Z与T统计量对比：厚尾分布与置信波动

几何画板及绘图内容加载中，请稍等……

2024-04-20 23:14

所属专题 Statistics Demonstrations

该范例通过可视化对比 Z 统计量与 T 统计量的分布差异，重点展示在有限样本条件下 t 分布的厚尾现象会导致更大的波动与更宽的区间。核心数学概念包括：标准差与标准误（sd、sePop、se）、t 统计量/Z 统计量的构造、抽样分布（由随机抽样得到的 samMean、samSD）、样本量对估计不确定性的影响，以及厚尾分布在推断中的作用。作图与交互要点：利用 Sequence 生成样本量与抽样次数（n、m、numSam），用 Random 生成随机样本并得到样本均值与样本标准差（sam、samMean、samSD），进一步计算标准误（samSE）及两类统计量（Tstatistic 与 Zstatistic）。通过 DotPlot 绘制抽样结果的离散散点分布，并用按钮与文本（button1、text1、text2、text3、text4、text5）展示 Z/T 的对比结论；l1、l2 等对象用于辅助标注或参考线。适用于高中到大学初阶的概率统计与回归推断教学，也适合作为“抽样分布—标准误—置信区间/显著性”的动态演示。

GeoGebra 命令及参数

Sequence(Sample[pop, n], k, 1, numSam) → sam, {{99.28, 101.38, 106.11, 102.02, 94.32}, {95.66, 99.79, 98.31, 95.5, 102.22}, {101.95, 100.65, 95.29, 99, 94.66}, {95.8, 95.72, 96.89, 102.74, 95.33}, {97.61, 101.92, 95.39, 95.17, 100.94}, {99.74, 97.93, 94.88, 87.62, 95.58}, {104.93, 93.96, 94.88, 101.68, 100.47}, {102.29, 101.61, 98.47, 97.27, 98.44}, {98.44, 106.38, 96.1, 94.63, 96.34}, {107.49, 103.49, 98.92, 91.59, 95.21}, {97.09, 98.89, 103.34, 96.54, 98.99}, {108.03, 102.52, 99.53, 97.56, 101.45}, {89.88, 100.59, 100.74, 98.19, 109}, {99.05, 104.37, 103.27, 103.14, 100.32}, {99.7, 103.03, 103.51, 94.95, 97.79}, {105.66, 104.07, 101.18, 103.46, 99.01}, {96.31, 99.6, 101.82, 101.29, 106.58}, {98.88, 100.17, 96.56, 105.39, 107.91}, {100.26, 96.51, 96.88, 101.88, 96.21}, {101.76, 99.47, 99, 101.44, 100.8}, {106.29, 95.78, 98.87, 97.96, 96.34}, {106.02, 101.88, 100.35, 101.16, 99.98}, {98.69, 100.8, 95.11, 92.56, 94.95}, {98.16, 100.26, 99.62, 98.36, 105.71}, {96.07, 98.98, 97.85, 100.81, 100.86}, {99.74, 100.88, 100.48, 104.05, 107.3}, {110.23, 103.07, 98.73, 100.17, 100.35}, {99.21, 96.21, 103.79, 103.04, 107.29}, {97.42, 101.15, 100.33, 99.25, 104.78}, {95.55, 101.02, 102.27, 95.71, 96.03}, {98.46, 106.27, 100.11, 96.5, 100.42}, {95.92, 103.93, 106.99, 106.9, 95.96}, {94.92, 105.8, 100.12, 100.44, 105.05}, {102.62, 93.7, 105.3, 94.26, 98.67}, {96.36, 105.46, 101.43, 101.45, 92.27}, {100.17, 89.97, 104.16, 98.19, 94.54}, {97.62, 106.73, 98.2, 95.45, 91.03}, {97.39, 102.26, 96.94, 100.9, 96.72}, {102.69, 102.11, 99.18, 92.4, 95.03}, {94.98, 98.16, 105.8, 98.36, 102.49}, {100.81, 91.98, 91.64, 102.94, 96.1}, {101.05, 104.99, 101.87, 100.71, 102.37}, {96.44, 103.59, 102.29, 98.94, 99.76}, {102.3, 104.01, 105.83, 103.43, 97.6}, {98.2, 104.32, 102.5, 99.29, 103.1}, {104.52, 100.8, 96.76, 102.71, 105.67}, {97.86, 99.21, 102.08, 98.69, 101.34}, {96.09, 95.1, 101.06, 98.38, 90.6}, {98.37, 107.76, 97.14, 97.44, 103.54}, {105.13, 101.03, 103.93, 105.79, 96.9}, {100.03, 102.86, 97.94, 99.98, 104.94}, {94.59, 100.26, 99.29, 96.75, 91.43}, {103.51, 98.6, 102.84, 99.65, 99.12}, {96.51, 97.97, 99.13, 96.53, 89.96}, {100.33, 93.04, 105.08, 99.87, 97.19}, {101.01, 103.09, 101.21, 101.39, 101.06}, {101.51, 94.69, 101.12, 98.34, 96.65}, {98.52, 104.47, 89.61, 95.73, 105.01}, {97.86, 102.16, 96.98, 102.28, 105.3}, {98.07, 101.27, 96.18, 98.67, 102.67}, {102.49, 93.26, 98.66, 104.07, 100.81}, {99.13, 104.27, 99.25, 99.06, 104.67}, {107, 98.15, 107.57, 95.97, 94.28}, {92.35, 94.42, 93.47, 104.16, 97.17}, {100.65, 98.1, 104.56, 98.46, 98.87}, {104.04, 100.1, 104.07, 104.55, 92.71}, {98.64, 104.24, 99.64, 98.08, 104.73}, {97.18, 97.45, 104.26, 94.58, 107.05}, {99.82, 95.7, 102.57, 97.72, 104.69}, {106.54, 97.9, 96.1, 97.37, 102.93}, {102.53, 93.17, 102.25, 95.83, 99.13}, {100.68, 98.22, 106.78, 97.93, 99.08}, {103.74, 102.28, 97.92, 99.82, 105.45}, {101.84, 97.85, 102.23, 96.17, 98.57}, {104.14, 96.98, 106.5, 97.78, 87.72}, {109.52, 95.7, 108.75, 94.91, 102.37}, {99.91, 100.67, 99.45, 97.88, 99.94}, {98.41, 102.27, 98.11, 98.27, 97.7}, {95.23, 96.45, 97.94, 98.64, 99.97}, {97.76, 96.36, 100.7, 98.54, 99.69}, {96.4, 104.31, 101.19, 103.36, 104.02}, {101.21, 107.25, 103.35, 94.54, 103.72}, {91.46, 102.99, 99.33, 100.57, 108.66}, {99.54, 100.43, 94.69, 100.68, 104.58}, {96.58, 96.95, 106.58, 109.01, 98.07}, {101.45, 99.17, 102.74, 99.94, 99.79}, {99.13, 96.19, 96.18, 106.19, 89.42}, {98.31, 103.93, 98.95, 100.33, 98.75}, {95.29, 104.32, 96.86, 94.39, 97.16}, {103.24, 97.33, 103.23, 102.93, 98.58}, {104.59, 99.29, 106.49, 95.34, 97.12}, {93.4, 100.16, 95.27, 92.55, 99.79}, {96.07, 103.98, 99.24, 101.84, 101.83}, {102.99, 105.31, 99.48, 103.51, 90.47}, {99.56, 96.12, 96.36, 105.91, 99.9}, {94.56, 93.78, 98.43, 96.34, 96.51}, {98.02, 103.02, 95.25, 103.39, 97.3}, {97.4, 99.71, 99.87, 103.08, 103.91}, {104.02, 94.6, 99.26, 97.39, 94.09}, {96.19, 98.22, 101.08, 96.16, 98.54}, {95.05, 105.01, 108.49, 93.35, 98.78}, {106.59, 98.58, 97.08, 102.3, 101.46}, {93.52, 100.03, 104.92, 99.33, 102.27}, {102.43, 98.22, 102.42, 100.97, 99.01}, {101.65, 96.9, 100.07, 99.81, 97.43}, {97.98, 98.79, 98.09, 95.96, 97.8}, {102.75, 96.53, 103.01, 102.65, 100.86}, {100.51, 106.98, 97.77, 100.79, 93.93}, {96.13, 97.72, 97.2, 99.55, 102.66}, {102.84, 101.88, 109.77, 97.15, 101.53}, {99.12, 104.02, 89.05, 100.01, 102.37}, {100.26, 101.32, 98.57, 106.21, 96.1}, {106.15, 104.45, 94.17, 103.36, 107.01}, {96, 105.15, 102.95, 106.01, 101.32}, {101.03, 100.17, 101.69, 97.49, 106.41}, {96.89, 103.69, 100.51, 101.02, 104.54}, {100.92, 106.29, 104.36, 103.83, 101.2}, {94.3, 105.3, 95.72, 100.92, 95.6}, {101.65, 96.15, 103.3, 98.01, 98.74}, {98.67, 99.88, 98.34, 100.09, 97.07}, {94.26, 95.18, 94.43, 102.01, 96.66}, {97.06, 99.82, 98.89, 103.69, 98.36}, {96.4, 95.58, 104.22, 97.45, 96.8}, {102.42, 94.64, 87.05, 110.77, 97.75}, {103.96, 95.05, 101.88, 97.2, 96.13}, {107.17, 94.03, 99.82, 95.63, 105.32}, {104.36, 102.25, 103.82, 98.8, 97.65}, {99.26, 102.24, 104.65, 95.94, 95.42}, {95.92, 100.38, 104.11, 102.44, 100.2}, {99.05, 106.7, 106.33, 108.49, 97.94}, {104.99, 97.17, 99.06, 105.02, 95.34}, {104.75, 106.27, 98.58, 97.04, 95.21}, {99.65, 96.56, 97.01, 98.27, 112.32}, {100.5, 106.58, 106.26, 87.75, 103.32}, {108.2, 97.53, 104.65, 99.47, 106.06}, {106.7, 101.02, 97.01, 100.35, 97.76}, {99.46, 103.11, 99.37, 96.44, 96.06}, {100.08, 107.87, 100.57, 99.32, 107.11}, {97.66, 103.74, 104.98, 104.76, 101.12}, {101.57, 105.91, 102.37, 100.03, 101.45}, {110.06, 101.68, 99.56, 105, 105.18}, {100.35, 100.24, 97.67, 99.23, 92.19}, {99.09, 94.31, 98.1, 101.76, 96.97}, {100.05, 102.05, 101.54, 101.03, 106.36}, {105.37, 102.34, 98.53, 105.73, 99.41}, {87.62, 98.31, 96.88, 99.76, 101.45}, {95.83, 97.88, 101.61, 102.88, 103.83}, {98.99, 104.7, 102.16, 100.05, 106.15}, {97.93, 100.78, 98.36, 105.55, 100.03}, {107.33, 101.48, 100.76, 95.25, 102.22}, {97.95, 95.32, 101.7, 101.4, 100.24}, {107.11, 96.43, 100.75, 99.76, 98}, {103.24, 100.35, 101.59, 103.73, 96.12}, {99.41, 99.38, 101.72, 91.46, 106.36}, {94.09, 100.68, 102.9, 100.62, 99.71}, {101.89, 100.16, 97.28, 97.25, 94.89}, {105.32, 96.38, 107.58, 97.78, 98.84}, {102.61, 98.63, 100.59, 99.04, 103.73}, {95.93, 105.93, 95.74, 98.78, 97.48}, {101.27, 97.32, 97.62, 94.57, 97.96}, {105.46, 100.08, 100.09, 104.93, 95.09}, {108.66, 102.16, 107.91, 99.48, 97.06}, {100.2, 102.17, 102.34, 98.66, 100.27}, {104.27, 98.25, 99.96, 97.92, 102.73}, {100.57, 103.36, 103.16, 102.29, 97.06}, {111.01, 99.34, 95.26, 93.23, 88.47}, {99.85, 99.87, 95.99, 102.49, 104.18}, {103.72, 97.39, 96.91, 97.34, 99.27}, {96.06, 95.17, 93.85, 101.73, 91.98}, {102.42, 96.03, 103.52, 97.18, 96.25}, {102.91, 98.16, 99.51, 106.25, 98.21}, {95.04, 101.3, 100.26, 102.15, 98.25}, {100.08, 103.47, 99.48, 102.95, 104.03}, {97.17, 97.53, 101.84, 100.8, 101.82}, {101.95, 103.56, 101.06, 105.75, 103.46}, {96.65, 100.01, 97.77, 97.28, 99.92}, {103.37, 101.98, 97.32, 97.62, 101.72}, {97.13, 104.77, 98.43, 94.58, 95.48}, {99.87, 100.77, 99.69, 100.94, 95.58}, {100.62, 99.95, 95.17, 104.51, 95.26}, {94.44, 98.56, 102.88, 100.85, 106.73}, {99.05, 92.99, 103.34, 94.17, 96.47}, {96.65, 98.06, 101.21, 99.17, 100.43}, {98.28, 95.57, 98.31, 99.06, 103.48}, {94.13, 99.87, 101.83, 101.92, 101.14}, {97.29, 99.56, 106.29, 96.46, 101.26}, {104.04, 98.77, 99.18, 95.92, 103.33}, {94.48, 105.77, 101.24, 100.85, 101.72}, {96.96, 102.66, 100.13, 101.15, 102.34}, {100.31, 99.58, 101.78, 103.78, 100.15}, {98.5, 90.07, 102.99, 100.12, 105.18}, {97.23, 100.43, 99.09, 95.49, 102.49}, {98.77, 95.09, 106.25, 97.63, 98.35}, {106.98, 97.33, 91.04, 101.06, 106.34}, {105.99, 106.45, 96.25, 92.06, 98.9}, {100.48, 99.94, 109.42, 106.59, 96.77}, {100.1, 102.15, 101.72, 96.93, 99.8}, {94.28, 106.66, 94.64, 105.5, 104.65}, {98.42, 98.44, 95.44, 100.29, 99.27}, {102.43, 98.47, 101.02, 98.04, 97.69}}
Sequence(mean[Element[sam, k]], k, 1, Length[sam]) → samMean
Sequence(stdev[Element[sam, k]], k, 1, Length[sam]) → samSD
DotPlot(Tstatistic, true) → l1
DotPlot(Zstatistic, true) → l2
Random(1, 2) → a
Random(1, 2) → b
Random(1, 2) → c
Random(1, 2) → d
Random(1, 2) → e
Random(1, 2) → f
Random(1, 2) → g
Random(1, 2) → h
Random(1, 2) → i
Random(1, 2) → j
Random(1, 2) → k
Random(1, 2) → l
Random(1, 2) → o
Random(1, 2) → p
Random(1, 2) → q
Random(1, 2) → r
Random(1, 2) → s

AI 绘图提示语参考

首先，生成多个随机样本，每个样本包含五个数据点。然后，计算每个样本的均值和标准差。接着，基于这些样本计算T统计量和Z统计量，并绘制两个点图：一个展示T统计量的分布，另一个展示Z统计量的分布。最后，通过对比两个点图，展示在有限样本下，由于t分布的厚尾效应，Z统计量的分布更集中，而T统计量的分布更分散。

以上提示语为根据原始绘图命令整理而来，仅供参考。