﻿ 基于因子分析的高中生成绩评价研究

# 基于因子分析的高中生成绩评价研究Research on High School Student Achievement Evaluation Based on the Factor Analysis

Abstract: High school student grades, as an important content of teaching management, are an important digital index to evaluate the comprehensive quality of high school students. It can reflect students’ mastery of knowledge, various abilities and teachers’ teaching level, and also provides an important reference factor for students’ future development. Therefore, the mathematical model research on the comprehensive evaluation of students’ performance is beneficial to make accurate and comprehensive evaluation of students' performance. Factor analysis comprehensive evaluation model is a kind of comprehensive evaluation model for students’ scores based on factor analysis. Through the principal component analysis of students’ scores and the application of factor analysis with the help of SPSS software, the comprehensive factor scores of students are calculated and the scores are ranked. This comprehensive evaluation model can make up for the shortcomings of some high school students’ ranking by directly calculating their total scores, and the evaluation results are more reasonable, fair and scientific.

1. 引言

2. 因子分析的基本理论

${X}_{i}={a}_{i1}{F}_{1}+{a}_{i2}{F}_{2}+\cdots +{a}_{ij}{F}_{j}+\cdots +{a}_{im}{F}_{m}+{\epsilon }_{i}$$i=1,2,\cdots ,p$$j=1,2,\cdots ,m$ ; (1)

$A=\left[\begin{array}{ccc}{a}_{11}& \cdots & {a}_{1m}\\ ⋮& \ddots & ⋮\\ {a}_{p1}& \cdots & {a}_{pm}\end{array}\right]=\left({A}_{1},{A}_{2},\cdots ,{A}_{m}\right)$$m\le p$. (2)

$X=\left[\begin{array}{c}{X}_{1}\\ {X}_{2}\\ \begin{array}{c}⋮\\ {X}_{P}\end{array}\end{array}\right]$$F=\left[\begin{array}{c}{F}_{1}\\ {F}_{2}\\ \begin{array}{c}⋮\\ {F}_{m}\end{array}\end{array}\right]$$\epsilon =\left[\begin{array}{c}{\epsilon }_{1}\\ {\epsilon }_{2}\\ \begin{array}{c}⋮\\ {\epsilon }_{P}\end{array}\end{array}\right]$ (3)

$\sum =AT{T}^{\prime }F+\Phi$ (4)

(1) $E\left({F}^{*}\right)=E\left({T}^{\prime }F\right)={T}^{\prime }E\left(F\right)=0$$\text{Var}\left({F}^{*}\right)=\text{Var}\left({T}^{\prime }F\right)={T}^{\prime }\text{Var}\left(F\right)T={I}_{m}$

(2) $E\left(\epsilon \right)=0$$Var\left(\epsilon \right)=diag\left({\Phi }_{1},{\Phi }_{2},\cdots ,{\Phi }_{p}\right)$

(3) $\mathrm{cov}\left({F}^{*},\epsilon \right)=\mathrm{cov}\left({T}^{\prime }F,\epsilon \right)={T}^{\prime }\mathrm{cov}\left(F,\epsilon \right)=0$

$\mathrm{cov}\left({x}_{i},{f}_{j}\right)=\mathrm{cov}\left({a}_{i1}{f}_{1}+{a}_{i2}{f}_{2}+\cdots +{a}_{im}{f}_{m}+{\epsilon }_{i},{f}_{j}\right)={\sum }_{k=1}^{m}{a}_{ik}\mathrm{cov}\left({f}_{k},{f}_{j}\right)$ (5)

${r}_{{x}_{i},{f}_{j}}=\mathrm{cov}\left({x}_{i},{x}_{j}\right)=\mathrm{cov}\left({\sum }_{k=1}^{m}{a}_{ik}{f}_{k},{\sum }_{k=1}^{m}{a}_{jk}{f}_{k}\right)={\sum }_{k=1}^{m}{a}_{ik}{a}_{jk}$ (6)

${h}_{i}^{2}={\sum }_{j=1}^{m}{a}_{ij}^{2}$ (7)

$\text{Var}\left({x}_{i}\right)={\sum }_{j=1}^{m}\text{Var}\left({a}_{ij}{f}_{j}\right)+\text{Var}\left({\epsilon }_{i}\right)={\sum }_{j=1}^{m}{a}_{ij}^{2}+{\sigma }_{i}^{2}={h}_{i}^{2}+{\sigma }_{i}^{2}$ (8)

$1={h}_{i}^{2}+{\sigma }_{i}^{2}$ (9)

${g}_{j}^{2}={\sum }_{i=1}^{p}{a}_{ij}^{2}$ (10)

${\stackrel{^}{F}}_{j}={b}_{j0}+{b}_{j1}{X}_{1}+\cdots +{b}_{jp}{X}_{p}$$\left(j=1,\cdots ,m\right)$ (11)

$\begin{array}{c}{a}_{ij}={r}_{{X}_{i},{F}_{j}}=E\left({X}_{i}{F}_{j}\right)=E\left[{X}_{i}\left({b}_{j1}{X}_{1}+\cdots +{b}_{jp}{X}_{p}\right)\right]\\ ={b}_{j1}E\left({X}_{i}{X}_{1}\right)+\cdots +{b}_{jp}E\left({X}_{i}{X}_{jp}\right)={b}_{j1}{r}_{i1}+\cdots +{b}_{jp}{r}_{ip}\end{array}$ (12)

$B=\left[\begin{array}{cccc}{b}_{11}& {b}_{12}& \cdots & {b}_{1p}\\ {b}_{21}& {b}_{22}& \cdots & {b}_{2p}\\ ⋮& ⋮& \ddots & ⋮\\ {b}_{m1}& {b}_{m2}& \cdots & {b}_{mp}\end{array}\right]$，矩阵形式可表示为 $A=R{B}^{\prime }$$B={A}^{\prime }{R}^{-1}$

$\stackrel{^}{F}=\left[\begin{array}{c}{\stackrel{^}{F}}_{1}\\ ⋮\\ {\stackrel{^}{F}}_{m}\end{array}\right]=\left[\begin{array}{c}{{b}^{\prime }}_{1}X\\ ⋮\\ {{b}^{\prime }}_{m}X\end{array}\right]=BX={A}^{\prime }{R}^{-1}X$ (13)

3. 因子分析的一般步骤

1) 要想用因子分析法解决一个问题，首先要利用KMO检验和Bartlett球形检验对数据进行检验。如果KMO值越接近于1，越适合做因子分析。如果KMO值小于0.5，则不适合做因子分析。当Bartlett检验统计量p值小于0.05时，则变量适合做因子分析。

2) 一般利用主成分分析的方法是提取公因子。

3) 对样本进行因子分析，通过分析因子的方差贡献率信息和旋转后的因子载荷分布，来确定变量和因子间相关关系。

4) 计算因子综合得分，我们可以直观量化结果来确定样本的综合排序。

4. 实证研究

(一) 适宜性检验。本文利用SPSS对处理后的数据进行KMO检验和巴特利特球形(如表1)，检验结果为KMO = 0. 578 > 0.5，勉强适合做因子分析。P = 0.000 < 0.05这表明样本取样度合理，变量间的相关性较强，适合做因子分析。

Table 1. KMO and Bartlett’s test

(二) 公因子选取与解释。本文用主成分分析法，选取特征值大于1的5个公因子。表2是各个公因子对于总方差的解释程度，其累计方差贡献率为61.592%，能够反映原始数据的大部分信息。即利用因子分析的方法将原问题中13门科目指标变量通过5个公共因子代替，对样本数据做到了较大程度的降维。

Table 2. Total variance explained

Table 3. Rotated factor matrix

$\begin{array}{c}{F}_{1}=-0.272{X}_{1}+0.317{X}_{2}-0.244{X}_{3}+0.432{X}_{4}+0.131{X}_{5}-0.117{X}_{6}-0.007{X}_{7}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-0.103{X}_{8}+0.078{X}_{9}-0.009{X}_{10}+0.078{X}_{11}+0.031{X}_{12}-0.052{X}_{13}\end{array}$

$\begin{array}{c}{F}_{2}=0.089{X}_{1}-0.015{X}_{2}-0.147{X}_{3}+0.097{X}_{4}+0.105{X}_{5}-0.161{X}_{6}+0.189{X}_{7}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+0.107{X}_{8}-0.046{X}_{9}-0.489{X}_{10}+0.519{X}_{11}-0.178{X}_{12}+0.074{X}_{13}{F}_{3}\\ =-0.066{X}_{1}+0.085{X}_{2}-0.049{X}_{3}-0.233{X}_{4}+0.394{X}_{5}+0.597{X}_{6}+0.229{X}_{7}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+0.022{X}_{8}+0.148{X}_{9}+0.122{X}_{10}-0.012{X}_{11}-0.189{X}_{12}+0.102{X}_{13}{F}_{4}\end{array}$

$\begin{array}{l}=-0.069{X}_{1}-0.230{X}_{2}-0.198{X}_{3}-0.095{X}_{4}-0.010{X}_{5}+0.017{X}_{6}-0.074{X}_{7}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+0.603{X}_{8}+0.273{X}_{9}-0.187{X}_{10}-0.078{X}_{11}+0.386{X}_{12}-0.017{X}_{13}{F}_{5}\\ =-0.027{X}_{1}-0.032{X}_{2}+0.069{X}_{3}+0.144{X}_{4}+0.181{X}_{5}-0.112{X}_{6}+0.554{X}_{7}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-0.056{X}_{8}-0.039{X}_{9}+0.107{X}_{10}+0.164{X}_{11}+0.204{X}_{12}-0.588{X}_{13}\end{array}$

Table 4. Factor score coefficient matrix

(三) 综合评价。通过因子得分系数矩阵，计算出因子得分函数，并得到学生在各公因子中的得分，在此基础上以各公因子的方差贡献率为权重并利用线性组合建立学生成绩综合模型，其模型如下：

$S=0.1776{F}_{1}+0.1208{F}_{2}+0.11001{F}_{3}+0.10714{F}_{4}+0.10036{F}_{5}$

Table 5. Comprehensive evaluation

5. 结论与展望

[1] 魏劲如. 基于因子分析的基金平价研究[J]. 统计学与应用, 2019, 81(1): 31-38.

[2] 汪冬华. 多元统计分析与SPSS应用[M]. 上海: 华东理工大学出版社, 2010: 208-218.

[3] 林海明. 因子分析应用中一些常见问题的解析[J]. 统计与决策, 2012(15): 65-69.

[4] 王小丽, 李林芝, 简太敏. 多元统计分析在大学生综合成绩评价中的应用[J]. 产业与科技论坛, 2018, 17(12): 117-119.

[5] 刘访华, 余瑞君. 基于因子分析的学生成绩评价对提高本科教学质量的启示[J]. 中国人民大学教育学刊, 2013, 11(4): 15-21.

[6] 张启贤, 陈欣, 刘新平. 基于因子分析下的学生成绩综合评价模型研究[J]. 西安文理学院学报(自然科学版), 2008(11): 1-6.

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