--- title: 2026-08-27 · 10:30 date: 2026-08-27 unlisted: true publish: true --- 水群时想到好玩的东西。 **《民族比例不正常群聊分析》** 癫子群聊里活跃聊天的一共 10 个人，统计民族为：土家族 5、苗族 1、汉族 4。户籍均为重庆户口，看似民族比例不正常。 设 $X$ 为 10 人中的少数民族人数。如果这 10 个人是独立随机抽出来的，那么 $X \sim B(10, p)$： $$P(X \ge 6) = \sum_{k=6}^{10} \binom{10}{k} p^k (1-p)^{10-k}$$ 七普重庆少数民族占比 6.77%，代入 $p = 0.0677$： $$P(X \ge 6) \approx 1.6 \times 10^{-5} \approx \frac{1}{63000}$$ 六万三千分之一，已经很小了。 --- 但是如果加入老家信息再算一次。 统计结果为：老家分别在黔江、酉阳（两人）、巫溪（近巴东、恩施片）、湖南慈利、贵州。 | 癫子 | 老家指向 | $p_i$ | | ------ | ----------- | ------- | | 土家族 ×1 | 黔江 | 0.71 | | 土家族 ×2 | 酉阳 | 0.93 ×2 | | 土家族 ×1 | 巫溪（近巴东、恩施片） | 0.50 | | 土家族 ×1 | 湖南慈利 | 0.61 | | 苗族 ×1 | 贵州 | 0.80 | | 汉族 ×4 | 重庆 | 0.03 ×4 | 期望少数民族人数约 4.6（$E = 0.71 + 0.93 \times 2 + 0.50 + 0.61 + 0.80 + 0.03 \times 4 = 4.6$）： $$P(X \ge 6) \approx 0.19$$ 回到较正常概率。 最后算一下分布程度，也就是人员分布是否平衡。这次用县政府驻地级坐标（WGS84，误差约 1~2 km）和椭球大圆距离重新算。 设 $n$ 人的两两椭球大圆距离为 $d_{ij}$，平均两两距离 $\bar d$、标准差 $s_d$ 与变异系数 $CV$ 为： $$\bar d = \frac{1}{\binom{n}{2}} \sum_{i 1$ 说明总体偏"拉开"而非堆叠，但点集中在东西两端、凸包巨大，只能作参考。 结论：人员分布是"西边一小簇 + 东边一条带"的双团结构，民族和空间位置高度绑定。 ![[assets/2026-08-27/民族比例散点图-地形版.png]] --- **Code** ```python from itertools import combinations import math import numpy as np from geopy.distance import geodesic from scipy.spatial import ConvexHull # (地名, 经度, 纬度, 分组) 分组: T=土家, H=汉族, M=苗族 people = [ ("黔江", 108.7707, 29.5332, "T"), ("酉阳A", 108.7677, 28.8411, "T"), ("酉阳B", 108.7677, 28.8411, "T"), ("巫溪", 109.5701, 31.3984, "T"), ("慈利", 111.1394, 29.4296, "T"), ("永川", 105.9274, 29.3561, "H"), ("丰都", 107.7308, 29.8635, "H"), ("璧山", 106.2278, 29.5923, "H"), ("主城", 106.5516, 29.5630, "H"), ("凯里", 107.9771, 26.5829, "M"), ] names = [p[0] for p in people] group = {p[0]: p[3] for p in people} # 两两椭球大圆距离 (km), geopy 参数顺序为 (纬度, 经度) dist = {} for i, j in combinations(range(len(people)), 2): a = (people[i][2], people[i][1]) b = (people[j][2], people[j][1]) dist[(names[i], names[j])] = geodesic(a, b).km D = np.array(sorted(dist.values())) mean_d = D.mean() std_d = D.std(ddof=0) print("mean=%.1f median=%.1f std=%.1f CV=%.2f min=%.1f max=%.1f km" % (mean_d, np.median(D), std_d, std_d / mean_d, D[0], D[-1])) # 基尼系数 G = sum((2 * k - len(D) - 1) * D[k] for k in range(len(D))) / (len(D) ** 2 * mean_d) print("Gini=%.3f" % G) # 局部等距投影, 参考纬度 29.5N ref = math.radians(29.5) def to_xy(lon, lat): return (lon - 105.0) * 111.32 * math.cos(ref), (lat - 28.0) * 111.32 # 平均中心与标准距离 xy = [to_xy(p[1], p[2]) for p in people] cx = np.mean([x for x, y in xy]) cy = np.mean([y for x, y in xy]) clon = cx / (111.32 * math.cos(ref)) + 105.0 clat = cy / 111.32 + 28.0 sd = math.sqrt(np.mean([geodesic((clat, clon), (p[2], p[1])).km ** 2 for p in people])) print("mean center=%.3fE %.3fN standard distance=%.1f km" % (clon, clat, sd)) # 分组两两距离统计 def pair_stats(pred): vals = [v for (a, b), v in dist.items() if pred(group[a], group[b])] return np.mean(vals), np.std(vals, ddof=0), len(vals) mT = pair_stats(lambda g1, g2: g1 == "T" and g2 == "T") mH = pair_stats(lambda g1, g2: g1 == "H" and g2 == "H") mM = pair_stats(lambda g1, g2: g1 != "H" and g2 != "H") mB = pair_stats(lambda g1, g2: (g1 == "H") != (g2 == "H")) print("Tujia within: mean=%.1f pair_std=%.1f n=%d" % mT) print("Han within: mean=%.1f pair_std=%.1f n=%d" % mH) print("minority within: mean=%.1f pair_std=%.1f n=%d" % mM) print("between: mean=%.1f pair_std=%.1f n=%d" % mB) # 分组标准距离 (到组平均中心的距离) def group_sd(groups): sel = [p for p in people if group[p[0]] in groups] gx = np.mean([to_xy(p[1], p[2])[0] for p in sel]) gy = np.mean([to_xy(p[1], p[2])[1] for p in sel]) clon = gx / (111.32 * math.cos(ref)) + 105.0 clat = gy / 111.32 + 28.0 return math.sqrt(np.mean([geodesic((clat, clon), (p[2], p[1])).km ** 2 for p in sel])) print("Han std dist=%.1f" % group_sd({"H"})) print("Tujia std dist=%.1f" % group_sd({"T"})) print("minority std dist=%.1f" % group_sd({"T", "M"})) # 分组重心 def center_of(groups): sel = [p for p in people if group[p[0]] in groups] gx = np.mean([to_xy(p[1], p[2])[0] for p in sel]) gy = np.mean([to_xy(p[1], p[2])[1] for p in sel]) return gx / (111.32 * math.cos(ref)) + 105.0, gy / 111.32 + 28.0 cH = center_of({"H"}) cT = center_of({"T"}) cMin = center_of({"T", "M"}) print("Han center=%.3fE %.3fN" % cH) print("Tujia center=%.3fE %.3fN" % cT) print("minority center=%.3fE %.3fN" % cMin) print("Han-Tujia dist=%.1f km" % geodesic((cH[1], cH[0]), (cT[1], cT[0])).km) print("Han-minority dist=%.1f km" % geodesic((cH[1], cH[0]), (cMin[1], cMin[0])).km) # 最近邻指数 R (凸包面积法) pts = np.array([to_xy(p[1], p[2]) for p in people]) hull = ConvexHull(pts) verts = pts[hull.vertices] area = 0.5 * abs(sum(verts[i, 0] * verts[(i + 1) % len(verts), 1] - verts[(i + 1) % len(verts), 0] * verts[i, 1] for i in range(len(verts)))) obs_nn = np.mean([min(geodesic((people[i][2], people[i][1]), (people[j][2], people[j][1])).km for j in range(len(people)) if j != i) for i in range(len(people))]) exp_nn = 0.5 * math.sqrt(area / len(people)) print("hull=%.0f km2 NN obs=%.1f exp=%.1f R=%.2f" % (area, obs_nn, exp_nn, obs_nn / exp_nn)) ```