# Combining Students’ Grades and Achievements on the National Assessment of Knowledge: A Fuzzy Logic Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Basic Definitions

#### 2.2. The Usage of Fuzzy Logic in Education

#### 2.3. Research Aims

## 3. Materials and Methods

#### 3.1. Methodology

#### 3.2. Data Collection

- Teacher-given grades on oral evaluations in mathematics;
- Teacher-given grades on written evaluations in mathematics;
- Scores on the national assessment INVALSI;
- Students’ gender.

#### 3.3. Data Filtering

- Grade 8: 619 students presented missing data on both teacher-given grades and were therefore excluded from the analyses. The remaining sample consisted of students who had at least one teacher-assigned grade (oral or written). We did not make any distinction between oral or written grades, since in Italian middle school educational system, students receive only one grade for mathematics, which refers to both oral and written evaluations;
- Grade 10: 3008 students were excluded due to missing data. The remaining sample consisted of students who had at least one teacher-assigned grade. For the students who had both oral and written grades, only the oral grade was considered, since it includes a broader range of evaluations;
- Grade 13: 6174 students presented missing data on both teacher-given grades. The filtering procedure is the same as for grade 10.

#### 3.3.1. Teacher-Given Grades

#### 3.3.2. The INVALSI Test

#### 3.4. Application of Fuzzy Logic

#### 3.4.1. Fuzzification of Teacher-Given Grades

#### 3.4.2. Fuzzification of Students’ Attainments on the National Assessment INVALSI

#### 3.4.3. Rules and Inference

- 4.
- It is possible to have the “Very high” (VH) level only if both tests are very high (VH), so the VH levels of [7] have been adapted to H;
- 5.
- High performance (H) on the INVALSI tests can only produce at least medium (M) ratings;
- 6.
- The final grade is high (H) only if both ratings are high (H) or one rating is very high (VH) and the other is at least medium (M).

#### 3.4.4. Defuzzification

#### 3.5. Quantitative Analysis

- 7.
- The Spearman’s ρ coefficient to compute the correlation between data [50,51], e.g., to what extent are teacher-given grades and fuzzy final grades correlated. The requirement is that each variable is measured on at least ordinal scale; the usage of this coefficient does not make any assumption regarding the distribution of the variables.
- 8.
- The Wilcoxon W signed-rank test for paired samples to check for differences between two categories [52], e.g., the difference between the traditional and fuzzy final grades. The assumptions of the test are that samples are random samples, which are mutually independent, and that the measurement scale is at least ordinal.
- 9.
- The Kruskal–Wallis χ
^{2}-test to check for differences among three or more categories [53] with the Dwass-Steel-Critchlow-Fligner (DSCF) post-hoc test, e.g., whether fuzzy final grades differ among different teacher-given grades. The assumptions are similar to those for the Wilcoxon test.

^{2}coefficient [54] for the Kruskal–Wallis test and the rank biserial correlation coefficient r

_{rb}[55] for the Wilcoxon test. Effect sizes lower than 0.10 are considered small, those around 0.25 are considered middle, and those greater than 0.40 are considered large [56,57].

## 4. Results

#### 4.1. Differences between Classical and Fuzzy Logic Assessment

#### 4.1.1. Grade 8

^{8}; p < 0.001) with a big effect size (r

_{rb}= 0.922).

^{2}(7) = 15,035; p < 0.001; ε

^{2}= 0.517). The DSCF pairwise comparisons between school grades are all statistically significant (p < 0.05), except for the comparison among students with teacher-given grade 3 and 4 (W = 3.89; p = 0.108).

#### 4.1.2. Grade 10

^{8}; p < 0.001) with a medium effect size (r

_{rb}= 0.600).

^{2}(9) = 13,150; p < 0.001; ε

^{2}= 0.401). The DSCF pairwise comparisons between school grades are all statistically significant (p < 0.05), except for the comparison among students with teacher-given grade 6 and 7 (W = 4.43; p = 0.055).

#### 4.1.3. Grade 13

^{8}; p < 0.001) with a medium effect size (r

_{rb}= 0.690).

^{2}(9) = 11,351; p < 0.001; ε

^{2}= 0.373). The DSCF pairwise comparisons between school grades are all statistically significant (p < 0.05), except for the comparison among students with teacher-given grade 1 and 2 (W = 4.37; p = 0.063).

#### 4.2. Differences in Fuzzy Logic Assessment among Students of Grade 8, 10, and 13

^{2}(2) = 51.2; p < 0.001), although the effect size is extremely low (ε

^{2}= 5.55 × 10

^{−4}). The DSCF test has shown that there are statistically significant differences in fuzzy grades between the 8th and 10th grade students (W = −8.572; p < 0.001), as well as between the 8th and 13th grade students (W = −8.983; p < 0.001). However, there are no statistically significant differences between the 10th and 13th grade students (W = −0.824; p = 0.830). Due to the low effect size, results indicate that students’ fuzzy achievements are hardly dependent on students’ classes. Moreover, high school students have similar fuzzy achievements, thus fuzzy grades are almost uniformly distributed in all grades.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Gao, X.; Li, P.; Shen, J.; Sun, H. Reviewing assessment of student learning in interdisciplinary STEM education. Int. J. STEM Educ.
**2020**, 7, 24. [Google Scholar] [CrossRef] - Mellati, M.; Khademi, M. Exploring teachers’ assessment literacy: Impact on learners’ writing achievements and implications for teacher development. Austr. J. Teach. Educ.
**2018**, 43, 1–18. [Google Scholar] [CrossRef] - Voskoglou, M.G. Fuzzy logic as a tool for assessing students’ knowledge and skills. Educ. Sci.
**2013**, 3, 208–221. [Google Scholar] [CrossRef] - Sripan, R.; Suksawat, B. Propose of fuzzy logic-based students’ learning assessment. In Proceedings of the ICCAS 2010, Gyeonggi-do, Korea, 27–30 October 2010; IEEE: Manhattan, NY, USA, 2010; pp. 414–417. [Google Scholar]
- Krouska, A.; Troussas, C.; Sgouropoulou, C. Fuzzy logic for refining the evaluation of learners’ performance in online engineering education. Eur. J. Eng. Sci. Tech.
**2019**, 4, 50–56. [Google Scholar] - Gokmen, G.; Akinci, T.Ç.; Tektaş, M.; Onat, N.; Kocyigit, G.; Tektaş, N. Evaluation of student performance in laboratory applications using fuzzy logic. Procedia Soc.
**2010**, 2, 902–909. [Google Scholar] [CrossRef] - Petrudi, S.H.J.; Pirouz, M.; Pirouz, B. Application of fuzzy logic for performance evaluation of academic students. In Proceedings of the 2013 13th Iranian Conference on Fuzzy Systems (IFSC), Qazvin, Iran, 27–29 August 2013; IEEE: Manhattan, NY, USA, 2013; pp. 1–5. [Google Scholar]
- Namli, N.A.; Şenkal, O. Using the fuzzy logic in assessing the programming performance of students. Int. J. Assess. Tool. Educ.
**2018**, 5, 701–712. [Google Scholar] [CrossRef] - Yadav, R.S.; Soni, A.K.; Pal, S. A study of academic performance evaluation using Fuzzy Logic techniques. In Proceedings of the 2014 International Conference on Computing for Sustainable Global Development (INDIACom), New Delhi, India, 5–7 March 2014; IEEE: Manhattan, NY, USA, 2014; pp. 48–53. [Google Scholar]
- Ivanova, V.; Zlatanov, B. Application of fuzzy logic in online test evaluation in English as a foreign language at university level. In Proceedings of the 45th International Conference on Application of Mathematics in Engineering and Economics (AMEE’19), Sozopol, Bulgaria, 7–13 June 2019; AIP Publishing: Long Island, NY, USA, 2019; Volume 2172, p. 040009. [Google Scholar]
- Eryılmaz, M.; Adabashi, A. Development of an intelligent tutoring system using bayesian networks and fuzzy logic for a higher student academic performance. Appl. Sci.
**2020**, 10, 6638. [Google Scholar] [CrossRef] - Ivanova, V.; Zlatanov, B. Implementation of fuzzy functions aimed at fairer grading of students’ tests. Educ. Sci.
**2019**, 9, 214. [Google Scholar] [CrossRef] - Amelia, N.; Abdullah, A.G.; Mulyadi, Y. Meta-analysis of student performance assessment using fuzzy logic. Indones. J. Sci. Technol.
**2019**, 4, 74–88. [Google Scholar] [CrossRef] - Chrysafiadi, K.; Troussas, C.; Virvou, M. Combination of fuzzy and cognitive theories for adaptive e-assessment. Expert Syst. Appl.
**2020**, 161, 113614. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] - Bissey, S.; Jacques, S.; Le Bunetel, J.C. The fuzzy logic method to efficiently optimize electricity consumption in individual housing. Energies
**2017**, 10, 1701. [Google Scholar] [CrossRef] - Liu, H.; Jeffery, C.J. Moonlighting Proteins in the Fuzzy Logic of Cellular Metabolism. Molecules
**2020**, 25, 3440. [Google Scholar] [CrossRef] [PubMed] - Thalmeiner, G.; Gáspár, S.; Barta, Á.; Zéman, Z. Application of Fuzzy Logic to Evaluate the Economic Impact of COVID-19: Case Study of a Project-Oriented Travel Agency. Sustainability
**2021**, 13, 9602. [Google Scholar] [CrossRef] - Khalil, S.; Hassan, A.; Alaskar, H.; Khan, W.; Hussain, A. Fuzzy Logical Algebra and Study of the Effectiveness of Medications for COVID-19. Mathematics
**2021**, 9, 2838. [Google Scholar] [CrossRef] - Xue, Z.; Dong, Q.; Fan, X.; Jin, Q.; Jian, H.; Liu, J. Fuzzy Logic-Based Model That Incorporates Personality Traits for Heterogeneous Pedestrians. Symmetry
**2017**, 9, 239. [Google Scholar] [CrossRef] - Zadeh, L.A.; Aliev, R.A. Fuzzy Logic Theory and Applications: Part I and Part II; World Scientific Publishing: Singapore, 2018. [Google Scholar]
- Yadav, R.S.; Singh, V.P. Modeling academic performance evaluation using soft computing techniques: A fuzzy logic approach. Int. J. Comput. Sci. Eng.
**2011**, 3, 676–686. [Google Scholar] - Azam, M.H.; Hasan, M.H.; Hassan, S.; Abdulkadir, S.J. Fuzzy type-1 triangular membership function approximation using fuzzy C-means. In Proceedings of the 2020 International Conference on Computational Intelligence (ICCI), Bandar Seri Iskandar, Malaysia, 8–9 October 2020; IEEE: Manhattan, NY, USA, 2020; pp. 115–120. [Google Scholar]
- Bakar, N.A.; Rosbi, S.; Bakar, A.A. Robust estimation of student performance in massive open online course using fuzzy logic approach. Int. J. Eng. Technol.
**2020**, editor issue. 143–152. [Google Scholar] - Bai, Y.; Wang, D. Fundamentals of fuzzy logic control—fuzzy sets, fuzzy rules and defuzzifications. In Advanced Fuzzy Logic Technologies in Industrial Applications; Bai, Y., Zhuang, H., Wang, D., Eds.; Springer: London, UK, 2006; pp. 17–36. [Google Scholar]
- Saliu, S. Constrained subjective assessment of student learning. J. Sci. Educ. Technol.
**2005**, 14, 271–284. [Google Scholar] [CrossRef] - Doz, D.; Felda, D.; Cotič, M. Assessing Students’ Mathematical Knowledge with Fuzzy Logic. Educ. Sci.
**2022**, 12, 266. [Google Scholar] [CrossRef] - INVALSI. Servizio Statistico. Available online: https://invalsi-serviziostatistico.cineca.it/ (accessed on 1 June 2022).
- INVALSI. Rapproto Prove INVALSI 2019. Available online: https://invalsi-areaprove.cineca.it/docs/2019/rapporto_prove_invalsi_2019.pdf (accessed on 1 June 2022).
- Cardone, M.; Falzetti, P.; Sacco, C. INVALSI Data for School System Improvement: The Value Added. Available online: https://www.invalsi.it/download2/wp/wp43_Falzetti_Cardone_Sacco.pdf (accessed on 1 June 2022).
- INVALSI. Istruzioni Informazioni Contest Scuola Secondaria Secondo Grado. Available online: https://invalsi-areaprove.cineca.it/docs/2020/02_2020_Istruzioni_informazioni_contesto_Scuola_secondaria_secondo%20_grad.pdf (accessed on 1 June 2022).
- DLgs 62/2017. Available online: https://www.gazzettaufficiale.it/eli/id/2017/05/16/17G00070/sg (accessed on 1 June 2022).
- RD 653/1925. Available online: https://www.normattiva.it/uri-res/N2Ls?urn:nir:stato:legge:1925-05-04;653 (accessed on 1 June 2022).
- DLgs 297/1994. Available online: https://www.gazzettaufficiale.it/eli/id/1994/05/19/094G0291/sg (accessed on 1 June 2022).
- D 254/2012. Available online: https://www.gazzettaufficiale.it/eli/id/2013/02/05/13G00034/sg (accessed on 1 June 2022).
- DPR 89/2010. Available online: https://www.gazzettaufficiale.it/eli/id/2010/06/15/010G0111/sg (accessed on 1 June 2022).
- INVALSI. Quadro di Riferimento 2018. Available online: https://invalsi-areaprove.cineca.it/docs/file/QdR_MATEMATICA.pdf (accessed on 1 June 2022).
- INVALSI. Rapproto Prove INVALSI 2018. Available online: https://www.invalsi.it/invalsi/doc_evidenza/2018/Rapporto_prove_INVALSI_2018.pdf (accessed on 1 June 2022).
- INVALSI. Rapproto Prove INVALSI 2017. Available online: https://www.invalsi.it/invalsi/doc_eventi/2017/Rapporto_Prove_INVALSI_2017.pdf (accessed on 1 June 2022).
- Organization for Economic Co-Operation and Development [OECD]. Technical Report PISA 2018. Available online: https://www.oecd.org/pisa/data/pisa2018technicalreport/Ch.09-Scaling-PISA-Data.pdf (accessed on 1 June 2022).
- Trends in International Mathematics and Science Study [TIMSS]. Scaling Methodology. Available online: https://timssandpirls.bc.edu/timss2019/methods/pdf/T19_MP_Ch11-scaling-methodology.pdf (accessed on 2 June 2022).
- Pastori, G.; Pagani, V. What do you think about INVALSI tests? School directors, teachers and students from Lombardy describe their experience. J. Educ. Cult. Psychol. Stud.
**2016**, 13, 97–117. [Google Scholar] [CrossRef] - Thukral, S.; Rana, V. Versatility of fuzzy logic in chronic diseases: A review. Med. Hypotheses
**2019**, 122, 150–156. [Google Scholar] [CrossRef] [PubMed] - MATLAB. Fuzzy Logic Toolbox. Available online: https://it.mathworks.com/products/fuzzy-logic.html (accessed on 2 June 2022).
- MATLAB. Breve Riepilogo su R2020b. Available online: https://it.mathworks.com/products/new_products/release2020b.html (accessed on 2 June 2022).
- The Jamovi Project. Jamovi (Version 2.2.5) [Computer Software]. Available online: https://www.jamovi.org (accessed on 1 June 2022).
- Arnastauskaitė, J.; Ruzgas, T.; Bražėnas, M. An Exhaustive Power Comparison of Normality Tests. Mathematics
**2021**, 9, 788. [Google Scholar] [CrossRef] - Gerald, B. A brief review of independent, dependent and one sample t-test. Int. J. Appl. Math. Theor. Phys.
**2018**, 4, 50–54. [Google Scholar] [CrossRef] - Hopkins, S.; Dettori, J.R.; Chapman, J.R. Parametric and nonparametric tests in spine research: Why do they matter? Glob. Spine J.
**2018**, 8, 652–654. [Google Scholar] [CrossRef] [PubMed] - Schober, P.; Boer, C.; Schwarte, L.A. Correlation coefficients: Appropriate use and interpretation. Anesth. Analg.
**2018**, 126, 1763–1768. [Google Scholar] [CrossRef] - Akoglu, H. User’s guide to correlation coefficients. Turk. J. Emerg. Med.
**2018**, 18, 91–93. [Google Scholar] [CrossRef] - Grzegorzewski, P.; Śpiewak, M. The sign test and the signed-rank test for interval-valued data. Int. J. Itell. Syst.
**2019**, 34, 2122–2150. [Google Scholar] [CrossRef] - Johnson, R.W. Alternate Forms of the One-Way ANOVA F and Kruskal–Wallis Test Statistics. J. Stat. Data Sci. Educ.
**2022**, 30, 82–85. [Google Scholar] [CrossRef] - Albers, C.; Lakens, D. When power analyses based on pilot data are biased: Inaccurate effect size estimators and follow-up bias. J. Exp. Soc. Psychol.
**2018**, 74, 187–195. [Google Scholar] [CrossRef] - Liu, X.S.; Carlson, R.; Kelley, K. Common language effect size for correlations. J. Gen. Psychol.
**2019**, 146, 325–338. [Google Scholar] [CrossRef] [PubMed] - Lovakov, A.; Agadullina, E.R. Empirically derived guidelines for effect size interpretation in social psychology. Eur. J. Soc. Psychol.
**2021**, 51, 485–504. [Google Scholar] [CrossRef] - Funder, D.C.; Ozer, D.J. Evaluating effect size in psychological research: Sense and nonsense. Adv. Methods Pract. Psychol. Sci.
**2019**, 2, 156–168. [Google Scholar] [CrossRef] - Quadlin, N. The mark of a woman’s record: Gender and academic performance in hiring. Am. Sciol. Rev.
**2018**, 83, 331–360. [Google Scholar] [CrossRef] - Kanetaki, Z.; Stergiou, C.; Bekas, G.; Jacques, S.; Troussas, C.; Sgouropoulou, C.; Ouahabi, A. Grade Prediction Modeling in Hybrid Learning Environments for Sustainable Engineering Education. Sustainability
**2022**, 14, 5205. [Google Scholar] [CrossRef] - Chung, S.J.; Choi, L.J. The development of sustainable assessment during the COVID-19 pandemic: The case of the English language program in South Korea. Sustainability
**2021**, 13, 4499. [Google Scholar] [CrossRef] - Bowers, A.J. Towards measures of different and useful aspects of schooling: Why schools need both teacher-assigned grades and standardized assessments. In Classroom Assessment and Educational Measurement; Routledge: New York, NY, USA, 2019; pp. 209–223. [Google Scholar]
- Gershenson, S.; Thomas, B.; Fordham Institute. Grade Inflation in High Schools (2005–2016). Available online: https://files.eric.ed.gov/fulltext/ED598893.pdf (accessed on 16 June 2022).
- Herppich, S.; Praetorius, A.K.; Förster, N.; Glogger-Frey, I.; Karst, K.; Leutner, D.; Behrmann, L.; Böhmer, M.; Ufer, S.; Klug, J.; et al. Teachers’ assessment competence: Integrating knowledge-, process-, and product-oriented approaches into a competence-oriented conceptual model. Teach. Teach. Educ.
**2018**, 76, 181–193. [Google Scholar] [CrossRef] - Marsh, J.A.; Farrell, C.C. How leaders can support teachers with data-driven decision making: A framework for understanding capacity building. Educ. Manag. Adm. Leadersh.
**2015**, 43, 269–289. [Google Scholar] [CrossRef] - Stronge, J.H.; Tucker, P.D. Handbook on Teacher Evaluation: Assessing and Improving Performance; Routledge: New York, NY, USA, 2020. [Google Scholar]
- Ferretti, F.; Funghi, S.; Martignone, F. How Standardised Tests Impact on Teacher Practices: An Exploratory Study of Teachers’ Beliefs. In Theorizing and Measuring Affect in Mathematics Teaching and Learning; Springer: Cham, Germany, 2020; pp. 139–146. [Google Scholar]
- Eriksson, K.; Helenius, O.; Ryve, A. Using TIMSS items to evaluate the effectiveness of different instructional practices. Instr. Sci.
**2019**, 47, 1–18. [Google Scholar] [CrossRef] - Westphal, A.; Vock, M.; Kretschmann, J. Unraveling the relationship between teacher-assigned grades, student personality, and standardized test scores. Front. Psychol.
**2021**, 12, 627440. [Google Scholar] [CrossRef] - Bergbauer, A.B.; Hanushek, E.A.; Woessmann, L. Testing. Available online: https://www.nber.org/system/files/working_papers/w24836/w24836.pdf (accessed on 14 July 2022).

**Figure 2.**General diagram illustrating the methodology used in the present study. * Data from school year 2018–2019 was used.

**Figure 6.**The output surface for the fuzzy inference system, plotting the output variable “Final Grade” against the first two input variables, i.e., “INVALSI” and “Grade”.

Grade | Original Sample | Final Sample | % of Original Sample |
---|---|---|---|

8 | 29,675 | 29,056 | 97.9% |

10 | 35,802 | 32,794 | 91.6% |

13 | 36,589 | 30,415 | 83.1% |

Grade | Gender | Frequency (f) | Percentage Frequency (%f) |
---|---|---|---|

8 | Male | 14,983 | 51.6% |

Female | 14,073 | 48.4% | |

10 | Male | 15,663 | 47.8% |

Female | 17,131 | 52.2% | |

13 | Male | 14,785 | 48.6% |

Female | 15,630 | 51.4% |

Level | Membership Function |
---|---|

Very low (VL) | $\mathrm{Tri}\left(x,1,1,3\right)$ |

Low (L) | $\mathrm{Tri}\left(x,1,3,5\right)$ |

Medium (M) | $\mathrm{Trap}\left(x,3,5,6,8\right)$ |

High (H) | $\mathrm{Tri}\left(x,6,8,10\right)$ |

Very high (VH) | $\mathrm{Tri}\left(x,8,10,10\right)$ |

Level | Membership Function |
---|---|

Very low (VL) | $\mathrm{Gauss}\left(x,120,40\right)$ |

Low (L) | $\mathrm{Gauss}\left(x,160,40\right)$ |

Medium (M) | $\mathrm{Gauss}\left(x,200,40\right)$ |

High (H) | $\mathrm{Gauss}\left(x,240,40\right)$ |

Very high (VH) | $\mathrm{Gauss}\left(x,280,40\right)$ |

Teacher-Given Grades | ||||||
---|---|---|---|---|---|---|

VL | L | M | H | VH | ||

INVALSI | VL | VL | VL | L | L | M |

L | VL | L | L | M | M | |

M | L | L | M | M | H | |

H | M | M | M | H | H | |

VH | M | M | H | H | VH |

Mathematics Grade | INVALSI | Fuzzy Grade | |
---|---|---|---|

Mean | 6.79 | 201 | 5.62 |

Standard deviation | 1.42 | 38.5 | 1.78 |

Median | 7 | 200 | 6 |

Minimum | 3 | 66.5 | 1 |

Maximum | 10 | 326 | 10 |

Skewness (SE) | 0.160 (0.0144) | 0.247 (0.0144) | −0.0461 (0.0144) |

Kurtosis (SE) | −0.693 (0.0287) | −0.0442 (0.0287) | −0.390 (0.0287) |

Fuzzy Grades | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

School Grade | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total |

3 | 3 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 (0.0%) |

4 | 0 | 384 | 2 | 202 | 420 | 7 | 0 | 0 | 0 | 0 | 1015 (3.5%) |

5 | 0 | 0 | 2860 | 0 | 0 | 1792 | 0 | 16 | 0 | 0 | 4668 (16.1%) |

6 | 0 | 0 | 3276 | 0 | 0 | 4274 | 0 | 49 | 0 | 0 | 7599 (26.2%) |

7 | 0 | 0 | 18 | 0 | 2412 | 4064 | 0 | 26 | 0 | 0 | 6520 (22.4%) |

8 | 0 | 0 | 35 | 0 | 0 | 2533 | 0 | 2710 | 0 | 0 | 5278 (18.2%) |

9 | 0 | 0 | 0 | 0 | 11 | 1270 | 717 | 0 | 1353 | 0 | 3351 (11.5%) |

10 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 300 | 0 | 310 | 613 (2.1%) |

Total | 3 (0.0%) | 384 (1.3%) | 6200 (21.3%) | 202 (.7%) | 2843 (9.8%) | 12943 (48.0%) | 717 (2.5%) | 3101 (10.7%) | 1353 (4.7%) | 310 (1.1%) | 29056 (100.0%) |

Fuzzy Grade | |||||
---|---|---|---|---|---|

Teacher-Given Grade | Mean | Standard Deviation | Median | Minimum | Maximum |

3 | 2.50 | 0.905 | 3 | 1 | 3 |

4 | 3.67 | 1.36 | 4 | 2 | 6 |

5 | 4.17 | 1.47 | 3 | 3 | 8 |

6 | 4.72 | 1.51 | 6 | 3 | 8 |

7 | 5.63 | 0.523 | 6 | 3 | 8 |

8 | 7.01 | 1.05 | 8 | 3 | 8 |

9 | 7.42 | 1.35 | 7 | 5 | 9 |

10 | 9.00 | 1.02 | 10 | 6 | 10 |

Mathematics Grade | INVALSI | Fuzzy Grade | |
---|---|---|---|

Mean | 6.15 | 206 | 5.54 |

Standard deviation | 1.45 | 39.1 | 1.60 |

Median | 6 | 203 | 6 |

Minimum | 1 | 72.3 | 1 |

Maximum | 10 | 314 | 10 |

Skewness (SE) | 0.004 (0.0135) | 0.226 (0.0135) | −0.293 (0.0135) |

Kurtosis (SE) | −0.233 (0.0271) | −0.053(0.0271) | 0.009 (0.0271) |

Fuzzy Grades | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

School Grade | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total |

1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 (0.0%) |

2 | 3 | 89 | 0 | 14 | 11 | 1 | 0 | 0 | 0 | 0 | 118 (0.4%) |

3 | 102 | 0 | 683 | 0 | 0 | 51 | 0 | 0 | 0 | 0 | 835 (2.5%) |

4 | 0 | 519 | 0 | 536 | 1951 | 227 | 0 | 0 | 0 | 0 | 3233 (9.9%) |

5 | 0 | 0 | 2282 | 0 | 0 | 3652 | 0 | 209 | 0 | 0 | 6143 (18.7%) |

6 | 0 | 0 | 2550 | 0 | 0 | 6560 | 566 | 0 | 0 | 9676 (29.5%) | |

7 | 0 | 0 | 17 | 0 | 2008 | 4654 | 0 | 180 | 0 | 0 | 6859 (20.9%) |

8 | 0 | 0 | 17 | 0 | 0 | 1745 | 0 | 2362 | 0 | 0 | 4124 (12.6%) |

9 | 0 | 0 | 0 | 0 | 8 | 634 | 202 | 0 | 785 | 0 | 1629 (5.0%) |

10 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 98 | 0 | 69 | 173 (0.5%) |

Total | 108 (0.3%) | 608 (1.9%) | 5549 (16.9%) | 550 (1.7%) | 3978 (12.1%) | 17539 (53%) | 202 (0.6%) | 3415 (10.4%) | 785 (2.4%) | 69 (0.2%) | 32794 (100.0%) |

Fuzzy Grade | |||||
---|---|---|---|---|---|

Teacher-Given Grade | Mean | Standard Deviation | Median | Minimum | Maximum |

1 | 1.00 | .000 | 1 | 1 | 1 |

2 | 2.53 | 1.10 | 2 | 1 | 6 |

3 | 2.94 | 1.02 | 3 | 1 | 6 |

4 | 4.42 | 1.16 | 5 | 2 | 6 |

5 | 4.95 | 1.54 | 6 | 3 | 8 |

6 | 5.33 | 1.47 | 6 | 3 | 8 |

7 | 5.75 | .599 | 6 | 3 | 8 |

8 | 7.13 | 1.02 | 8 | 3 | 8 |

9 | 7.56 | 1.42 | 7 | 5 | 9 |

10 | 8.73 | 1.10 | 8 | 6 | 10 |

Mathematics Grade | INVALSI | Fuzzy Grade | |
---|---|---|---|

Mean | 6.33 | 204 | 5.54 |

Standard deviation | 1.46 | 39.8 | 1.62 |

Median | 6 | 201 | 6 |

Minimum | 1 | 69.5 | 1 |

Maximum | 10 | 341 | 10 |

Skewness (SE) | −0.005 (0.014) | 0.211 (0.014) | −0.205 (0.014) |

Kurtosis (SE) | −0.215 (0.0281) | −0.177(0.0281) | −0.053 (0.0281) |

Fuzzy Grades | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

School Grade | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total |

1 | 4 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5 (0.0%) |

2 | 1 | 50 | 0 | 13 | 12 | 2 | 0 | 0 | 0 | 0 | 78 (0.3%) |

3 | 59 | 0 | 534 | 0 | 0 | 68 | 0 | 0 | 0 | 0 | 661 (2.4%) |

4 | 0 | 449 | 0 | 320 | 1458 | 228 | 0 | 0 | 0 | 0 | 2455 (10.5%) |

5 | 0 | 0 | 2024 | 0 | 0 | 2907 | 0 | 178 | 0 | 0 | 5109 (16.8%) |

6 | 0 | 0 | 2889 | 0 | 0 | 5500 | 0 | 506 | 0 | 0 | 8895 (29.2%) |

7 | 0 | 0 | 22 | 0 | 2379 | 4141 | 0 | 416 | 0 | 0 | 6688 (22.0%) |

8 | 0 | 0 | 42 | 0 | 0 | 2062 | 0 | 2141 | 0 | 0 | 4245 (14.0%) |

9 | 0 | 0 | 0 | 0 | 18 | 877 | 298 | 13 | 783 | 0 | 1989 (6.5% |

10 | 0 | 0 | 0 | 0 | 0 | 5 | 0 | 134 | 10 | 141 | 290 (1.0%) |

Total | 64 (.2%) | 499 (1.6%) | 5512 (18.1%) | 333 (1.1%) | 3867 (12.7%) | 15790 (51.9%) | 298 (1.0%) | 3118 (10.3%) | 793 (2.6%) | 141 (0.5%) | 30415 (100.0%) |

Fuzzy Grade | |||||
---|---|---|---|---|---|

Teacher-Given Grade | Mean | Standard Deviation | Median | Minimum | Maximum |

1 | 1.40 | 0.894 | 1 | 1 | 3 |

2 | 2.88 | 1.31 | 2 | 1 | 6 |

3 | 3.13 | 1.13 | 3 | 1 | 6 |

4 | 4.41 | 1.24 | 5 | 2 | 6 |

5 | 4.88 | 1.57 | 6 | 3 | 8 |

6 | 5.14 | 1.55 | 6 | 3 | 8 |

7 | 5.68 | 0.608 | 6 | 3 | 8 |

8 | 6.98 | 1.07 | 8 | 3 | 8 |

9 | 7.33 | 1.40 | 7 | 5 | 9 |

10 | 8.97 | 1.05 | 9 | 6 | 10 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Doz, D.; Felda, D.; Cotič, M.
Combining Students’ Grades and Achievements on the National Assessment of Knowledge: A Fuzzy Logic Approach. *Axioms* **2022**, *11*, 359.
https://doi.org/10.3390/axioms11080359

**AMA Style**

Doz D, Felda D, Cotič M.
Combining Students’ Grades and Achievements on the National Assessment of Knowledge: A Fuzzy Logic Approach. *Axioms*. 2022; 11(8):359.
https://doi.org/10.3390/axioms11080359

**Chicago/Turabian Style**

Doz, Daniel, Darjo Felda, and Mara Cotič.
2022. "Combining Students’ Grades and Achievements on the National Assessment of Knowledge: A Fuzzy Logic Approach" *Axioms* 11, no. 8: 359.
https://doi.org/10.3390/axioms11080359