Fuzzy Cluster Analysis on Influencing Factors of College Student Scores

Student scores are affected by a massive number of factors. The massive data contain lots of valuable information. Cluster analysis, which aims to allocate data objects with similar properties and features to the same cluster, can effectively distinguish the key influencing factors, facilitating the design of pertinent measures to improve student scores.

The data on different students involve many fuzzy concepts (e.g. attention from parents, learning interest, frequency of independent completion of homework, and dormitory atmosphere) that cannot be defined or classified by the set theory in classic mathematics. FCA can provide realistic mathematical description of these uncertain data, and mine out the factors affecting student scores from them [13].

2.1 Fuzzy processing of original data

The original data on the influencing factors of student scores were divided into Boolean data, numeric data, generic data, and null data. The four kinds of data were initialized by membership function.

2.1.1 Membership function of Boolean data

Boolean attributes are relatively simple. In this analysis, only two factors exist as Boolean data: “Preview before class?” and “Participation in clubs and student union?”.

Let U be the entire data domain, n be the total number of data in U, and N is the number of yes or no. Then, the membership function of Boolean data can be defined as [14]:

$u(a)=\left\{ \begin{matrix}   \frac{N(0)}{\text{n}}\text{        a=0}  \\   \frac{N(1)}{\text{n}}\text{        a=}1  \\\end{matrix} \right.$    (1)

2.1.2 Membership function of numeric data

Many factors exist as numeric data, such as monthly number of exchanges with tutor, weekly number of self-studies, and mean exchange time with tutor. The numerical attribute values can be classified, putting the same attribute values into the same class [15].

Let U be the entire data domain, n be the total number of data in U, I be the total number of classes, C_ibe the i-th class, and N(C_i) be the number of attribute values in class C_i [16]. Then, membership function of numeric data can be defined as:

$u({{C}_{i}})\text{=}\frac{N({{C}_{i}})}{n}$      (2)

2.1.3 Membership function of generic data

Generic attribute values are classification attributes, e.g. education of parents, attention from parents, learning interest, learning satisfaction, and frequency of independent completion of homework. The value of each attribute is the common value of a class out of a limited number of classes. Once the same attribute values are allocated to the same class, the membership function will focus on the proportion of each class of attribute values in the total set of classes [17].

Let U be the entire data domain, n be the total number of data in U, J be the number of attribute classes, C_j be the j-th class, and N(C_i) be the number of attribute values in class C_j. Then, the membership function of generic data can be defined as [18]:

$u({{C}_{j}})\text{=}\frac{N({{C}_{j}})}{n}$      (3)

2.1.4 Membership function of null data

Each null data corresponds to the features of its attribute value. Null values may appear in all the previous three types of data. If the ratio of the number of nulls to the number of total elements in an attribute value surpasses the preset threshold Z₀, then the attribute will not be considered in cluster analysis; if the ratio is below the threshold, the attribute will be classified into three levels (high, medium, and low), corresponding to the membership levels (high, medium, and low).

Let C_ij be the value of the j-th attribute of the i-th element; r₀ be the said ratio; h₀ is the high-level threshold; l₀ is the low-level threshold. Then, the membership function of null data can be defined as:

$u({{C}_{ij}})=\left\{ \begin{matrix}   \min ,\text{        }{{r}_{0}}\le {{l}_{0}}  \\   mid,\text{ }{{l}_{0}}<{{r}_{0}}<{{h}_{0}}  \\   \max ,\text{       }{{h}_{0}}\ge {{r}_{0}}  \\\end{matrix} \right.$      (4)

2.1.5 Fuzzy processing of the data on influencing factors

The fuzzy processing of the data on influencing factors of student scores was explained through an example. For convenience, 14 attributes were selected as classification attributes. To diversify the original data, an online questionnaire survey was conducted among students from different colleges, who learn on different online education platforms. A total of 248 questionnaires were returned, among which 235 were valid. Due to the sheer volume of data, this paper only presents the results on six factors of the first 50 respondents. But the subsequent analysis still deals with 12 factors of 235 respondents (Table 1).

2.1.6 Initialization of attribute values

(1) Boolean attribute values

1) Preview before class?

$u(Y\text{es})\text{=}\frac{60}{235}\text{=}0.2553$  

$u(No)\text{=}\frac{165}{235}\text{=}0.7021$

2) Participation in clubs and student union?

$u(Y\text{es})\text{=}\frac{146}{235}\text{=}0.6213$

$u(N\text{o})\text{=}\frac{83}{235}\text{=}0.3532$

(2) Generic attribute values

1) Education of parents

$u(Primary\text{ }school\text{ }graduate)\text{=}\frac{45}{235}\text{=}0.1915$

$u(Junior\text{ }high\text{ }school\text{ }graduate)\text{=}\frac{54}{235}\text{=}0.2298$

$u(Senior\text{ }high\text{ }school\text{ }graduate)\text{=}\frac{56}{235}\text{=}0.2383$

$u(Bachelor)\text{=}\frac{36}{235}\text{=}0.1532$

$u(Master)\text{=}\frac{22}{235}\text{=}0.0936$

$u(Doctor)\text{=}\frac{20}{235}\text{=}0.0851$

2) Attention from parents

$u(Strong)\text{=}\frac{66}{235}\text{=}0.2809$

$u(Moderate)\text{=}\frac{54}{235}\text{=}0.2298$

$u(Slight)\text{=}\frac{57}{235}\text{=}0.2426$

$u(Weak)\text{=}\frac{45}{235}\text{=}0.1915$

3) Learning interest

$u(Strong)\text{=}\frac{50}{235}\text{=}0.2127$

$u(Neutral)\text{=}\frac{84}{235}\text{=}0.3574$

$u(Weak)\text{=}\frac{83}{235}\text{=}0.3532$

4) Weekly nonattendance

$u(Rare)\text{=}\frac{86}{235}\text{=}0.3660$

$u(Never)\text{=}\frac{88}{235}\text{=}0.3745$

$u(Occasional)\text{=}\frac{48}{235}\text{=}0.2043$

5) Learning satisfaction

$u(Strong)\text{=}\frac{50}{235}\text{=}0.2128$

$u(Slight)\text{=}\frac{125}{235}\text{=}0.5319$

$u(Weak)\text{=}\frac{63}{235}\text{=}0.2681$

6) Frequency of independent completion of homework

$u(Strongly\text{ }high)\text{=}\frac{56}{235}\text{=}0.2383$

$u(Slightly\text{ }low)\text{=}\frac{61}{235}\text{=}0.2596$

$u(Slightly\text{ }high)\text{=}\frac{63}{235}\text{=}0.2681$

$u(Strongly\text{ }low)\text{=}\frac{41}{235}\text{=}0.1475$

7) Influence of roommates

$u(Positive\text{ }influence)\text{=}\frac{50}{235}\text{=}0.6383$

$u(Negative\text{ }influence)\text{=}\frac{47}{235}\text{=}0.2$

$u(No\text{ }influence)\text{=}\frac{131}{235}\text{=}0.5574$

(3) Numeric attribute values

1) Monthly number of exchanges with tutor can be divided into the following intervals depending on the attribute values:

d₁:[0, 3]; d₂:[4,8]; d₃:[9,12]

$u({{d}_{1}})\text{=}\frac{118}{235}\text{=}0.5021$

$u({{d}_{2}})\text{=}\frac{66}{235}\text{=}0.2808$

$u({{d}_{3}})\text{=}\frac{45}{235}\text{=}0.2255$

2) Weekly number of self-studies can be divided into the following intervals depending on the attribute values:

d₁:[0, 2]; d₂:[3,5]; d₃:[6,7]

$u({{d}_{1}})\text{=}\frac{58}{235}\text{=}0.2468$

$u({{d}_{2}})\text{=}\frac{125}{235}\text{=}0.5319$

$u({{d}_{3}})\text{=}\frac{54}{235}\text{=}0.2298$

3) Mean time of exchange with tutor can be divided into the following intervals depending on the attribute values:

d₁:[0, 1]; d₂:[1,1.8]; d₃:[1.8,2.5]

$u({{d}_{1}})\text{=}\frac{66}{235}\text{=}0.2809$

$u({{d}_{2}})\text{=}\frac{93}{235}\text{=}0.3957$

$u({{d}_{3}})\text{=}\frac{67}{235}\text{=}0.2851$

4) NCEE results can be divided into the following intervals depending on the attribute values:

d₁:[400, 500]; d₂:[500,600]; d₃:[600,750]

$u({{d}_{1}})\text{=}\frac{33}{235}\text{=}0.1404$

$u({{d}_{2}})\text{=}\frac{99}{235}\text{=}0.4596$

$u({{d}_{3}})\text{=}\frac{92}{235}\text{=}0.3915$

(4) Null attribute values

There is no null attribute value in the original data.

2.1.7 Data initialization

The initial data on influencing factors of student scores are shown in Table 2.

2.2 Clustering of initial data

The initial data were clustered by fuzzy matrix. Let U be the universe (Table 3) containing |U| elements. Then, the clustering was implemented in the following steps.

(1) Establishing the fuzzy similarity matrix R of the universe U

Let |U| be the order of R. The elements r_ij of matrix R can be calculated by the Euclidean distance formula:

${{r}_{ij}}\text{=}\left\{ \begin{matrix}   1&i=j  \\   \sqrt{\frac{1}{\text{m}}\sum\limits_{\text{k=}1}^{\text{m}}{{{({{S}_{ik}}\text{-}{{S}_{jk}})}^{2}}}}&i\ne j  \\\end{matrix} \right.$       (5)

where, m is the number of attributes; S_ikis the attribute value of the i-th row and k-th column [19]. By formula (5), the matrix R can be obtained as Table 3.

Table 2. The initial data on influencing factors of student scores

2.3 Clustering

(1) Taking λ=0.817, the influencing factors of student scores were divided into six groups: {A₁, A₂}; {A₃}; {A₁₂, A₄}; {A₅, A₆, A₈, A₉, A₇}; {A₁₀, A₁₁}; {A₁₃}.

Group 1 includes attention from parents, and education of parents;

Group 2 includes NCEE results;

Group 3 includes mean time of exchange with tutor, and monthly number of exchanges with tutor;

Group 4 includes frequency of independent completion of homework, learning interest, weekly nonattendance, preview before class? and weekly number of self-studies [20];

Group 5 includes learning satisfaction, and influence of roommates;

Group 6 includes participation in clubs and student union?

(2) Taking λ=0.773, the influencing factors of student scores were divided into four groups [21]:

{A₁, A₂}; {A₃}; {A₁₀, A₁₂, A₅, A₄, A₅, A₆, A₈, A₉, A₇}; {A₁₃, A₁₁}.

Group 1 includes attention from parents, and education of parents;

Group 2 includes NCEE results;

Group 3 includes mean time of exchange with tutor, monthly number of exchanges with tutor, frequency of independent completion of homework, learning interest, weekly nonattendance, preview before class? weekly number of self-studies, and learning satisfaction;

Group 4 includes participation in clubs and student union? and influence of roommates.

To sum up, it is important to divide the factors affecting student scores under certain conditions. The greater the λ values, the more refined the divisions, and the better the pertinence. The FCA can excellently divide the influencing factors, facilitating the subsequent screening of the key factors and formulation of countermeasures.

This section performs ANOVA on the elements of each potential factor, aiming to find the element that contributes the greatest to the potential factor. The ANOVA results provide valuable reference for improving teaching quality and student scores.

4.1 Multi-way ANOVA of family factor

As shown in Table 11 below, A1 had the most significant effect in family factor, i.e. A1 has greater impact on student scores than A2.

4.2 Multi-way ANOVA of exchange factor

As shown in Table 12 below, A12 had the most significant effect in exchange factor, i.e. A12 has greater impact on student scores than A4.

4.3 Multi-way ANOVA of learner factor

As shown in Table 13 below, A15 had the most significant effect in learner factor, i.e. A5 has greater impact on student scores than A5, A6, A7, A8, and A9.

4.4 Multi-way ANOVA of classmate factor

As shown in Table 14 below, A10 had the most significant effect in classmate factor. But there is no reason to conclude that A11 does not have a significant effect on student scores.

4.5 One-way ANOVA of campus factor

As shown in Table 15 below, there is no evidence that A13 has or does not have significant effect on student scores.

4.6 One-way ANOVA of exam factor

As shown in Table 16 below, there was significant difference in intra-subject means. Under the significance level of 0.05, the F ratio was 0.436, greater than the corresponding p-value of 0.009. Hence, the original hypothesis that NCEE results have a significant effect on student scores was rejected.

4.7 Discussion

Through the above ANOVAs, it is learned that, among the presupposed factors, A1, A12, A5, A10, and A3 are the leading factors affecting college student scores. Further ANOVA reveals that A3 and A10 are the two top influencing factors of college student scores. The above analysis shows that the improvement of teaching quality requires the concerted efforts from the school, teachers, and students. To promote the development of the school and students, the school management should invest more in hardware facilities and soft power, creating a favorable learning, working, and living environment for students and teachers. The teachers should teach students in accordance with their aptitude, continuously improve their teaching skills, and adopt various teaching methods and means, making the students more interested, and proactive in learning. The students must concentrate their energy in learning, and lay a solid foundation for advanced professional courses.