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相关系数的图形直观与线性关系解读：GeoGebra 交互示例

相关系数的图形直观与线性关系解读：GeoGebra 交互示例
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2023-10-25 06:17
函数解析几何概率统计滑块动画函数图像
该范例通过散点与参考直线（含引导线）展示数据如何随线性趋势变化，从而直观理解“相关系数”反映的对应程度与方向。核心数学概念包括：相关系数（通常为皮尔逊相关系数 PMCC）、线性关系与线性拟合、回归直线斜率对趋势方向的指示、以及通过频数/采样构造数据点并进行统计解读。作图要点：利用 Shuffle、Sequence、Sample、KeepIf 等命令对点与数据进行生成/筛选；用 Line、Segment 与 Polygon 构建参考直线、线段与轮廓元素；使用 Slope 获取斜率信息，并通过 Translate、Corner、Element 将坐标关系映射到视图；借助 Text、Label 进行相关系数与问题提示的标注；再结合 PMCC 计算相关系数，并可用 Slider Animation（如 labelSlider）与移动引导线（hGuideline、vGuideline、MoveGuideLine）让相关趋势动态变化。对象层面包含 example、v1s、v2s、frequencies、r、v1、v2、slp、f、Cor1、Cor3 等用于表示数据、频数与相关度结果，适合用于 9-12 年级概率统计课程中开展模型预测、区分相关与因果的交互式教学与探究活动。

GeoGebra 命令及参数

  1. Shuffle(Sequence[-4, -1, 0.2]) → v1s, {-2.8, -3.5999999999999996, -2.5999999999999996, -4, -2, -1.2000000000000002, -1.6, -1, -1.8, -3, -3.1999999999999993, -1.4000000000000001, -3.3999999999999995, -2.1999999999999993, -3.8, -2.3999999999999995}
  2. Shuffle(Sequence[0, 4, 0.2]) → v2s, {0.4, 2.2, 3.6000000000000005, 3, 3.8000000000000007, 1.2, 1.7999999999999998, 1.5999999999999999, 1, 0.6000000000000001, 2.8000000000000007, 3.4000000000000004, 3.2, 0, 0.2, 0.8, 4, 2.4000000000000004, 1.4, 2, 2.6000000000000005}
  3. Shuffle({1 / 10, 1 / 5, 1 / 3, 1 / 2, 1, 3, 4, 5, 9, 10}) → frequencies, {3, 0.2, 0.5, 4, 10, 0.3333333333333333, 5, 0.1, 9, 1}
  4. Corner(5) → AppSize
  5. Element(v1s, Mod[example, Length[v1s]] + 1) → v1
  6. Element(v2s, Mod[example, Length[v2s]] + 1) → v2
  7. Line((-5, v1), (5, v2)) → g
  8. Slope(g) → slp
  9. Sequence(RandomNormal[0, (((1 - r^(2)) * (r^(2) + 0.1)) * 5)], ii, 1, 100) → l2, {-0.24865600393024015, -0.4427163928876197, -0.06436210133034098, -0.7246319666843015, 0.1518131634784586, 0.2783136548947105, -0.08877464593912159, -0.24058088448183162, -0.05712717781762687, -0.06423043757026925, 0.39825486588386466, 0.7075312066655958, -0.3262376427966622, -0.5723245120750319, -0.27928180563388555, -0.07049671607224141, 0.11852623574981205, -0.1463807178102759, 0.32525153572259174, -0.1842690692292607, -0.4747902410008965, 0.0832416549280435, 0.5311078737767596, 0.19730645485737308, -0.1296478081407259, 0.18735619858140565, 0.31400317616087886, -0.12257898186089448, 0.6087135165018176, -1.279155909882137, 0.2827494470950619, -0.23100518632541264, -0.08981478843637765, -0.9689171778513928, 0.23848140358592615, 1.0245434582157533, -0.6040482362219812, 0.7308526648335587, -0.2128144312592865, 0.23769152491819634, 0.9317973687320863, -0.7499051450311507, -0.04395514429968953, 0.5825030431754739, -0.9129946765796846, 0.13420294834548918, 0.0964497163074819, -0.32540220629108846, -0.27157799250002157, -0.7984497778844687, 0.08825112557947422, 1.1558360033689312, -0.2746119731854489, 0.5162099281346505, -0.3332501829911684, -0.3812971389014234, 0.15746428357187267, 0.11959749773574901, 0.01122581881267802, -0.39624077657157447, -0.3585295005697764, -0.8226532437499681, -0.7656341447433536, 0.12230599920670075, 0.1320565815170991, -0.40488749810086977, 0.13333504974605417, -0.6523812111577524, 0.5031527678652897, 0.08227681019297829, -0.3407671265640928, 0.30434097036692154, 0.06790200065394335, -0.33507998440307635, -0.43360831610784045, 0.627308460110241, 0.002199833979304936, 0.06352829010100344, 0.5439244093003046, -0.3204633620044284, -0.6497569867185523, 0.24045007334481863, -0.31569812366433053, 0.12094517804052977, 1.4901306986156224, 0.08139998093222266, -0.011035814245865114, 0.14073615260172131, 0.7974913927879635, -0.047794937202976014, 0.05481483990259745, 0.03779061937276644, 0.6867733019435083, -0.41580916304592114, 0.11181834570481163, -0.11922560710612219, 0.1317711898668424, -0.021782237243583923, 0.033727151690141906, -0.3620900224723636}
  10. Sample(Sequence[-5, 5, 0.1], 100, true) → l1, {4.299999999999999, -1.9, -2.5999999999999996, -4.400000000000002, -1.0999999999999992, -4.700000000000001, 4.299999999999999, 2.3000000000000003, 1.4000000000000004, 3.1, -5, -2.4999999999999996, -3.5999999999999996, 4.6999999999999975, 1.2000000000000002, -1, -0.30000000000000016, -3.1999999999999993, 4.1, 3.2, 0.6, 3.900000000000001, 1.5000000000000004, 4.199999999999999, 3.4000000000000004, -4.100000000000003, 1.1, -0.5000000000000001, 1, 0.1, 1.9000000000000008, 0.7999999999999999, -0.30000000000000016, -4.700000000000001, -1.3999999999999995, -2.8, -4.800000000000001, -2.4999999999999996, -0.9, 3.2, -2.1999999999999993, 2.2, 3.2, -1, -3.3999999999999995, -0.5000000000000001, 3.7000000000000006, 0.4, -0.30000000000000016, -5, 1.7000000000000006, -3.3999999999999995, 1.5000000000000004, 3.1, -0.10000000000000014, -0.6000000000000001, 4.1, 1.8000000000000007, -1.5999999999999996, -3.9, -3, 0.1, 4.6999999999999975, -4.600000000000001, 4.499999999999998, 3.900000000000001, 3.900000000000001, 3.3000000000000003, 4.899999999999997, 4.199999999999999, -0.8, -3.8, -1, -4.500000000000002, 3.8000000000000007, 4.6999999999999975, 4.599999999999998, 2.2, 0.8999999999999999, 3.900000000000001, -3.5999999999999996, -0.6000000000000001, 3.5000000000000004, 0.4, -1.4999999999999996, -4.500000000000002, 4.399999999999999, 4.299999999999999, 0.7999999999999999, 3.4000000000000004, 1, 0.30000000000000004, -3.8, 3.900000000000001, 3.7000000000000006, 4.199999999999999, -4.3000000000000025, 4.199999999999999, 0.30000000000000004, 2.900000000000001}
  11. Element(frequencies, Mod[example, Length[frequencies]] + 1) → frec
  12. Translate(vGuideline, Vector[(Vector[Corner[1], Corner[2]] / 800 * 16)]) → vGuideline'
  13. Polygon({ViewCorner1_{1}, (x(ViewCorner3_{1}), y(ViewCorner1_{1})), ViewCorner3_{1}, (x(ViewCorner1_{1}), y(ViewCorner3_{1}))}) → q1_{1}
  14. Segment((x(ViewCorner1_{1}), 0), (x(ViewCorner3_{1}), 0)) → xAxisFake_{1}
  15. Segment((0, y(ViewCorner1_{1})), (0, x(ViewCorner3_{1}))) → yAxisFake_{1}
  16. Text("x", (x(ViewCorner3_{1}), 0.1), true, true, 1, 0) → text1_{1}
  17. Text("y", (0, y(ViewCorner3_{1})), true, true, 0, 1) → text2_{1}
  18. Translate(hGuideline, Vector[(Vector[Corner[1], Corner[4]] / 500 * 16)]) → hGuideline'
  19. Zip((xx, (r * f(xx)) + (h1(xx) * (1 - abs(r))) + ii), xx, l1, ii, l2) → l3
  20. KeepIf(abs(y(PP)) ≤ 5, PP, l3) → l4
  21. PMCC(l4) → a

AI 绘图提示语参考

这是一个交互式图形，展示两个数据集的散点图和拟合线。用户可以动态调整数据点，观察相关系数的变化。图形包含散点数据、拟合线、斜率、相关系数计算结果和标签。构造顺序：先生成两个数据集的点，然后绘制拟合线，计算斜率，再生成随机数据点并拟合，最后添加标签和坐标轴。
以上提示语为根据原始绘图命令整理而来，仅供参考。