기술통계

기술통계 박창이 서울시립대학교통계학과 박창이 ( 서울시립대학교통계학과 ) 기술통계 1 / 17

친구수에대한히스토그램 I from matplotlib import pyplot as plt from collections import Counter num_friends = [100,49,41,40,25,21,21,19,19,18,18,16, 15,15,15,15,14,14,13,13,13,13,12, 12,11,10,10,10,10,10,10,10,10,10,10, 10,10,10,10,10,9,9,9,9,9,9,9,9,9,9,9, 9,9,9,9,9,9,9,8,8,8,8,8,8,8,8,8,8,8, 8,8,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,6, 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, 6,6,6,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, 5,5,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4, 4,4,4,4,3,3,3,3,3,3,3,3,3,3,3,3,3,3, 3,3,3,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2, 2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1] 박창이 ( 서울시립대학교통계학과 ) 기술통계 2 / 17

친구수에대한히스토그램 II friend_counts = Counter(num_friends) xs = range(101) ys = [friend_counts[x] for x in xs] plt.bar(xs, ys) plt.axis([0,101,0,25]) plt.title("histogram of Friend Counts") plt.xlabel("# of friends") plt.ylabel("# of people") plt.show() 박창이 ( 서울시립대학교통계학과 ) 기술통계 3 / 17

친구수에대한히스토그램 III 박창이 ( 서울시립대학교통계학과 ) 기술통계 4 / 17

몇가지통계치 num_points = len(num_friends) # 204 largest_value = max(num_friends) # 100 smallest_value = min(num_friends) # 1 sorted_values = sorted(num_friends) smallest_value = sorted_values[0] # 1 second_smallest_value = sorted_values[1] # 1 second_largest_value = sorted_values[-2] # 49 박창이 ( 서울시립대학교통계학과 ) 기술통계 5 / 17

중심위치 I 평균 >>> from future import division >>> def mean(x): return sum(x) / len(x) >>> print("mean(num_friends)", mean(num_friends)) mean(num_friends) 7.333333333333333 박창이 ( 서울시립대학교통계학과 ) 기술통계 6 / 17

중심위치 II 중앙값 >>> def median(v): """finds the middle-most value of v""" n = len(v) sorted_v = sorted(v) midpoint = n // 2 if n % 2 == 1: # if odd, return the middle value return sorted_v[midpoint] else: # if even, return the average of the middle values lo = midpoint - 1 hi = midpoint return (sorted_v[lo] + sorted_v[hi]) / 2 >>> print("median(num_friends)", median(num_friends)) median(num_friends) 6.0 박창이 ( 서울시립대학교통계학과 ) 기술통계 7 / 17

중심위치 III 분위수 >>> def quantile(x, p): """returns the pth-percentile value in x""" p_index = int(p * len(x)) return sorted(x)[p_index] >>> print("quantile(num_friends, 0.10)", quantile(num_friends, 0.10)) quantile(num_friends, 0.10) 1 >>> print("quantile(num_friends, 0.25)", quantile(num_friends, 0.25)) quantile(num_friends, 0.25) 3 >>> print("quantile(num_friends, 0.75)", quantile(num_friends, 0.75)) quantile(num_friends, 0.75) 9 >>> print("quantile(num_friends, 0.90)", quantile(num_friends, 0.90)) quantile(num_friends, 0.90) 13 박창이 ( 서울시립대학교통계학과 ) 기술통계 8 / 17

중심위치 IV 최빈값 >>> def mode(x): """returns a list, might be more than one mode""" counts = Counter(x) max_count = max(counts.values()) return [x_i for x_i, count in counts.items() if count == max_count] >>> print("mode(num_friends)", mode(num_friends)) mode(num_friends) [6, 1] 박창이 ( 서울시립대학교통계학과 ) 기술통계 9 / 17

산포 I 범위 >>> def data_range(x): return max(x) - min(x) >>> print("data_range(num_friends)", data_range(num_friends)) data_range(num_friends) 99 박창이 ( 서울시립대학교통계학과 ) 기술통계 10 / 17

산포 II 분산 >>> def sum_of_squares(vec): return sum([elem**2 for elem in vec]) >>> def de_mean(x): """translate x by subtracting its mean (so the result has mean 0)"" x_bar = mean(x) return [x_i - x_bar for x_i in x] >>> def variance(x): """assumes x has at least two elements""" n = len(x) deviations = de_mean(x) return sum_of_squares(deviations) / (n - 1) >>> print("variance(num_friends)", variance(num_friends)) variance(num_friends) 81.54351395730716 박창이 ( 서울시립대학교통계학과 ) 기술통계 11 / 17

산포 III 표준편차 >>> def standard_deviation(x): return math.sqrt(variance(x)) >>> print("standard_deviation(num_friends)", standard_deviation(num_friends)) standard_deviation(num_friends) 9.03014473623248 사분위수범위 >>> def interquartile_range(x): return quantile(x, 0.75) - quantile(x, 0.25) >>> print("interquartile_range(num_friends)", interquartile_range(num_friends)) interquartile_range(num_friends) 6 박창이 ( 서울시립대학교통계학과 ) 기술통계 12 / 17

상관관계 I 공분산 >>> daily_minutes = [1,68.77,51.25,52.08,38.36,44.54,57.13,51.4,41.42, 31.22,34.76,54.01,38.79,47.59,49.1,27.66,41.03, 36.73,48.65,28.12,46.62,35.57,32.98,35,26.07,23.77, 39.73,40.57,31.65,31.21,36.32,20.45,21.93,26.02, 27.34,23.49,46.94,30.5,33.8,24.23,21.4,27.94,32.24, 40.57,25.07,19.42,22.39,18.42,46.96,23.72,26.41, 26.97,36.76,40.32,35.02,29.47,30.2,31,38.11,38.18, 36.31,21.03,30.86,36.07,28.66,29.08,37.28,15.28, 24.17,22.31,30.17,25.53,19.85,35.37,44.6,17.23,13.47, 26.33,35.02,32.09,24.81,19.33,28.77,24.26,31.98, 25.73,24.86,16.28,34.51,15.23,39.72,40.8,26.06,35.76, 34.76,16.13,44.04,18.03,19.65,32.62,35.59,39.43, 14.18,35.24,40.13,41.82,35.45,36.07,43.67,24.61, 20.9,21.9,18.79,27.61,27.21,26.61,29.77,20.59, 27.53,13.82,33.2,25,33.1,36.65,18.63,14.87,22.2, 36.81,25.53,24.62,26.25,18.21,28.08,19.42,29.79, 박창이 ( 서울시립대학교통계학과 ) 기술통계 13 / 17

상관관계 II 32.8,35.99,28.32,27.79,35.88,29.06,36.28,14.1,36.63, 37.49,26.9,18.58,38.48,24.48,18.95,33.55,14.24,29.04, 32.51,25.63,22.22,19,32.73,15.16,13.9,27.2,32.01,29.27 33,13.74,20.42,27.32,18.23,35.35,28.48,9.08,24.62, 20.12,35.26,19.92,31.02,16.49,12.16,30.7,31.22,34.65, 13.13,27.51,33.2,31.57,14.1,33.42,17.44,10.12,24.42, 9.82,23.39,30.93,15.03,21.67,31.09,33.29,22.61,26.89, 23.48,8.38,27.81,32.35,23.84] >>> import numpy as np >>> def covariance(x, y): n = len(x) return np.dot(de_mean(x), de_mean(y)) / (n - 1) >>> print("covariance(num_friends, daily_minutes)", covariance(num_friends, daily_minutes)) covariance(num_friends, daily_minutes) 22.4254351396 박창이 ( 서울시립대학교통계학과 ) 기술통계 14 / 17

상관관계 III 상관계수 : 이상치제거전 >>> def correlation(x, y): stdev_x = standard_deviation(x) stdev_y = standard_deviation(y) if stdev_x > 0 and stdev_y > 0: return covariance(x, y) / stdev_x / stdev_y else: return 0 # if no variation, correlation is zero >>> print("correlation(num_friends, daily_minutes)", correlation(num_friends, daily_minutes)) correlation(num_friends, daily_minutes) 0.247369573665 박창이 ( 서울시립대학교통계학과 ) 기술통계 15 / 17

상관관계 IV 상관계수 : 이상치제거후 >>> outlier = num_friends.index(100) # index of outlier >>> num_friends_good = [x for i, x in enumerate(num_friends) if i!= outlier] >>> daily_minutes_good = [x for i, x in enumerate(daily_minutes) if i!= outlier] >>> print("correlation(num_friends_good, daily_minutes_good)", correlation(num_friends_good, daily_minutes_good)) correlation(num_friends_good, daily_minutes_good) 0.573679211567 박창이 ( 서울시립대학교통계학과 ) 기술통계 16 / 17

참고 많이사용되는통계함수를포함하는모듈 SciPy: http://docs.scipy.org pandas: http://pandas.pydata.org StatModels: statmodels.sourceforge.net 박창이 ( 서울시립대학교통계학과 ) 기술통계 17 / 17