学生学习状况的评价模型

word文档可自由复制编辑word文档可自由复制编辑word文档可自由复制编辑北方民族大学第八届数学建模竞赛竞赛论文竞赛分组：竞赛题目：组员：所在学院:数学与信息科学学院制版北方民族大学第八届数学建模竞赛承诺书为保证竞赛的公平、公正，维护竞赛的严肃性，在竞赛期间，我们承诺遵守以下竞赛规定：只在本参赛队的三人之间进行问题的讨论，绝不与本参赛队外的其他人讨论与竞赛题目相关的任何问题，不抄袭、剽窃他人的成果，引用的参考文献在答卷中进行标注。承诺人签名：承诺人所在分组：2014JMZ02承诺人所在学院：数学与信息科学学院2014年6月23日学生学习状况的评价模型摘要本文通过对学生的实际成绩、进步程度以及学生每学期的综合排名三个方面来综合评价学生的学习情况。针对题目中所提出的三个问题，本文用相应的方法和数学软件做出了合理的解释。对于问题一：①用到Excel做描述性统计分析，对学生整体作分析，得出以下结论：学生三个学期考试的及格率大于等于89%且呈上升趋势；学生三个学期的的平均分和中位数都在71~75分之间，说明绝大部分学生具备较好的学习能力且整体学习能力逐渐提高。②用spss软件对三个学期的学生成绩做了“人数——分数直方图”，从图上可以看出学生的成绩近似服从正态分布。也反映了一个问题：优等生较少。针对问题二：我们采用了层次分析法，把实际成绩、进步程度以及综合排名做为准则层，找出各因素所占权重，进而对所有的学生做了系统的排名。注：本学期的进步度=本学期成绩-上学期成绩，综合排名用学生学期排名得分（排名前10%的得分为100，处于10%-60%之间的得分为80，处于60%-90%之间的得分为60，后10%的得分为40）表示。得出组合权向量为ω=(0.080,0.160,0.080,0.140,0.419,0.012,0.069,0.040)。针对问题三：我们对前三个学期成绩做了多元线性回归预测模型，用Excel做线性回归分析，最终调整后的拟合系数为0.722914274，能够较好的拟合，得出的预测模型为：y=0.377253547x+0.347485052x+20.99585486，很好的预测了第四学期和第五学期的成绩。 1 2关键字：描述性统计分析层次分析法多元线性回归权重组合权向量目录一、问题重述............................................3二、模型的假设及符号说明.................................32.1模型的假设................................................32.2符号说明..................................................3模型的建立与求解....................................43.1对学生整体成绩的分析......................................43.2对学生学习状况的评价......................................63.2.1数据处理.....................................................................................3.2.2层次分析模型...............................................................................3.3对以后两学期成绩的预测....................................93.3.1多元线性回归模型............................................................................3.4预测接下来两学期的学习状况...............................113.4.2学习状况评价预测............................................................................模型结果分析与检验.................................12五、模型的优化与推广...................................12六、参考文献...........................................13七、附录...............................................14问题重述现行的评价方式单纯的根据“绝对分数”评价学生的学习状况，忽略了基础条件的差异；只对基础条件较好的学生起到促进作用，对基础条件相对薄弱的学生很难起到鼓励作用。为了激励优秀学生取得更好的成绩，同时对基础薄弱的学生树立信心，建立合理的数学模型来解决这一问题是有必要的。题目中提到以下三个问题：1.请根据附件数据，对这些学生的整体情况进行分析说明；2.请根据附件数据，全面、客观、合理的评价这些学生的学习状况；3.根据你的评价情况，请预测这些学生后面一个学期或两个学期的学习情况。针对以上问题由于附件中只给出了300名学生连续三个学期的成绩，如果从多个因素着手就会脱离客观现实，具有不可操作性。因此我们主要着眼于学生的实际成绩、进步程度以及综合排名，本文所使用的两个模型只针对这三个因素展开。模型的假设及符号说明2.1模型的假设①.成绩、进步程度和排名都实行百分制；②.每个学生的学习能力保持不变；③.每个学生的学习考试环境相同；④.综合排名分等级并且计分，方式如下：将学生实际成绩排名后，取前10%的学生分数记为100，接下来50%的学生分数记为80，再接下来30%的学生成绩记为60，最后10%的学生成绩记为40；2.2符号说明x第一学期的实际成绩1第二学期的实际成绩2第三学期的实际成绩ω组合权向量k第j个权占的权重jC第j个学期的实际成绩（j=1,2,3）j第j-2个学期的进步程度（j=4,5）第j-5个学期的综合排名分数（j=6,7,8）S第i个学生三个学期的综合成绩iC第i个学生第j个学期的实际成绩i,jrank学生每学期实际成绩排名a，b，c多元线性回归系数三、模型的建立与求解3.1对学生整体成绩的分析⑴.根据附件中所给的300组数据，用Excel做描述性统计分析并做相应的处理得出下表:表一：学生三学期成绩分析表及格率89%89.70%92.30%平均分71.2756412172.717008572.40156629中位数72.7242346974.8572.98357143标准差9.25801059811.291106619.647327595最低分24.34375016.25最高分最高分与最低89.4590.2535714390.61584906分差值65.1062590.2535714374.36584906观测数300300300由表一我们可以得出以下结论：①学生三个学期考试的及格率大于等于89%且呈上升趋势，说明约九成的学生都能及格，绝大部分学生具备较好的学习能力，上升趋势说明整体学习能力逐渐提高。②学生三个学期的的平均分和中位数都在71~75分之间，说明整体的学习情况还不错。③第二学期的标准差比第一学期的标准差大，说明第二学期总体成绩较向两端分散；第三学期的标准差比带一学期的大，但比第二学期的小，说明第三学期总体成绩比起第一学期要分散，比起第二学期要集中。④从最高分与最低分的差值来看，有增大的趋势。⑵.对学生三个学期的成绩用spss软件做出人数——分数直方图如下：图一:300图一:300名学生第一学期人数——分数直方图图二:300名学生第二学期人数——分数直方图图三:300名学生第三学期人数——分数直方图由上面三个图可以直观的得出以下结论：①这300个人当中85分以上的人数较少，且优等生的分数不高，最高分不到91分，学校需加强对优等生的拔尖培养。②学生成绩分布近似服从正态分布，数据比较合理。③第三个图与前两个图相比70分左右分布的人较多，不及格的人明显减少，体现了教学的良性发展。3.2对学生学习状况的评价3.2.1数据处理其中进步程度表达式为：CCC,CCC；4 2 1 5 3 2综合成绩排名表达式为： rank 100， 3000.1; 80, 0.1rank0.6; 300C= 其中j(6,7,8).j rank 60, 0.6 0.9; 300 rank 40, 0.9 1 3003.2.2层次分析模型学习状况评价实际成绩进步程度第一学期成绩第二学期成绩第三学期成绩第二学期进步程度第三学期进步程度综合排名第一学期综合排名第二学期综合排名第三学期综合排名目标层准则层方案层图四：层次分析模型构造优先关系矩阵矩阵如下： 1 1 123 121 AB:214BC:212 11 11 1 11 34 2 111541BC:13BC:512 231 31 41 2 运用matlab软件，所有构造的矩阵都通过了一致性检验，并且得到B层各指标相对于A层的权值从左到右依次为（0.3196,0.5584,0.122），C层各指标相对于B层各指标的权值从左到右依次为（0.25,0.50,0.25,0.25,0.75,0.0974,0.5696,0.3331），因此组合权向量为：ω=(0.080,0.160,0.080,0.140,0.419,0.012,0.069,0.040)，即如下面的关系图：学习状况学习状况评价第一学期成绩0.080第二学期成绩0.160第三学期成绩0.080第二学期进步程度0.140第三学期进步程度0.419第一学期综合排名0.012第二学期综合排名0.069第三学期综合排名0.040图五因此对于学号为i的学生的学习状况的综合评定定量表示如下：S8kC，其中(j1,2,3...8),i(1,2,3...300) i jj j1 根据该表达式算出结果并排名，下表列出的是学号为1~20的20名学生的信息（所有学生的信息参见附表）：表二：学号为1~20的20名学生的信息学学学学进进排排排综综生期期期步步名名名合合序123度度分分分评排号成 成 成 1 2 值1值值3价 名 绩 绩 绩 266.87554.371470.5266-12.503616.155260406030.590119762.12561.571465.9792-0.55364.407860606029.129124174.37579.108974.61504.7339 -4.493980808033.036412962.93860.339369.4200-2.59829.080760606030.944018782.07578.350081.9151-3.72503.5651100808036.54753968.17570.025071.22501.8500 1.200060606030.377820652.37553.250057.83930.8750 4.589340404024.222628470.22564.915770.2500-5.30935.334360606030.376320779.30085.332179.66606.0321 -5.6661801008035.90085071.52568.467973.9650-3.05715.497180608032.769413977.35077.403675.13110.0536 -2.272480808033.318412275.40080.935782.26795.5357 1.332280808036.57633860.50047.207156.5050-13.29299.297960404024.028328568.67563.932174.0755-4.742910.143360608033.295212354.29564.821464.458010.5264-0.363440606028.213125989.45088.671490.6158-0.77861.944410010010041.3984275.12578.653683.66603.5286 5.0125808010038.36211978.07579.403681.21501.3286 1.811480808036.07284873.05073.914374.85500.8643 0.940780808033.853810779.92586.903685.26086.9786 -1.64278010010039.268111表1：学号为1~20的20名学生的信息:由表1的计算结果可以看出：在这20名学生中，16号学生综合成绩最好，考试成绩突出且其学习状态也比较稳定；而13号学生综合成绩最差，该同学虽然第三学期进步较大，但其第二学期退步很大，又因为第二学期的考试难度是这三学期中最大的，该同学三次成绩也比较低，说明该同学学习状况较差，对于这种情况，老师有必要采取一定的措施，帮助该同学摆脱差的学习状况。因此，通过层次分析模型可以客观、全面、正确地评价学生的学习状况。3.3对以后两学期成绩的预测3.3.1多元线性回归模型用Cij来表示第i个学生第j个学期的实际成绩，在理想化的情况下假设学生第三学期的成绩由前两学期决定，此三者符合一定的线性关系，建立如下线性方程：Ca*Cb*Cc.i3 i1 i2 用spss得出线性系数：a=0.467887b=0.322979c=15.56655则有多元线性回归方程：C0.467887*C0.322979*C15.56655. i3 i1 i2 回归的结果如下：表三回归统计 MultipleR 0.775552 RSquare 0.601481AdjustedRSquare0.598798标准误差 6.110668 观测值 300其中，我们看到调整的R平方为0.5987980.6，说明该模型可以模拟真实成绩的60%，效果不是太好，且用matlab画出残差分析图，可见大部分残差在[-9,9]之间波动，有部分数据在此区间之外，这部分学生成绩波动较大（星型图标所示的数据），因此不能用线性模型来拟合，需要剔除该类数据来优化模型，提高模型的逼真性。.图六：总评分数残差通过对数据残差进行筛选，剔除部分异常值点（残差值在[-9,9]之间的值），其分别为学生序列为21、60、76、100、123、154、174、184、196、205、220、221、227、237、239、247、256、268、282、290的学生成绩，得到的新的数据将在附录中给出。根据剔除部分极值得到的新的数据，再次进行多元线性拟合，得到的效果如下：表四回归统计 MultipleR 0.851410918 RSquare 0.724900551AdjustedRSquare0.722914274标准误差 3.760439965 观测值 280其中，调整的R平方达到了72%的精度，说明模型的逼真性有了很大的提高。新的模型线性系数分别为：a0.3772535,b0.3474851,c20.9958549线性拟合方程为：C=0.3772535*C+0.3474851*C+20.9958549.i,j i,j-1 i,j-2运用此模型可以模拟出第四、五学期的成绩。从预测值来看，与真实值的差值在7左右波动，在不考虑极少数异常的情况下，该模型能够很好地预测学生后几学期的成绩，下面给出部分预测数据，具体详见附录。表五：10名学生第四学期和第五学期的预测成绩学生序号学生序号学期四成绩学期五成绩166.014622970.54137531267.150696369.22062245376.76757675.82021351467.881476870.7725947579.017941279.35616825672.162657472.94118361761.182893564.0760716869.896363371.78585808980.870465879.151428281072.527328874.10157618根据预测的成绩值，同前几学期相比，总体平均分逐渐上涨，说明大部分学生成绩有稳步的提升。同时，数据的方差变小，说明学生间的差距在逐渐减小，学生总体情况良好，达到了教学的目的。3.4预测接下来两学期的学习状况3.4.1数据处理引用层次分析模型，利用多元线性模型预测出的接下来两学期的成绩，进行加权分析，其中各权的权重均已在层次分析模型中得出，第四五学期的进步程度在理想状态下等于第二三学期既不读的加权平均，即：0.25*CC*0.75 4 53.4.2学习状况评价预测由此得出接下来两学期的学习综合评定量的值：S0.3196*C0.5584*[0.25*CC*0.75]i,j i,j 4 50.122*[0.0974*C0.5696*C0.3331*C] 6 7 8j(4,5)将学生数据代入该方程，得出每个学生接下来两学期的综合评定量（见附录）。由预测值进行排名可以看到，与前几学期相比，接下来两学期，部分学生进步较大，学习状况良好，整体来看，较前几学期综合评定分有所提高，说明整体的学习状况有提高的趋势。模型结果分析与检验针对上述模型得出的结果，我们用300名学生的综合成绩做折线图如下：19241924293439114274053667992105118131144157170183196209222235248261274图七:300名学生的综合评价成绩由上图可以看出大多数学生的水平是比较相近的，但仍有个别学生的成绩比较低，老师应该加强对这部分学生的关心。模型的优化与推广层次分析法优点：层次分析法利用权重关系比较进行分析可以学生学习情况综合评价指标权重值的科学性和可信性，从而能够很好的反应学生实际的学习情况，避免了传统的将各项分数相加求和的不合理做法，从而使教育管理者能更全面地了解学生的学习状态，从而进行有效地教学管理。缺点：此方法仍在一定程度上受主观因素的影响，如一开始各个因素的各项指标权重是已经确定再进行求解的，这里就有一定的主观性。改进：在对刚开始的各个因素的各项指标权重赋值上，可以根据不同学校的标准进行设定，或者查阅相关的资料进行确定。多元线性回归预测优点：用多变量线性回归模型，通过多组数据，可直观、快速分析出三者之间的线性关系。回归分析可以准确的剂量各个因素之间的相关程度与拟合程度的高低，提高预测方程式的效果。缺点：可能忽略了交互效应和非线性的因果关系，拟合程度差会导致预测效果差。如一开始调整后的拟合系数只有0.5左右，拟合程度较低。改进：对原始数据进行筛选，排除一些异常值后得到的调整后的拟合系数为0.7左右，大大提高了预测效果。六、参考文献姜启源谢金星叶俊编著《数学模型》高等教育出版社2003年8月第三版；熊启才曹吉利张东生赵临龙编著《数学模型方法及应用》重庆大学出版社2005.3；周义仓郝孝良编著《数学建模实验》西安交通大学出版社1999七、附录 平均排名分 学期四综合评学期五综合评序号学期四成绩学期五成绩平均进步度值 价 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