | 4.232 | 6.178 | 0.227 | 0.152 | 0.843 | 6.908 | 0.006 | 0.458 | 0.529 | 0.033 | 0.044 | 3.526 | 3.261 | 3.274 | 3.524 ^ 2.0806 | | ||
| - | ^ 35 | 4.477 | 4.372 | 0.426 | 1.401 | 0.203 | 1.518 | 2.164 | 0.189 | 4.024 | 1.966 | 0.292 | 2.966 | 1.336 | 0.042 | 3.341 | 2.119 | 5.539 | 2.587 | 3.110 | 1.310 ^ 2.1691 | | ||
| - | ^ 36 | 0.111 | 1.992 | 6.206 | 2.709 | 1.945 | 0.147 | 1.115 | 0.636 | 0.950 | 2.772 | 0.312 | 4.643 | 0.093 | 0.424 | 0.804 | 0.674 | 6.441 | 1.963 | 1.737 | 4.805 ^ 2.0240 | | ||
| - | ^ 37 | 0.828 | 8.049 | 3.812 | 3.834 | 0.053 | 6.521 | 1.561 | 0.599 | 0.432 | 2.556 | 0.112 | 1.862 | 3.195 | 0.389 | 1.659 | 0.658 | 2.626 | 0.121 | 1.938 | 0.144 ^ 2.0475 | | ||
| - | ^ 38 | 0.134 | 0.311 | 0.058 | 1.465 | 0.294 | 3.914 | 0.204 | 2.654 | 0.153 | 0.376 | 0.230 | 7.209 | 0.465 | 7.334 | 0.902 | 2.677 | 2.145 | 1.197 | 1.498 | 1.465 ^ 1.7343 | | ||
| - | ^ 39 | 4.235 | 1.310 | 3.097 | 0.834 | 0.274 | 2.496 | 1.203 | 3.559 | 1.680 | 7.648 | 5.120 | 2.622 | 4.531 | 0.059 | 2.174 | 0.118 | 4.891 | 0.625 | 3.175 | 0.072 ^ 2.4862 | | ||
| - | ^ 40 | 0.734 | 1.906 | 0.666 | 0.639 | 1.421 | 0.850 | 9.692 | 2.509 | 0.903 | 3.566 | 3.268 | 0.667 | 0.166 | 2.066 | 1.862 | 0.787 | 1.445 | 0.879 | 0.660 | 2.905 ^ 1.8796 | | ||
| - | ^ 41 | 0.503 | 1.413 | 3.400 | 3.427 | 2.182 | 0.068 | 0.120 | 1.586 | 3.585 | 0.062 | 0.701 | 8.047 | 0.048 | 3.140 | 1.423 | 1.779 | 0.200 | 2.865 | 1.319 | 5.208 ^ 2.0538 | | ||
| - | ^ 42 | 0.380 | 0.200 | 3.991 | 2.283 | 1.155 | 3.369 | 1.128 | 0.403 | 3.137 | 5.723 | 6.428 | 0.917 | 1.471 | 0.084 | 5.994 | 3.125 | 0.006 | 0.043 | 1.495 | 1.114 ^ 2.1223 | | ||
| - | ^ 43 | 0.970 | 1.037 | 5.398 | 0.345 | 4.075 | 1.359 | 1.828 | 0.497 | 4.956 | 0.712 | 2.346 | 3.880 | 1.634 | 0.112 | 1.425 | 2.680 | 3.155 | 1.507 | 1.556 | 0.070 ^ 1.9771 | | ||
| - | ^ 44 | 11.129 | 3.531 | 1.407 | 0.336 | 1.321 | 5.667 | 3.609 | 4.771 | 4.931 | 1.399 | 0.227 | 0.671 | 2.316 | 1.506 | 0.396 | 1.389 | 2.935 | 1.228 | 0.986 | 0.326 ^ 2.5041 | | ||
| - | ^ 45 | 0.737 | 0.801 | 3.970 | 1.944 | 0.060 | 0.907 | 0.352 | 1.573 | 1.134 | 1.951 | 1.173 | 0.995 | 2.906 | 2.076 | 0.057 | 2.165 | 1.163 | 0.159 | 0.244 | 1.072 ^ 1.2720 | | ||
| - | ^ 46 | 0.523 | 3.368 | 0.618 | 1.431 | 1.430 | 0.606 | 3.026 | 0.109 | 2.260 | 3.059 | 0.146 | 0.945 | 1.762 | 1.175 | 0.672 | 0.537 | 2.533 | 1.692 | 0.612 | 0.022 ^ 1.3263 | | ||
| - | ^ 47 | 0.935 | 9.402 | 1.524 | 2.165 | 1.017 | 3.351 | 2.447 | 1.711 | 0.839 | 0.118 | 0.047 | 1.477 | 1.390 | 1.590 | 0.171 | 0.871 | 2.205 | 0.312 | 7.225 | 2.636 ^ 2.0717 | | ||
| - | ^ 48 | 0.437 | 0.685 | 5.529 | 1.408 | 1.518 | 0.865 | 1.596 | 2.714 | 1.101 | 1.082 | 0.370 | 3.146 | 0.701 | 4.896 | 0.315 | 3.114 | 2.833 | 0.343 | 4.350 | 5.247 ^ 2.1125 | | ||
| - | ^ 49 | 2.072 | 4.611 | 0.110 | 0.591 | 4.959 | 1.516 | 4.977 | 2.151 | 0.048 | 2.434 | 2.353 | 0.747 | 0.820 | 0.842 | 0.414 | 3.097 | 3.533 | 1.408 | 2.632 | 1.959 ^ 2.0637 | | ||
| - | ^ 50 | 0.029 | 6.070 | 0.646 | 2.171 | 2.641 | 0.665 | 0.088 | 1.003 | 1.532 | 4.137 | 10.775 | 0.559 | 2.243 | 4.651 | 0.244 | 1.140 | 1.935 | 0.555 | 2.314 | 1.707 ^ 2.2553 | | ||
| - | ^ 51 | 0.455 | 1.838 | 0.058 | 2.623 | 0.576 | 0.073 | 0.192 | 1.512 | 0.486 | 1.188 | 1.429 | 0.342 | 0.076 | 1.165 | 0.093 | 0.260 | 7.326 | 2.719 | 0.779 | 0.278 ^ 1.1734 | | ||
| - | ^ 52 | 1.710 | 0.575 | 4.100 | 0.794 | 2.207 | 3.705 | 9.652 | 2.578 | 0.759 | 0.862 | 2.981 | 0.417 | 3.555 | 0.544 | 0.098 | 0.975 | 0.381 | 2.296 | 2.569 | 0.705 ^ 2.0732 | | ||
| - | ^ 53 | 3.679 | 1.539 | 2.471 | 0.051 | 1.637 | 0.426 | 0.905 | 2.947 | 3.250 | 6.314 | 0.515 | 3.187 | 1.054 | 1.492 | 5.527 | 3.685 | 0.947 | 0.520 | 1.130 | 0.141 ^ 2.0709 | | ||
| - | ^ 54 | 3.985 | 0.053 | 7.856 | 0.194 | 1.378 | 7.327 | 0.633 | 1.535 | 1.224 | 1.096 | 2.191 | 4.572 | 2.148 | 2.215 | 1.209 | 1.312 | 1.286 | 0.693 | 0.484 | 0.650 ^ 2.1021 | | ||
| - | ^ 55 | 0.597 | 2.408 | 1.536 | 0.206 | 1.746 | 2.348 | 0.769 | 1.225 | 1.884 | 0.534 | 0.280 | 1.353 | 1.216 | 0.038 | 0.309 | 2.536 | 0.036 | 5.341 | 0.337 | 6.104 ^ 1.5402 | | ||
| - | ^ 56 | 1.291 | 0.400 | 0.899 | 3.010 | 0.452 | 0.965 | 1.363 | 1.095 | 0.009 | 5.845 | 0.018 | 2.315 | 0.882 | 0.794 | 1.077 | 0.499 | 1.600 | 0.147 | 0.350 | 0.319 ^ 1.1665 | | ||
| - | ^ 57 | 4.678 | 3.136 | 3.460 | 0.713 | 1.425 | 0.178 | 0.183 | 0.529 | 0.239 | 6.277 | 0.404 | 3.198 | 0.656 | 0.260 | 2.568 | 0.668 | 4.165 | 1.118 | 1.156 | 3.755 ^ 1.9383 | | ||
| - | ^ 58 | 1.432 | 0.069 | 0.212 | 0.028 | 1.415 | 0.856 | 0.516 | 3.524 | 1.580 | 0.536 | 0.434 | 3.149 | 0.128 | 2.616 | 1.454 | 0.274 | 7.262 | 6.596 | 2.314 | 0.081 ^ 1.7238 | | ||
| - | ^ 59 | 2.150 | 1.156 | 0.222 | 1.829 | 0.780 | 2.645 | 0.234 | 0.485 | 0.933 | 1.108 | 0.242 | 2.210 | 0.861 | 4.167 | 1.636 | 0.967 | 0.641 | 5.392 | 1.114 | 1.332 ^ 1.5052 | | ||
| - | ^ 60 | 2.612 | 0.576 | 0.083 | 1.391 | 2.341 | 0.408 | 1.399 | 0.541 | 0.678 | 0.361 | 2.816 | 1.260 | 0.208 | 1.033 | 1.792 | 1.354 | 2.238 | 0.132 | 0.390 | 1.767 ^ 1.1690 | | ||
| - | ^ 61 | 0.701 | 0.912 | 0.510 | 2.674 | 0.483 | 1.590 | 4.171 | 2.011 | 4.038 | 2.645 | 0.248 | 0.419 | 1.208 | 3.503 | 1.306 | 0.940 | 0.712 | 5.544 | 0.292 | 4.875 ^ 1.9391 | | ||
| - | ^ 62 | 1.568 | 1.829 | 0.526 | 0.469 | 1.255 | 1.467 | 1.594 | 1.673 | 0.452 | 1.542 | 0.927 | 0.152 | 1.369 | 3.874 | 0.149 | 5.414 | 0.484 | 1.830 | 5.333 | 3.140 ^ 1.7524 | | ||
| - | ^ 63 | 1.637 | 2.635 | 4.276 | 0.326 | 11.681 | 2.053 | 1.084 | 3.981 | 0.773 | 0.769 | 0.492 | 0.123 | 0.973 | 0.960 | 1.924 | 1.316 | 2.999 | 1.772 | 2.886 | 1.439 ^ 2.2050 | | ||
| - | ^ 64 | 0.248 | 4.613 | 1.522 | 0.420 | 0.849 | 0.053 | 0.506 | 1.400 | 0.329 | 6.751 | 0.430 | 1.736 | 0.041 | 1.893 | 4.997 | 0.737 | 0.163 | 0.082 | 2.536 | 7.939 ^ 1.8623 | | ||
| - | ^ 65 | 5.306 | 1.164 | 0.934 | 2.964 | 1.062 | 2.758 | 5.257 | 0.485 | 5.342 | 1.458 | 8.977 | 0.325 | 4.605 | 1.349 | 0.532 | 0.021 | 0.337 | 0.333 | 0.078 | 1.027 ^ 2.2157 | | ||
| - | ^ 66 | 1.410 | 1.574 | 0.905 | 0.571 | 0.579 | 1.836 | 1.149 | 0.255 | 0.438 | 0.826 | 1.735 | 1.065 | 1.930 | 0.885 | 8.831 | 0.610 | 0.214 | 0.098 | 0.124 | 1.185 ^ 1.3110 | | ||
| - | ^ 67 | 7.571 | 1.087 | 5.222 | 1.202 | 2.425 | 0.861 | 1.867 | 0.755 | 6.939 | 1.826 | 0.384 | 1.689 | 0.197 | 2.882 | 0.898 | 0.524 | 2.606 | 0.378 | 1.387 | 1.346 ^ 2.1023 | | ||
| - | ^ 68 | 0.044 | 3.975 | 0.279 | 0.189 | 2.197 | 0.469 | 2.781 | 0.889 | 0.008 | 3.187 | 0.155 | 3.615 | 0.875 | 0.883 | 4.788 | 1.094 | 1.214 | 3.568 | 2.890 | 4.534 ^ 1.8817 | | ||
| - | ^ 69 | 7.203 | 1.205 | 0.615 | 0.640 | 0.418 | 2.775 | 0.331 | 3.244 | 4.246 | 1.367 | 1.205 | 6.738 | 3.241 | 2.286 | 0.619 | 4.081 | 1.437 | 1.466 | 2.106 | 1.384 ^ 2.3304 | | ||
| - | ^ 70 | 0.518 | 3.629 | 0.603 | 2.158 | 4.221 | 4.436 | 7.151 | 1.086 | 0.499 | 2.951 | 0.330 | 2.244 | 2.293 | 2.066 | 3.645 | 5.574 | 1.684 | 1.084 | 1.145 | 5.183 ^ 2.6250 | | ||
| - | ^ 71 | 2.200 | 1.666 | 2.616 | 0.092 | 4.738 | 1.955 | 0.539 | 0.942 | 0.541 | 0.120 | 0.853 | 2.006 | 8.365 | 5.076 | 0.330 | 0.516 | 0.800 | 1.067 | 0.102 | 1.241 ^ 1.7883 | | ||
| - | ^ 72 | 3.619 | 0.519 | 2.280 | 4.521 | 0.583 | 4.615 | 3.822 | 2.343 | 1.409 | 0.378 | 2.574 | 3.074 | 0.663 | 1.472 | 0.589 | 0.299 | 0.110 | 0.597 | 2.421 | 3.696 ^ 1.9792 | | ||
| - | ^ 73 | 0.959 | 0.616 | 3.791 | 1.879 | 4.467 | 1.564 | 1.501 | 1.325 | 0.278 | 5.751 | 3.586 | 0.358 | 1.080 | 2.482 | 1.551 | 0.941 | 5.097 | 1.630 | 9.033 | 1.636 ^ 2.4763 | | ||
| - | ^ 74 | 0.134 | 0.757 | 2.667 | 1.850 | 3.609 | 1.944 | 0.003 | 3.802 | 1.689 | 0.266 | 2.569 | 3.699 | 0.495 | 0.491 | 1.427 | 0.974 | 1.045 | 2.689 | 0.768 | 3.716 ^ 1.7297 | | ||
| - | ^ 75 | 2.488 | 0.077 | 2.175 | 1.559 | 4.838 | 3.287 | 0.257 | 5.162 | 11.153 | 2.587 | 2.773 | 2.728 | 3.675 | 2.413 | 2.412 | 0.593 | 0.661 | 2.485 | 7.879 | 2.142 ^ 3.0672 | | ||
| - | ^ 76 | 3.284 | 2.970 | 0.659 | 1.022 | 0.362 | 4.640 | 3.348 | 0.588 | 1.042 | 1.232 | 3.904 | 1.577 | 0.935 | 8.953 | 2.726 | 1.566 | 3.461 | 0.276 | 0.802 | 5.276 ^ 2.4312 | | ||
| - | ^ 77 | 0.501 | 0.277 | 0.238 | 2.603 | 5.811 | 1.792 | 1.630 | 0.934 | 6.613 | 0.125 | 0.592 | 0.048 | 1.707 | 0.580 | 0.383 | 3.684 | 3.683 | 0.956 | 2.621 | 3.471 ^ 1.9125 | | ||
| - | ^ 78 | 5.974 | 3.450 | 1.478 | 0.801 | 1.414 | 0.183 | 0.748 | 2.007 | 0.690 | 0.271 | 0.629 | 0.116 | 0.260 | 4.797 | 4.983 | 1.397 | 0.128 | 0.104 | 3.387 | 2.308 ^ 1.7563 | | ||
| - | ^ 79 | 2.795 | 2.704 | 0.756 | 1.531 | 0.869 | 7.059 | 1.165 | 1.567 | 0.953 | 1.090 | 0.915 | 0.704 | 0.060 | 0.219 | 0.126 | 1.280 | 0.947 | 1.599 | 3.994 | 1.503 ^ 1.5918 | | ||
| - | ^ 80 | 2.811 | 0.004 | 5.020 | 2.256 | 2.335 | 8.548 | 3.081 | 2.117 | 4.352 | 2.696 | 1.892 | 1.765 | 1.280 | 2.302 | 1.319 | 0.296 | 2.545 | 0.091 | 1.193 | 3.753 ^ 2.4828 | | ||
| - | ^ 81 | 0.153 | 3.576 | 0.536 | 2.279 | 0.546 | 2.142 | 0.591 | 0.174 | 0.016 | 7.545 | 0.309 | 1.869 | 0.366 | 4.315 | 1.086 | 1.619 | 0.872 | 0.850 | 0.818 | 0.831 ^ 1.5247 | | ||
| - | ^ 82 | 1.201 | 1.797 | 1.227 | 0.329 | 0.060 | 0.774 | 0.303 | 1.165 | 0.964 | 0.453 | 1.561 | 0.033 | 8.099 | 0.174 | 4.889 | 0.915 | 0.183 | 0.065 | 2.762 | 4.567 ^ 1.5761 | | ||
| - | ^ 83 | 6.606 | 5.031 | 1.112 | 0.809 | 1.318 | 0.325 | 1.094 | 2.099 | 1.173 | 3.151 | 3.894 | 1.816 | 2.158 | 1.536 | 1.411 | 0.605 | 0.555 | 3.149 | 0.946 | 0.328 ^ 1.9558 | | ||
| - | ^ 84 | 2.998 | 0.449 | 0.073 | 1.113 | 0.548 | 3.084 | 1.680 | 0.363 | 0.183 | 1.413 | 0.899 | 0.561 | 1.332 | 2.156 | 3.080 | 4.324 | 1.304 | 0.273 | 0.592 | 1.705 ^ 1.4065 | | ||
| - | ^ 85 | 0.018 | 1.565 | 1.585 | 0.505 | 1.472 | 2.132 | 4.042 | 1.312 | 0.041 | 1.142 | 0.117 | 6.817 | 3.564 | 0.407 | 1.188 | 0.253 | 0.183 | 0.174 | 0.301 | 1.101 ^ 1.3960 | | ||
| - | ^ 86 | 0.627 | 3.079 | 0.437 | 1.255 | 16.305 | 4.288 | 2.463 | 0.770 | 1.208 | 1.726 | 2.018 | 1.578 | 0.848 | 4.735 | 2.395 | 0.974 | 0.492 | 0.068 | 1.646 | 6.194 ^ 2.6553 | | ||
| - | ^ 87 | 0.121 | 2.089 | 1.255 | 5.111 | 1.903 | 2.901 | 1.569 | 2.454 | 2.159 | 5.792 | 1.900 | 0.494 | 3.405 | 2.427 | 0.753 | 1.996 | 6.225 | 2.632 | 2.942 | 3.661 ^ 2.5895 | | ||
| - | ^ 88 | 4.436 | 3.507 | 1.807 | 4.369 | 1.503 | 0.410 | 0.006 | 2.275 | 1.941 | 8.112 | 5.200 | 3.903 | 0.334 | 6.883 | 0.485 | 0.627 | 0.111 | 0.595 | 3.355 | 0.075 ^ 2.4967 | | ||
| - | ^ 89 | 0.863 | 1.520 | 0.426 | 1.221 | 4.022 | 2.973 | 4.753 | 1.192 | 1.406 | 5.025 | 0.694 | 1.158 | 1.441 | 3.382 | 2.110 | 0.624 | 6.227 | 5.700 | 1.980 | 0.793 ^ 2.3755 | | ||
| - | ^ 90 | 1.942 | 1.399 | 1.291 | 1.122 | 0.236 | 1.906 | 0.288 | 0.759 | 1.024 | 4.984 | 2.775 | 0.092 | 5.645 | 1.823 | 0.728 | 0.254 | 2.979 | 1.924 | 0.539 | 0.696 ^ 1.6203 | | ||
| - | ^ 91 | 0.183 | 1.683 | 1.348 | 1.464 | 0.446 | 0.564 | 0.341 | 0.585 | 5.209 | 0.058 | 0.167 | 2.729 | 3.184 | 2.563 | 0.754 | 1.899 | 4.158 | 0.142 | 2.973 | 0.585 ^ 1.5518 | | ||
| - | ^ 92 | 0.925 | 1.083 | 0.362 | 2.391 | 2.882 | 1.356 | 0.533 | 0.542 | 1.016 | 6.305 | 3.881 | 8.728 | 0.008 | 2.696 | 0.597 | 0.004 | 3.989 | 4.892 | 2.375 | 2.053 ^ 2.3309 | | ||
| - | ^ 93 | 0.391 | 6.696 | 0.818 | 7.124 | 0.527 | 2.703 | 1.525 | 2.350 | 1.024 | 7.114 | 0.381 | 0.310 | 4.714 | 1.330 | 1.232 | 5.264 | 2.899 | 1.417 | 1.391 | 0.143 ^ 2.4677 | | ||
| - | ^ 94 | 0.573 | 3.221 | 0.748 | 2.306 | 0.143 | 0.916 | 7.048 | 3.305 | 1.703 | 1.865 | 1.646 | 0.015 | 2.853 | 0.890 | 3.999 | 0.399 | 5.494 | 0.738 | 3.286 | 2.296 ^ 2.1722 | | ||
| - | ^ 95 | 4.930 | 1.082 | 0.424 | 0.688 | 1.515 | 1.701 | 0.390 | 1.375 | 0.366 | 1.729 | 3.562 | 0.304 | 1.261 | 6.466 | 4.887 | 1.767 | 3.352 | 0.699 | 0.874 | 0.301 ^ 1.8837 | | ||
| - | ^ 96 | 1.462 | 0.142 | 1.370 | 4.436 | 3.364 | 3.347 | 1.207 | 0.700 | 1.814 | 1.740 | 1.992 | 3.814 | 3.720 | 0.378 | 0.561 | 5.845 | 4.491 | 0.394 | 6.303 | 1.982 ^ 2.4531 | | ||
| - | ^ 97 | 3.323 | 4.009 | 1.076 | 0.320 | 1.432 | 1.012 | 1.946 | 0.402 | 2.453 | 0.792 | 0.689 | 0.318 | 1.596 | 0.286 | 3.182 | 0.205 | 10.800 | 0.646 | 0.360 | 0.490 ^ 1.7669 | | ||
| - | ^ 98 | 2.229 | 3.460 | 0.908 | 0.530 | 0.489 | 1.146 | 0.081 | 1.380 | 2.909 | 1.100 | 3.694 | 1.069 | 6.089 | 0.397 | 1.210 | 1.979 | 1.459 | 1.184 | 1.528 | 0.367 ^ 1.6604 | | ||
| - | ^ 99 | 5.436 | 0.143 | 3.574 | 3.510 | 8.224 | 0.687 | 0.909 | 2.749 | 2.898 | 3.329 | 1.177 | 2.058 | 3.570 | 4.898 | 1.030 | 1.354 | 0.114 | 0.754 | 0.391 | 0.854 ^ 2.3830 | | ||
| - | ^ 100 | 0.027 | 0.404 | 0.736 | 1.755 | 2.121 | 1.928 | 2.104 | 0.712 | 0.771 | 1.791 | 1.871 | 1.767 | 0.644 | 2.571 | 0.256 | 0.104 | 4.168 | 1.339 | 0.360 | 0.132 ^ 1.2781 | | ||
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| - | [[표본평균]] 자료를 이용해 [[정규성 검정]]을 실시한 결과는 아래와 같다. | ||
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| - | {{:normal_distribution_test.png|}} | ||
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| - | [[p값]]이 상당히 크므로 [[표본평균]]은 [[정규분포]]를 따른다고 짐작할 수 있고, [[중심극한정리]]에 의해 [[표본평균]]은 아래와 같은 [[분포]]를 따른다. | ||
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| - | * $$ \overline{X} = N(2, 2^{2}/20) $$ | ||