[파이썬/통계] Python에 의한 기술 통계 : 모집단의 표본 추출 시뮬레이션

필요한 라이브러리 임포트

# 수치 계산에 사용하는 라이브러리입니다.
import numpy as np
import pandas as pd
import scipy as sp
from scipy import stats

# 그래프를 그리는 라이브러리입니다.
from matplotlib import pyplot as plt
import seaborn as sns
sns.set()

# 표시 자리수 지정입니다.
%precision 3
# 그래프를 jupyter Notebook 내에 표시하도록 하기 위한 지정입니다.
%matplotlib inline

 

물고기 다섯 마리만 있는 호수에서 표본 추출

 

- np.random.choice 함수는 배열의 요소 중에서 무작위로 샘플링을 수행하는 함수입니다. 위 코드에서는 fish_5 배열에서 크기가 3인 샘플을 비복원추출하여 뽑았습니다. 따라서 2, 3, 4, 5, 6 중에서 중복되지 않게 3개를 선택하여 반환하게 됩니다.

 

- np.random.seed(1)은 난수 발생 시 사용하는 시드 값을 1로 설정하는 함수입니다. 시드 값을 설정하면, 그 이후의 난수 발생에서 항상 동일한 순서로 난수가 발생합니다. 이를 통해 난수 발생 과정을 재현할 수 있습니다.

따라서 위 코드에서는 fish_5에서 중복되지 않는 크기가 3인 무작위 표본을 추출하고 있습니다. 이때 시드 값이 1로 고정되어 있기 때문에, 코드를 실행할 때마다 같은 표본이 추출됩니다.

 

- np.random.seed(1)
sp.mean(
    np.random.choice(fish_5, size = 3, replace = False)
) 는 설정된 시드값에 따라 같은 샘플링 결과가 나오도록 랜덤 시드를 설정한 후, fish_5 배열에서 중복되지 않도록 3개의 샘플을 추출한 다음, 추출된 샘플의 평균을 계산하는 코드입니다.

물고기가 더 많은 호수에서 표본 추출

표본에 쓰일 데이터 원본

length
5.297441538 
3.505566383 
3.572546017 
3.135979137 
4.689275271 
2.151478246 
5.393975941 
3.385805906 
4.251448956 
3.795960382 
5.167434390 
2.344919823 
3.737425123 
3.688032882 
4.904324009 
3.114404976 
3.857617128 
3.292328517 
4.029618058 
4.462822999 
3.113821674 
4.913102089 
4.718270187 
4.398458764 
4.717681386 
3.447892870 
3.897313836 
3.245922172 
3.781121427 
4.420784967 
3.441535935 
3.677856542 
3.445132385 
3.318494455 
3.457894963 
3.985641910 
3.100446390 
4.183636943 
5.325854459 
4.590419335 
3.842065270 
3.284498999 
3.397063600 
5.352020114 
4.036504771 
3.485341206 
4.148778533 
5.678806513 
4.092078577 
4.490379355 
4.236328674 
3.713519069 
3.080246362 
3.715848660 
3.828395486 
4.465874479 
4.668100522 
4.741918786 
4.224642754 
4.705088519 
3.391262190 
4.999762431 
4.406821120 
3.756917184 
4.387259098 
3.935231997 
4.902609100 
5.213678756 
5.747176957 
2.876723822 
2.838566095 
3.591542466 
4.124034460 
4.697898712 
4.248721080 
2.375322212 
3.750417383 
4.659278766 
4.180174388 
4.606419683 
3.817630373 
3.834915311 
4.145289429 
4.324380847 
4.154695808 
4.091156792 
3.458362821 
4.298347063 
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4.900889844 
4.956529981 
4.144163569 
3.695060146 
3.483951075 
4.335153009 
4.057766143 
3.720247246 
4.030726398 
3.498959801 
4.555150709 
3.637489069 
4.977036101 
4.319124060 
4.471448057 
3.118395180 
4.131523263 
4.589227176 
3.231553231 
3.782459321 
4.021925847 
2.895458334 
4.248339999 
4.673851918 
3.307027056 
4.276696645 
2.944206907 
3.964782331 
2.701009436 
4.894426085 
4.323458418 
3.976064022 
3.374623449 
5.016500616 
5.572105535 
2.506917553 
4.986376767 
5.300090275 
4.266652438 
3.034770517 
4.687622661 
3.850812083 
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3.010097212 
4.436957611 
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3.496131166 
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3.078785368 
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4.033106728 
3.846284944 
3.914257627 
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4.597124521 
4.420071662 
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4.058151635 
4.491322653 
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4.542745482 
3.747281936 
2.044660529 
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4.349473006 
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3.886452075 
3.900387802 
4.009741379 
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3.581422683 
3.196834245 
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3.855794121 
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4.718186985 
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3.814436157 
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3.545193375 
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3.098999934 
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3.734153823 
2.919852314 
4.888788882 
3.525786145 
3.004651922 
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4.495314854 
3.647242870 
5.123706984 
4.099244652 
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4.398656328 
5.244921934 
4.083459172 
3.018362011 
5.958564722 
3.558440518 
3.836454046 
3.434534034 
3.832802677 
4.190250987 
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5.431916485 
3.899257607 
3.007643032 
3.048353134 
3.462295606 
2.654192455 
4.656918969 
3.596552676 
3.746586209 
3.994274826 
2.876624384 
3.305584310 
4.536463081 
4.491449960 
3.640659673 
5.446642452 
2.949460876 
3.719338901 
3.810809697 
1.757583112 
5.548407594 
4.289346608 
3.158720524 
5.639475447 
4.465104239 
4.339986514 
3.509379165 
4.080910936 
2.773203683 
4.632875815 
3.695738585 
4.103208568 
4.959043767 
4.223970259 
4.206115897 
4.217360170 
3.408191544 
4.665713578 
5.232544110 
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4.704902328 
3.292790809 
3.300398957 
2.841160654 
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3.792106350 
5.117539835 
3.369214361 
3.645197613 
4.072258371 
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4.009034726 
4.137777943 
3.101119780 
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3.846277554 
3.950254882 
4.390318938 
3.450336727 
3.928070982 
3.757502925 
4.330190840 
4.624657730 
3.230171194 
4.465303317 
5.651182908 
2.816895392 
3.330541579 
3.290149531 
3.772138706 
5.296242513 
4.006490562 
3.439105600 
4.494065830 
3.515143890 
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4.240412874 
5.108673384 
3.465829730 
6.424534000 
4.656562213 
4.520331030 
3.954771254 
3.414341407 
3.300413818 
3.886826635 
3.356907317 
4.222309565 
3.333806605 
4.493488222 
4.761967981 
3.430173146 
4.951536188 
3.805118435 
4.921567573 
4.346910247 
4.895155795 
3.196839880 
3.910212377 
5.158877152 
3.500533627 
2.363302122 
2.439118426 
1.987282354 
2.301628926 
3.665928104 
5.020324727 
3.641415056 
4.255045571 
3.907650094 
4.002641146 
3.861006285 
3.856213073 
4.365339239 
3.053429980 
4.805244663 
4.733036627 
3.839483525 
4.641183559 
3.433776031 
3.565292881 
4.121010833 
3.843359053 
3.636199340 
3.456931828 
3.549048383 
4.748382916 
2.438530064 
4.278257962 
3.806324437 
4.579015754 
4.408539025 
1.766037663 
4.464290568 
4.255644083 
4.013310084 
3.620224022 
4.679557910 
3.664814126 
5.466020999 
4.448052435 
5.708915161 
3.366311660 
2.588699167 
4.568579196 
4.679095409 
4.024125963 
2.762695784 
3.636874746 
4.491006342 
3.848202905 
3.902847113 
3.855188462 
3.247408478 
3.568660666 
2.852636310 
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3.614855807 
4.378518123 
3.176913669 
4.632476865 
2.494753941 
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3.770024336 
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3.730320481 
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3.313486075 
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3.687409525 
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4.536210322 
3.416909869 
4.876458967 
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3.388321767 
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5.152023837 
3.563322759 
4.098664010 
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3.853446970 
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3.881042235 
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3.463006194 
2.437148914 
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3.151543179 
4.649718026 
3.505045064 
4.310804242 
2.534212766 
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3.964395487 
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3.851860240 
4.397842350 
3.520321162 
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3.261930987 
3.669443674 
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4.854226699 
3.116598166 
4.667821121 
3.163053411 
4.583190514 
2.474198733 
3.100606383 
3.587931301 
3.862378886 
5.136588048 
4.720200174 
5.258274136 
4.962692291 
3.769121000 
3.782372474 
4.852399001 
4.028141739 
3.869985420 
2.925840773 
3.910479916 
1.759256783 
3.630285842 
3.210566448 
4.550777253 
3.612211804 
3.732108765 
5.075278445 
3.155131837 
4.249541609 
2.477039569 
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4.324552412 
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3.275640817 
4.251944060 
4.250739855 
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4.314627574 
3.197648356 
3.037500295 
6.003929824 
5.534194626 
2.880572219 
4.356567498 
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4.108294857 
3.253052080 
3.849609992 
3.602232907 
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3.780734201 
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4.811996181 
4.981884598 
3.049326724 
3.627230698 
3.362091844 
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5.156922001 
3.613086467 
4.016625349 
2.915027728 
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3.705215403 
2.805677029 
3.521028318 
2.148778342 
3.970294209 
4.085931339 
4.226638762 
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3.754606669 
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3.840874409 
2.331568467 
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4.577495897 
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3.514806688 
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3.414647673 
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2.488142984 
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3.144931583 
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2.817632493 
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3.330670522 
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4.814156311 
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3.613452397 
3.966950215 
4.826810136 
4.534786858 
5.942143944 
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4.279117679 
4.488394572 
3.716135965 
3.212263083 
4.133586469 
4.388896151 
2.877696094 
3.577206032 
3.699891456 
4.208660113 
3.787068981 
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4.074411696 
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2.624565475 
5.323708843 
5.131827762 
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4.516919780 
4.481841323 
3.404971746 
4.127743054 
4.619371721 
4.244058739 
4.838254686 
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4.060499446 
3.380519598 
3.703672055 
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3.396744705

- 1만개의 데이터에서 10개의 표본 데이터를 랜덤으로 추출하고 결과값을 가져옴

- 샘플 표본의 평균값을 계산함

 

모집단 분포

- 모집단 평균 계산 코드

- 모집단 표준편차 계산 코드 (여기서 ddof 인수는 편향되지 않은 표본 표준편차를 계산하기 위해 필요한 것으로, 디폴트 값은 0입니다. 따라서 위 코드에서는 불편하지 않은 표본 표준편차를 계산하게 됩니다.)

- 모집단 분산 계산 코드

- 히스토그램을 그리는 Seaborn 라이브러리를 사용하여 10만 마리의 물고기 데이터(fish_100000)의 분포를 확인합니다. kde=False로 설정하여, 밀도 그래프를 그리지 않고, 빈도수를 나타내는 히스토그램만을 표시합니다. color='black'으로 설정하여 히스토그램의 색상을 검정색으로 지정합니다.

 

모집단 분포와 정규 분포의 확률 밀도 함수 비교

 

 

1부터 7까지 0.1씩 증가하는 값들로 이루어진 배열을 만드는 코드입니다. np.arange 함수는 start부터 stop까지 step만큼 증가하는 숫자들을 원소로 갖는 1차원 배열을 반환합니다. 이 때 stop은 배열에 포함되지 않습니다. 따라서 이 코드에서 반환되는 배열은 1부터 6.1까지 0.1씩 증가한 숫자들을 원소로 갖습니다.

 

 

 

위 코드는 평균(loc)이 4이고 표준편차(scale)가 0.8인 정규분포에서 x값에 해당하는 확률밀도를 계산하는 코드입니다.

np.arange(start=1, stop=7.1, step=0.1) 함수는 1부터 7까지 0.1 간격으로 숫자를 생성합니다. 이렇게 생성된 숫자들은 x 변수에 저장됩니다.

stats.norm.pdf(x=x, loc=4, scale=0.8) 함수는 x 값을 통해 정규분포에서 각각의 값이 나타날 확률밀도를 계산합니다. 예를 들어, x 값이 3일 때는 평균에서 더 멀리 떨어져 있기 때문에 낮은 확률밀도가 나올 것이고, x 값이 4일 때는 평균에서 더 가까워져 있기 때문에 높은 확률밀도가 나올 것입니다.

 

이 코드는 평균이 4이고 표준편차가 0.8인 정규 분포에서 x 값에 대한 확률 밀도 함수 값을 계산하고 이를 시각화하는 코드입니다.

np.arange(start = 1, stop = 7.1, step = 0.1) 코드는 1부터 7까지 0.1씩 증가하는 배열을 생성합니다. 이를 x 값으로 사용하여 stats.norm.pdf 함수를 사용하여 평균 4, 표준편차 0.8인 정규 분포의 확률 밀도 함수 값을 계산합니다. 계산된 값들을 plt.plot 함수를 사용하여 시각화합니다.

 

sns.distplot() 함수는 주어진 데이터의 분포를 시각화해주는 함수입니다. kde 옵션을 False로 설정하면 히스토그램을 그릴 수 있습니다.

norm_hist 옵션을 True로 설정하면 히스토그램이 아니라 밀도함수를 그릴 수 있습니다. 따라서 이때는 확률밀도함수를 그리기 위한 기댓값(loc)과 표준편차(scale)을 지정해줘야 합니다.

위 코드에서는 fish_100000 데이터의 밀도함수와 평균 4, 표준편차 0.8인 정규분포의 밀도함수를 함께 그려주었습니다.

 

데이터를 얻을 수 있는 프로세스 추상화

stats.norm.rvs() 함수는 주어진 평균(loc)과 표준편차(scale)를 갖는 정규분포에서 무작위 표본을 추출합니다. size 인자는 추출할 표본의 크기를 지정합니다. 위 코드에서는 평균 4, 표준편차 0.8을 갖는 정규분포에서 크기가 10인 무작위 표본을 추출하고 있습니다.

 

mean은 너무 많이 언급해서 알고 계시죠?! 평균값을 구하는 코드입니다.