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@ -181,7 +181,7 @@
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"\\frac{-300.91}{\\sqrt{250.55}\\sqrt{1508.34}}\n",
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"$$\n",
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"$$\n",
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"= \\frac{-300.91}{15.83*38.84} = -0.4895\n",
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"= \\frac{-300.91}{15.83×38.84} = -0.4895\n",
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"$$\n",
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"计算结果表明，伦敦市的月平均气温（t）与降水量（p）呈负相关，即异向相关"
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]
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@ -421,7 +421,7 @@
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"计算等级相关系数\n",
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"$$\n",
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"r_R = 1-\\frac{2\\sum D^2}{n(n^2-1)}\n",
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"= 1-\\frac{6*18}{10(10^2-1)}\n",
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"= 1-\\frac{6×18}{10(10^2-1)}\n",
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"=0.891\n",
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"$$\n",
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"**等级相关系数的显著性检验**\n",
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@ -503,7 +503,7 @@
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"由于每个评分老师对6篇论文的评定都无相同的等级：\n",
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"$$\n",
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"S=\\sum^6_{i=1}-\\frac{1}{6}(\\sum^6_{i=1}R_i)^2\n",
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"= 3192- \\frac{1}{6} * 126^2 = 546\n",
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"= 3192- \\frac{1}{6} × 126^2 = 546\n",
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"$$\n",
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"$$\n",
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"W = \\frac{S}{\\frac{1}{12}K^2(N^3-N)} = \\frac{546}{\\frac{1}{12}6^2(6^3-6)}\n",
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@ -532,11 +532,11 @@
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"丙T = (23 - 2)+(23 - 2) = 12\n",
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"$$\n",
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"S=\\sum^6_{i=1}-\\frac{1}{6}(\\sum^6_{i=1}R_i)^2\n",
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"=791.5-\\frac{1}{6} * 63^2 = 130.00\n",
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"=791.5-\\frac{1}{6} × 63^2 = 130.00\n",
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"$$\n",
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"$$\n",
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"W = \\frac{S}{\\frac{1}{12}[K^2(N^3-N)-K\\sum^K_{i=1}T_i]}\n",
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"=\\frac{130}{\\frac{1}{12}[3^2(6^3-6)-3*(6+12)]}\n",
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"=\\frac{130}{\\frac{1}{12}[3^2(6^3-6)-3×(6+12)]}\n",
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"=\\frac{130}{153} = 0.849\n",
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"$$\n",
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"由W=0.849可看出专家评定结果有较大的一致性"
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@ -577,6 +577,127 @@
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"print('p_value', p_value)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 质量相关分析\n",
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"质量相关是指一个变量为质，另一个变量为量，这两个变量之间的相关。如智商、学科分数、身高、体重等是表现为量的变量，男与女、优与劣、及格与不及格等是表现为质的变量。\n",
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"\n",
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"质与量的相关主要包括二列相关、点二列相关、多系列相关。"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 二列相关\n",
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"当两个变量都是正态连续变量。其中一个变量被人为地划分成二分变量(如按一定标推将属于正态连续变量的学科考试分数划分成及格与不及格,录取与未录取,把某一体育项目测验结果划分成通过与未通过,达标与末达标,把健康状况划分成好与差,等等),表示这两个变量之间的相关,称为二列相关\n",
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"\n",
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"**二列相关的使用条件:**\n",
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"<ul>\n",
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" <li>两个变量都是连续变量,且总体呈正态分布,或总体接近正态分布,至少是单峰对称分布。\n",
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" <li>两个变量之间是线性关系。\n",
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" <li>二分变量是人为划分的,其分界点应尽量靠近中值。\n",
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" <li>样本容量应当大于80。\n",
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"</ul>\n",
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"$$\n",
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"R = \\frac{\\overline{X}_p-\\overline{X}_q}{σ} × \\frac{pq}{Y}\n",
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"$$\n",
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"\n",
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"$$p 表示二分变量中某一类别频数的比率$$\n",
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"$$q 表示二分变量中另一类别频数的比率$$\n",
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"$$\\overline{X}_p 表示与二分变量中p类别相对应的连续变量的平均数$$\n",
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"$$\\overline{X}_q 表示与二分变量中q类别相对应的连续变量的平均数$$\n",
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"$$σ 表示连续变量的标准差$$\n",
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"$$Y 表示正态曲线下与p相对应的纵线高度$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**二列相关实例：**\n",
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"<br>10名考生成绩如下，包括总分和一道问答题，试求该问答题的区分度（6分以上为通过，包括6分）\n",
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"<img src=\"assets/20201121183005.png\" width=\"50%\">\n",
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"问答题，被人为的分成两类，通过和不通过，应求二列相关。\n",
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"<br>\n",
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"$$\n",
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"当p=0.6时，查正态分布表得到：x=0.25\n",
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"$$\n",
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"$$\n",
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"当x=0.25时，代入标准正态密度函数Y=\\frac{1}{\\sqrt{2π}}e^{-\\frac{x^2}{x}}\n",
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"得到：Y=0.3866\n",
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"$$\n",
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"$$\n",
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"\\overline{X}_p = 67.33, \\overline{X}_q=61.25,σ=6.12\n",
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"$$\n",
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"则可以通过公式计算得到二列相关系数：\n",
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"$$\n",
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"R=\\frac{\\overline{X}_p-\\overline{X}_q}{σ}×\\frac{pq}{Y}\n",
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"=\\frac{67.33-61.25}{6.12}×\\frac{0.6×0.4}{0.3866} ≈0.62\n",
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"$$\n",
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"区分度较高"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 点二列相关\n",
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"当两个变量其中一个是正态连续性变量,另一个是真正的二分名义变量(例如,男与女,已婚和未婚,色肓与非色盲,生与死,等等),这时,表示这两个变量之间的相关,称为点二列相关。\n",
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"$$\n",
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"R = \\frac{\\overline{X}_p-\\overline{X}_q}{σ} × \\sqrt{pq}\n",
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"$$\n",
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"\n",
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"$$p表示二分变量中某一类别频数的比率$$\n",
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"$$q 表示二分变量中另一类别频数的比率$$\n",
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"$$\\overline{X}_p 表示与二分变量中p类别相对应的连续变量的平均数$$\n",
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"$$\\overline{X}_q 表示与二分变量中q类别相对应的连续变量的平均数$$\n",
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"$$σ 表示连续变量的标准差$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**点二列相关实例：**\n",
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"<br>有50道选择题,每题2分,有20人的总成绩和第五题的情况,第五题与总分的相关程度如亻\n",
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"<img src=\"assets/20201121183858.png\" width=\"50%\">\n",
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"p(答对学生的比例) = 10/20=0.5，q=1-p=0.5\n",
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"$$\n",
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"\\overline{X}_p=88.4, \\overline{X}_q=74.8, σ=8.66\n",
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"$$\n",
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"$$\n",
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"R = \\frac{\\overline{X}_p-\\overline{X}_q}{σ} × \\sqrt{pq}\n",
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"= \\frac{88.4-74.8}{8.66}\\sqrt{0.5×0.8} = 0.785\n",
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"$$\n",
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"相关系数较高，第五题的情况与总分有一致性（区分度较高）"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"PointbiserialrResult(correlation=0.7849870641173371, pvalue=4.145927973490392e-05)"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"#拿上面的实例，对了就是1，错了是0，x是第5题的选答情况，y是分数\n",
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"x = [1,0,0,0,0,0,0,1,1,1,1,0,1,1,1,1,1,0,0,0]\n",
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"y = [84,82,76,60,72,74,76,84,88,90,78,80,92,94,96,88,90,78,76,74]\n",
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"stats.pointbiserialr(x,y) #可以看到相关系数值是0.7849，和上面的计算结果一致"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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benjas
5 years ago
