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중심극한정리 (Central Limit Theorem : CLT)

정의

$X_{1}, \ ... \ ,X_{n}$이 평균이 $\mu$이고 분산이 $\sigma^{2}$인 분포로 부터의 확률표본일 때, $X$의 분포와 관계없이 표본평균 $\overline{X}$의 분포는 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 확률표본으로 부터 랜덤하게 크기 20인 샘플을 100회 추출한 자료와 각 샘플군의 표본평균을 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

표본평균 자료를 이용해 정규성 검정을 실시한 결과는 아래와 같다.

p값이 상당히 크므로 표본평균은 정규분포를 따른다고 짐작할 수 있고, 중심극한정리에 의해 표본평균은 아래와 같은 분포를 따른다.

  • $$ \overline{X} = N(2, 2^{2}/20) $$