Pendahuluan

Latar Belakang Kasus

Studi kasus ini bertujuan untuk menganalisis faktor-faktor akademik yang memengaruhi performa belajar siswa dengan menggunakan data nilai ujian matematika, membaca, dan menulis yang diperoleh dari dataset Students Performance in Exams di Kaggle. Dalam konteks pendidikan, nilai matematika, membaca, dan menulis sering dianggap sebagai indikator utama kemampuan kognitif dan prestasi belajar secara keseluruhan. Melalui analisis PCA, studi kasus ini berupaya mereduksi tiga variabel tersebut menjadi beberapa komponen utama yang mampu menggambarkan struktur data secara lebih sederhana tanpa kehilangan informasi penting. Pendekatan ini digunakan untuk mengetahui apakah ketiga nilai ujian tersebut dapat direpresentasikan oleh satu faktor umum, misalnya academic performance index atau apakah terdapat pola variasi lain yang membedakan kemampuan siswa di berbagai aspek. Dengan demikian, studi kasus ini memberikan gambaran mengenai bagaimana ketiga kemampuan akademik tersebut saling berkaitan, serta sejauh mana masing-masing variabel berkontribusi dalam menjelaskan perbedaan performa.

Sumber Data

Sumber data dalam studi ini berasal dari dataset publik berjudul “Students Performance in Exams” yang tersedia di platform Kaggle. Dataset tersebut dikumpulkan sebagai bagian dari penelitian yang bertujuan memahami faktor-faktor yang memengaruhi hasil ujian siswa pada beberapa mata pelajaran inti. Data ini berisi lebih dari 1.000 observasi siswa dengan berbagai latar belakang demografis dan kebiasaan belajar. Meskipun dataset lengkap mencakup variabel seperti jenis kelamin, kelompok etnis, tingkat pendidikan orang tua, jenis makan siang, serta dukungan kursus, studi ini secara khusus memanfaatkan tiga variabel numerik utama, yaitu nilai matematika, membaca, dan menulis, yang merupakan indikator langsung dari performa akademik siswa. Data diunduh secara langsung melalui halaman resmi Kaggle pada tautan: https://www.kaggle.com/datasets/spscientist/students-performance-in- Berikut adalah contoh 10 data pertama

Data_StudentsPerformance <- data.frame(
  math_score = c(72,69,90,47,76,71,88,40,64,38),
  reading_score = c(72,90,95,57,78,83,95,43,64,60),
  writing_score = c(74,88,93,44,75,78,92,39,67,50)
)

# Menampilkan tabel
Data_StudentsPerformance
##    math_score reading_score writing_score
## 1          72            72            74
## 2          69            90            88
## 3          90            95            93
## 4          47            57            44
## 5          76            78            75
## 6          71            83            78
## 7          88            95            92
## 8          40            43            39
## 9          64            64            67
## 10         38            60            50

Latar Belakang Metode dan Tinjauan Pustaka

Analisis Principal Component Analysis (PCA) digunakan untuk mereduksi dimensi data dan mengidentifikasi struktur utama yang menjelaskan variasi terbesar dari tiga variabel akademik penting, yaitu math score, reading score, dan writing score, yang secara umum dianggap sebagai indikator inti performa akademik siswa. Ketiga variabel ini sering memiliki korelasi tinggi sehingga PCA berguna untuk menyederhanakan informasi menjadi satu atau dua komponen utama tanpa kehilangan makna pentingnya. Berdasarkan berbagai literatur statistik dan pendidikan, PCA banyak digunakan untuk mengukur indeks performa siswa karena mampu menggabungkan beberapa kemampuan akademik menjadi satu skor komposit yang lebih ringkas dan mudah dianalisis, sekaligus membantu mengungkap pola hubungan antara kemampuan numerik dan literasi siswa dalam konteks penilaian pendidikan.

Tujuan

Berikut ini adalah tujuan pada penelitian ini: 1. Menganalisis struktur hubungan antar variabel nilai siswa (math score, reading score, dan writing score) untuk mengetahui pola korelasi yang membentuk indeks performa akademik. 2. Mengidentifikasi komponen utama (principal components) yang mampu merangkum variasi terbesar dari ketiga nilai tersebut sehingga dapat menggambarkan performa siswa secara lebih ringkas dan informatif. 3. Menentukan kontribusi masing-masing variabel dalam membentuk indeks performa siswa sehingga dapat diketahui mata pelajaran mana yang paling dominan memengaruhi kinerja akademik secara keseluruhan.

Source Code

library(readr)
library(knitr)
library(kableExtra)

StudentsPerformance <- read_csv("C:/Users/Windows/Downloads/archive (2)/StudentsPerformance.csv")
## Rows: 1000 Columns: 8
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (5): gender, race/ethnicity, parental level of education, lunch, test pr...
## dbl (3): math score, reading score, writing score
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
View(StudentsPerformance)

StudentsPerformance %>%
  kable(caption = "Source Code") %>%
  kable_styling(full_width = FALSE, bootstrap_options = c("striped", "hover"))
Source Code
gender race/ethnicity parental level of education lunch test preparation course math score reading score writing score
female group B bachelor’s degree standard none 72 72 74
female group C some college standard completed 69 90 88
female group B master’s degree standard none 90 95 93
male group A associate’s degree free/reduced none 47 57 44
male group C some college standard none 76 78 75
female group B associate’s degree standard none 71 83 78
female group B some college standard completed 88 95 92
male group B some college free/reduced none 40 43 39
male group D high school free/reduced completed 64 64 67
female group B high school free/reduced none 38 60 50
male group C associate’s degree standard none 58 54 52
male group D associate’s degree standard none 40 52 43
female group B high school standard none 65 81 73
male group A some college standard completed 78 72 70
female group A master’s degree standard none 50 53 58
female group C some high school standard none 69 75 78
male group C high school standard none 88 89 86
female group B some high school free/reduced none 18 32 28
male group C master’s degree free/reduced completed 46 42 46
female group C associate’s degree free/reduced none 54 58 61
male group D high school standard none 66 69 63
female group B some college free/reduced completed 65 75 70
male group D some college standard none 44 54 53
female group C some high school standard none 69 73 73
male group D bachelor’s degree free/reduced completed 74 71 80
male group A master’s degree free/reduced none 73 74 72
male group B some college standard none 69 54 55
female group C bachelor’s degree standard none 67 69 75
male group C high school standard none 70 70 65
female group D master’s degree standard none 62 70 75
female group D some college standard none 69 74 74
female group B some college standard none 63 65 61
female group E master’s degree free/reduced none 56 72 65
male group D some college standard none 40 42 38
male group E some college standard none 97 87 82
male group E associate’s degree standard completed 81 81 79
female group D associate’s degree standard none 74 81 83
female group D some high school free/reduced none 50 64 59
female group D associate’s degree free/reduced completed 75 90 88
male group B associate’s degree free/reduced none 57 56 57
male group C associate’s degree free/reduced none 55 61 54
female group C associate’s degree standard none 58 73 68
female group B associate’s degree standard none 53 58 65
male group B some college free/reduced completed 59 65 66
female group E associate’s degree free/reduced none 50 56 54
male group B associate’s degree standard none 65 54 57
female group A associate’s degree standard completed 55 65 62
female group C high school standard none 66 71 76
female group D associate’s degree free/reduced completed 57 74 76
male group C high school standard completed 82 84 82
male group E some college standard none 53 55 48
male group E associate’s degree free/reduced completed 77 69 68
male group C some college standard none 53 44 42
male group D high school standard none 88 78 75
female group C some high school free/reduced completed 71 84 87
female group C high school free/reduced none 33 41 43
female group E associate’s degree standard completed 82 85 86
male group D associate’s degree standard none 52 55 49
male group D some college standard completed 58 59 58
female group C some high school free/reduced none 0 17 10
male group E bachelor’s degree free/reduced completed 79 74 72
male group A some high school free/reduced none 39 39 34
male group A associate’s degree free/reduced none 62 61 55
female group C associate’s degree standard none 69 80 71
female group D some high school standard none 59 58 59
male group B some high school standard none 67 64 61
male group D some high school free/reduced none 45 37 37
female group C some college standard none 60 72 74
male group B associate’s degree free/reduced none 61 58 56
female group C associate’s degree standard none 39 64 57
female group D some college free/reduced completed 58 63 73
male group D some college standard completed 63 55 63
female group A associate’s degree free/reduced none 41 51 48
male group C some high school free/reduced none 61 57 56
male group C some high school standard none 49 49 41
male group B associate’s degree free/reduced none 44 41 38
male group E some high school standard none 30 26 22
male group A bachelor’s degree standard completed 80 78 81
female group D some high school standard completed 61 74 72
female group E master’s degree standard none 62 68 68
female group B associate’s degree standard none 47 49 50
male group B high school free/reduced none 49 45 45
male group A some college free/reduced completed 50 47 54
male group E associate’s degree standard none 72 64 63
male group D high school free/reduced none 42 39 34
female group C some college standard none 73 80 82
female group C some college free/reduced none 76 83 88
female group D associate’s degree standard none 71 71 74
female group A some college standard none 58 70 67
female group D some high school standard none 73 86 82
female group C bachelor’s degree standard none 65 72 74
male group C high school free/reduced none 27 34 36
male group C high school standard none 71 79 71
male group C associate’s degree free/reduced completed 43 45 50
female group B some college standard none 79 86 92
male group C associate’s degree free/reduced completed 78 81 82
male group B some high school standard completed 65 66 62
female group E some college standard completed 63 72 70
female group D some college free/reduced none 58 67 62
female group D bachelor’s degree standard none 65 67 62
male group B some college standard none 79 67 67
male group D bachelor’s degree standard completed 68 74 74
female group D associate’s degree standard none 85 91 89
male group B high school standard completed 60 44 47
male group C some college standard completed 98 86 90
female group C some college standard none 58 67 72
female group D master’s degree standard none 87 100 100
male group E associate’s degree standard completed 66 63 64
female group B associate’s degree free/reduced none 52 76 70
female group B some high school standard none 70 64 72
female group D associate’s degree free/reduced completed 77 89 98
male group C high school standard none 62 55 49
male group A associate’s degree standard none 54 53 47
female group D some college standard none 51 58 54
female group E bachelor’s degree standard completed 99 100 100
male group C high school standard none 84 77 74
female group B bachelor’s degree free/reduced none 75 85 82
female group D bachelor’s degree standard none 78 82 79
female group D some high school standard none 51 63 61
female group C some college standard none 55 69 65
female group C bachelor’s degree standard completed 79 92 89
male group B associate’s degree standard completed 91 89 92
female group C some college standard completed 88 93 93
male group D high school free/reduced none 63 57 56
male group E some college standard none 83 80 73
female group B high school standard none 87 95 86
male group B some high school standard none 72 68 67
male group D some college standard completed 65 77 74
male group D master’s degree standard none 82 82 74
female group A bachelor’s degree standard none 51 49 51
male group D master’s degree standard none 89 84 82
male group C some high school free/reduced completed 53 37 40
male group E some college free/reduced completed 87 74 70
female group C some college standard completed 75 81 84
male group D bachelor’s degree free/reduced completed 74 79 75
male group C bachelor’s degree standard none 58 55 48
male group B some high school standard completed 51 54 41
male group E high school standard none 70 55 56
female group C associate’s degree standard none 59 66 67
male group D some college standard completed 71 61 69
female group D some high school standard none 76 72 71
female group C some college free/reduced none 59 62 64
female group E some college free/reduced completed 42 55 54
male group A high school standard none 57 43 47
male group D some college standard none 88 73 78
female group C some college free/reduced none 22 39 33
male group B some high school standard none 88 84 75
male group C associate’s degree free/reduced none 73 68 66
female group D bachelor’s degree standard completed 68 75 81
male group E associate’s degree free/reduced completed 100 100 93
male group A some high school standard completed 62 67 69
male group A bachelor’s degree standard none 77 67 68
female group B associate’s degree standard completed 59 70 66
male group D bachelor’s degree standard none 54 49 47
male group D some high school standard none 62 67 61
female group C some college standard completed 70 89 88
female group E high school free/reduced completed 66 74 78
male group B some college free/reduced none 60 60 60
female group B associate’s degree standard completed 61 86 87
male group D associate’s degree free/reduced none 66 62 64
male group B associate’s degree free/reduced completed 82 78 74
female group E some college free/reduced completed 75 88 85
male group B master’s degree free/reduced none 49 53 52
male group C high school standard none 52 53 49
female group E master’s degree standard none 81 92 91
female group C bachelor’s degree standard completed 96 100 100
male group C high school free/reduced completed 53 51 51
female group B master’s degree free/reduced completed 58 76 78
female group B high school standard completed 68 83 78
female group C some college free/reduced completed 67 75 70
male group A high school standard completed 72 73 74
male group E some high school standard none 94 88 78
female group D some college standard none 79 86 81
female group C associate’s degree standard none 63 67 70
female group C bachelor’s degree free/reduced completed 43 51 54
female group C master’s degree standard completed 81 91 87
female group B high school free/reduced completed 46 54 58
female group C associate’s degree standard completed 71 77 77
female group B master’s degree free/reduced completed 52 70 62
female group D some high school standard completed 97 100 100
male group C master’s degree free/reduced completed 62 68 75
female group C some college free/reduced none 46 64 66
female group E high school standard none 50 50 47
female group D associate’s degree standard none 65 69 70
male group C some high school free/reduced completed 45 52 49
male group C associate’s degree free/reduced completed 65 67 65
male group E high school standard none 80 76 65
male group D some high school standard completed 62 66 68
male group B some high school free/reduced none 48 52 45
female group C bachelor’s degree standard none 77 88 87
female group E associate’s degree standard none 66 65 69
male group D some college standard completed 76 83 79
female group B some high school standard none 62 64 66
male group D some college standard completed 77 62 62
female group C master’s degree standard completed 69 84 85
male group D associate’s degree standard none 61 55 52
male group C some high school free/reduced completed 59 69 65
male group E high school free/reduced none 55 56 51
female group B some college free/reduced none 45 53 55
female group B bachelor’s degree free/reduced none 78 79 76
female group C associate’s degree standard completed 67 84 86
female group D some college free/reduced none 65 81 77
male group C associate’s degree standard none 69 77 69
female group B associate’s degree standard none 57 69 68
male group C some college standard none 59 41 42
male group D some high school standard completed 74 71 78
male group E bachelor’s degree standard none 82 62 62
male group E high school standard completed 81 80 76
female group B some college free/reduced none 74 81 76
female group B some college free/reduced none 58 61 66
male group D some high school free/reduced completed 80 79 79
male group C some college free/reduced none 35 28 27
female group C high school free/reduced none 42 62 60
male group C associate’s degree free/reduced completed 60 51 56
male group E high school standard completed 87 91 81
male group B some high school standard completed 84 83 75
female group E associate’s degree free/reduced completed 83 86 88
female group C high school free/reduced none 34 42 39
male group B high school free/reduced none 66 77 70
male group B some high school standard completed 61 56 56
female group D high school standard completed 56 68 74
male group B associate’s degree standard none 87 85 73
female group C some high school free/reduced none 55 65 62
male group D some high school standard none 86 80 75
female group B associate’s degree standard completed 52 66 73
female group E master’s degree free/reduced none 45 56 54
female group C some college standard none 72 72 71
male group D high school standard none 57 50 54
male group A some high school free/reduced none 68 72 64
female group C some college standard completed 88 95 94
male group D some college standard none 76 64 66
male group C associate’s degree standard none 46 43 42
female group B bachelor’s degree standard none 67 86 83
male group E some high school standard none 92 87 78
male group C bachelor’s degree standard completed 83 82 84
male group D associate’s degree standard none 80 75 77
male group D bachelor’s degree free/reduced none 63 66 67
female group D some high school standard completed 64 60 74
male group B some college standard none 54 52 51
male group C associate’s degree standard none 84 80 80
male group D high school free/reduced completed 73 68 66
female group E bachelor’s degree standard none 80 83 83
female group D high school standard none 56 52 55
male group E some college standard none 59 51 43
male group D some high school standard none 75 74 69
male group C associate’s degree standard none 85 76 71
male group E associate’s degree standard none 89 76 74
female group B high school standard completed 58 70 68
female group B high school standard none 65 64 62
male group C high school standard none 68 60 53
male group A some high school standard completed 47 49 49
female group D some college free/reduced none 71 83 83
female group B some high school standard completed 60 70 70
male group D master’s degree standard none 80 80 72
male group D high school standard none 54 52 52
female group E some college standard none 62 73 70
female group C associate’s degree free/reduced none 64 73 68
male group C associate’s degree standard completed 78 77 77
female group B some college standard none 70 75 78
female group C master’s degree free/reduced completed 65 81 81
female group C some high school free/reduced completed 64 79 77
male group C some college standard completed 79 79 78
female group C some high school free/reduced none 44 50 51
female group E high school standard none 99 93 90
male group D high school standard none 76 73 68
male group D some high school free/reduced none 59 42 41
female group C bachelor’s degree standard none 63 75 81
female group D high school standard none 69 72 77
female group D associate’s degree standard completed 88 92 95
female group E some college free/reduced none 71 76 70
male group C bachelor’s degree standard none 69 63 61
male group C some college standard none 58 49 42
female group D associate’s degree free/reduced none 47 53 58
female group D some college standard none 65 70 71
male group B some college standard completed 88 85 76
male group C bachelor’s degree standard none 83 78 73
female group C some high school standard completed 85 92 93
female group E high school standard completed 59 63 75
female group C some high school free/reduced none 65 86 80
male group B bachelor’s degree free/reduced none 73 56 57
male group D high school standard none 53 52 42
male group D high school standard none 45 48 46
female group D bachelor’s degree free/reduced none 73 79 84
female group D some college free/reduced completed 70 78 78
female group B some high school standard none 37 46 46
male group B associate’s degree standard completed 81 82 82
male group E associate’s degree standard completed 97 82 88
female group B some high school standard none 67 89 82
male group B bachelor’s degree free/reduced none 88 75 76
male group E some high school standard completed 77 76 77
male group C associate’s degree standard none 76 70 68
male group D some high school standard none 86 73 70
male group C some high school standard completed 63 60 57
female group E bachelor’s degree standard none 65 73 75
male group D high school free/reduced completed 78 77 80
male group B associate’s degree free/reduced none 67 62 60
male group A some high school standard completed 46 41 43
male group E associate’s degree standard completed 71 74 68
male group C high school free/reduced completed 40 46 50
male group D associate’s degree free/reduced none 90 87 75
male group A some college free/reduced completed 81 78 81
male group D some high school free/reduced none 56 54 52
female group C associate’s degree standard completed 67 84 81
male group B associate’s degree standard none 80 76 64
female group C associate’s degree standard completed 74 75 83
male group A some college standard none 69 67 69
male group E some college standard completed 99 87 81
male group C some high school standard none 51 52 44
female group B associate’s degree free/reduced none 53 71 67
female group D high school free/reduced none 49 57 52
female group B associate’s degree standard none 73 76 80
male group B bachelor’s degree standard none 66 60 57
male group D bachelor’s degree standard completed 67 61 68
female group C associate’s degree free/reduced completed 68 67 69
female group C bachelor’s degree standard completed 59 64 75
male group C high school standard none 71 66 65
female group D master’s degree standard completed 77 82 91
male group C associate’s degree standard none 83 72 78
male group B bachelor’s degree standard none 63 71 69
female group D associate’s degree free/reduced none 56 65 63
female group C high school free/reduced completed 67 79 84
female group E high school standard none 75 86 79
female group C some college standard none 71 81 80
female group C some high school free/reduced none 43 53 53
female group C high school free/reduced none 41 46 43
female group C some college standard none 82 90 94
male group C some college standard none 61 61 62
male group A some college free/reduced none 28 23 19
male group C associate’s degree standard completed 82 75 77
female group B some high school standard none 41 55 51
male group C high school standard none 71 60 61
male group C associate’s degree standard none 47 37 35
male group E associate’s degree standard completed 62 56 53
male group B associate’s degree standard none 90 78 81
female group C bachelor’s degree standard none 83 93 95
female group B some college free/reduced none 61 68 66
male group D some high school standard completed 76 70 69
male group C associate’s degree standard none 49 51 43
female group B some high school free/reduced none 24 38 27
female group D some high school free/reduced completed 35 55 60
male group C high school free/reduced none 58 61 52
female group C high school standard none 61 73 63
female group B high school standard completed 69 76 74
male group D associate’s degree standard completed 67 72 67
male group D some college standard none 79 73 67
female group C high school standard none 72 80 75
male group B some college standard none 62 61 57
female group C bachelor’s degree standard completed 77 94 95
male group D high school free/reduced none 75 74 66
male group E associate’s degree standard none 87 74 76
female group B bachelor’s degree standard none 52 65 69
male group E some college standard none 66 57 52
female group C some college standard completed 63 78 80
female group C associate’s degree standard none 46 58 57
female group C some college standard none 59 71 70
female group B bachelor’s degree standard none 61 72 70
male group A associate’s degree standard none 63 61 61
female group C some college free/reduced completed 42 66 69
male group D some college free/reduced none 59 62 61
female group D some college standard none 80 90 89
female group B high school standard none 58 62 59
male group B some high school standard completed 85 84 78
female group C some college standard none 52 58 58
female group D some high school free/reduced none 27 34 32
male group C some college standard none 59 60 58
male group A bachelor’s degree free/reduced completed 49 58 60
male group C high school standard completed 69 58 53
male group C bachelor’s degree free/reduced none 61 66 61
female group A some high school free/reduced none 44 64 58
female group D some high school standard none 73 84 85
male group E some college standard none 84 77 71
female group C some college free/reduced completed 45 73 70
male group D some high school standard none 74 74 72
female group D some college standard completed 82 97 96
female group D bachelor’s degree standard none 59 70 73
male group E associate’s degree free/reduced none 46 43 41
female group D some high school standard none 80 90 82
female group D master’s degree free/reduced completed 85 95 100
female group A some high school standard none 71 83 77
male group A bachelor’s degree standard none 66 64 62
female group B associate’s degree standard none 80 86 83
male group C associate’s degree standard completed 87 100 95
male group C master’s degree free/reduced none 79 81 71
female group E some high school free/reduced none 38 49 45
female group A some high school free/reduced none 38 43 43
female group E some college standard none 67 76 75
female group E bachelor’s degree standard none 64 73 70
female group C associate’s degree free/reduced none 57 78 67
female group D high school standard none 62 64 64
male group D master’s degree standard none 73 70 75
male group E some high school free/reduced completed 73 67 59
female group D some college standard none 77 68 77
male group E some college standard none 76 67 67
male group C associate’s degree standard completed 57 54 56
female group C some high school standard completed 65 74 77
male group A high school free/reduced none 48 45 41
female group B high school free/reduced none 50 67 63
female group C associate’s degree standard none 85 89 95
male group B some high school standard none 74 63 57
male group D some high school standard none 60 59 54
female group C some high school standard completed 59 54 67
male group A some college standard none 53 43 43
female group A some college free/reduced none 49 65 55
female group D high school standard completed 88 99 100
female group C high school standard none 54 59 62
female group C some high school standard none 63 73 68
male group B associate’s degree standard completed 65 65 63
female group B associate’s degree standard none 82 80 77
female group D high school free/reduced completed 52 57 56
male group D associate’s degree standard completed 87 84 85
female group D master’s degree standard completed 70 71 74
male group E some college standard completed 84 83 78
male group D associate’s degree standard none 71 66 60
male group B some high school standard completed 63 67 67
female group C bachelor’s degree free/reduced completed 51 72 79
male group E high school standard none 84 73 69
male group C bachelor’s degree standard completed 71 74 68
male group C associate’s degree standard none 74 73 67
male group D some college standard none 68 59 62
male group E high school free/reduced completed 57 56 54
female group C associate’s degree free/reduced completed 82 93 93
female group D high school standard completed 57 58 64
female group D master’s degree free/reduced completed 47 58 67
female group A some high school standard completed 59 85 80
male group B some college free/reduced none 41 39 34
female group C some college free/reduced none 62 67 62
male group C bachelor’s degree standard none 86 83 86
male group C some high school free/reduced none 69 71 65
male group A some high school free/reduced none 65 59 53
male group C some high school free/reduced none 68 63 54
male group C associate’s degree free/reduced none 64 66 59
female group C high school standard none 61 72 70
male group C high school standard none 61 56 55
female group A some high school free/reduced none 47 59 50
male group C some high school standard none 73 66 66
male group C some college free/reduced completed 50 48 53
male group D associate’s degree standard none 75 68 64
male group D associate’s degree free/reduced none 75 66 73
male group C high school standard none 70 56 51
male group D some high school standard completed 89 88 82
female group C some college standard completed 67 81 79
female group D high school standard none 78 81 80
female group A some high school free/reduced none 59 73 69
female group B associate’s degree standard none 73 83 76
male group A some high school free/reduced none 79 82 73
female group C some high school standard completed 67 74 77
male group D some college free/reduced none 69 66 60
male group C high school standard completed 86 81 80
male group B high school standard none 47 46 42
male group B associate’s degree standard none 81 73 72
female group C some college free/reduced completed 64 85 85
female group E some college standard none 100 92 97
female group C associate’s degree free/reduced none 65 77 74
male group C some college free/reduced none 65 58 49
female group C associate’s degree free/reduced none 53 61 62
male group C bachelor’s degree free/reduced none 37 56 47
female group D bachelor’s degree standard none 79 89 89
male group D associate’s degree free/reduced none 53 54 48
female group E bachelor’s degree standard none 100 100 100
male group B high school standard completed 72 65 68
male group C bachelor’s degree free/reduced none 53 58 55
male group B some college free/reduced none 54 54 45
female group E some college standard none 71 70 76
female group C some college free/reduced none 77 90 91
male group A bachelor’s degree standard completed 75 58 62
female group C some college standard none 84 87 91
female group D associate’s degree free/reduced none 26 31 38
male group A high school free/reduced completed 72 67 65
female group A high school free/reduced completed 77 88 85
male group C some college standard none 91 74 76
female group C associate’s degree standard completed 83 85 90
female group C high school standard none 63 69 74
female group C associate’s degree standard completed 68 86 84
female group D some high school standard none 59 67 61
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female group D some college free/reduced none 77 86 86

Hasil dan Pembahasan

Berdasarkan hasil analisis Principal Component Analysis (PCA) yang ditampilkan pada output R, dapat dilihat bahwa komponen utama pertama (PC1) memiliki kontribusi yang sangat dominan dalam menjelaskan variansi data. Dari tabel proportion of variance, PC1 menjelaskan sekitar 90,61% dari total variansi, sementara PC2 hanya menjelaskan 7,88%, dan PC3 sebesar 1,50%. Hal ini menunjukkan bahwa ketiga variabel yaitu math score, reading score, dan writing score memiliki hubungan yang sangat kuat satu sama lain, sehingga mayoritas informasi dapat diringkas hanya ke dalam satu komponen utama.

Pada loading matrix, ketiga variabel memiliki loading yang besar dan negatif pada PC1 (math = -0.55, reading = -0.59, writing = -0.59), yang berarti bahwa ketiganya berkontribusi kuat dan searah dalam membentuk komponen utama pertama. Dengan demikian, PC1 dapat ditafsirkan sebagai faktor umum performa akademik siswa, karena ketiga nilai ujian bergerak bersama-sama dan mencerminkan kemampuan akademik secara keseluruhan.

Komponen kedua (PC2), meskipun kontribusinya kecil, menangkap variasi tambahan yang tidak terwakili oleh PC1. PC2 memiliki loading yang positif besar pada variabel reading score (0.359) dan negatif pada math score (-0.813), sehingga PC2 dapat diartikan sebagai dimensi yang menangkap perbedaan antara kemampuan numerik dan kemampuan verbal siswa. Namun, proporsi variansi yang dijelaskan PC2 terlalu kecil untuk menjadi faktor utama. Sementara itu, PC3 yang hanya menjelaskan 1,5% variansi tidak memberikan informasi berarti dan dapat diabaikan. Secara keseluruhan, hasil PCA menunjukkan bahwa ketiga nilai ujian sangat homogen dan secara garis besar dapat diringkas menjadi satu indeks performa akademik melalui PC1.

Berdasarkan Scree Plot yang disajikan, analisis Komponen Utama (PCA) dari tiga variabel nilai siswa yaitu math score, reading score, dan writing score mengindikasikan bahwa sebagian besar varians dalam data dijelaskan oleh Komponen Utama Pertama (PC1). Dengan menggunakan kriteria Kaiser (Nilai eigen > 1) maupun kriteria titik “Elbow” (di mana kurva mulai mendatar), PC1 adalah komponen yang perlu dipertahankan. PC1 memiliki nilai eigen yang signifikan (sekitar 2.4), sementara PC2 dan PC3 memiliki nilai yang mendekati nol dan menjelaskan sangat sedikit varians sisanya. Oleh karena ketiga nilai ujian ini diasumsikan berkorelasi positif, PC1 dapat diinterpretasikan sebagai faktor umum yang mendasari. Dengan demikian, PCA berhasil mereduksi tiga variabel terpisah menjadi satu dimensi yang paling informatif, yang secara efektif mewakili kualitas akademik siswa.

Biplot PCA menunjukkan bahwa ketiga variabel yaitu math score, reading score, dan writing score memiliki arah vektor yang hampir sejajar dan panjang yang mirip, sehingga ketiganya sangat berkorelasi dan bergerak searah. Artinya, siswa yang memiliki nilai tinggi pada satu mata pelajaran cenderung memiliki nilai tinggi pada mata pelajaran lainnya. Titik-titik data siswa juga tersebar rapat di sekitar pusat dan tidak membentuk kelompok terpisah, menandakan bahwa perbedaan performa lebih banyak muncul pada satu dimensi utama, yaitu PC1. Dengan demikian, biplot menegaskan bahwa seluruh variabel berkontribusi kuat terhadap satu faktor umum, yaitu performa akademik keseluruhan.

Penutup

Kesimpulan

Secara keseluruhan, hasil analisis PCA menunjukkan bahwa ketiga variabel yaitu math score, reading score, dan writing score memiliki hubungan yang sangat kuat dan saling bergerak searah, sehingga mayoritas informasi data dapat diringkas hanya ke dalam satu komponen utama. PC1 menjelaskan lebih dari 90% variansi total, menjadikannya representasi terbaik dari performa akademik keseluruhan siswa. Hasil loading yang besar dan searah memperkuat bahwa ketiga mata pelajaran tersebut berkontribusi hampir sama dalam membentuk dimensi utama ini. Sementara itu, PC2 dan PC3 hanya menjelaskan variansi yang sangat kecil, sehingga tidak memiliki makna penting dalam pemisahan karakteristik siswa. Biplot juga menunjukkan pola yang konsisten dengan temuan ini, di mana ketiga vektor variabel tampak sejajar dan titik observasi siswa terkonsentrasi tanpa pola pemisahan kelompok. Dengan demikian, dapat disimpulkan bahwa satu faktor umum sudah cukup untuk menggambarkan performa akademik siswa secara keseluruhan, dan PCA berhasil mereduksi data menjadi bentuk yang lebih sederhana tanpa kehilangan informasi penting.

Daftar Pustaka

Jolliffe, I. T., & Cadima, J. (2016). Principal Component Analysis: A Review and Recent Developments. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374(2065), 1–16. Abdi, H., & Williams, L. J. (2010). Principal Component Analysis. Wiley Interdisciplinary Reviews: Computational Statistics, 2(4), 433–459.

Syntax

library(tidyverse)
## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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## ✔ forcats   1.0.0     ✔ stringr   1.5.1
## ✔ ggplot2   4.0.1     ✔ tibble    3.2.1
## ✔ lubridate 1.9.4     ✔ tidyr     1.3.1
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
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## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
library(ggplot2)
library(psych)
## 
## Attaching package: 'psych'
## 
## The following objects are masked from 'package:ggplot2':
## 
##     %+%, alpha
# 1. Import Data
library(readr)
StudentsPerformance <- read_csv("C:/Users/Windows/Downloads/archive (2)/StudentsPerformance.csv")
## Rows: 1000 Columns: 8
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (5): gender, race/ethnicity, parental level of education, lunch, test pr...
## dbl (3): math score, reading score, writing score
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
View(StudentsPerformance)

colnames(StudentsPerformance)[6] <- "math_score"
colnames(StudentsPerformance)[7] <- "reading_score"
colnames(StudentsPerformance)[8] <- "writing_score"

# 2. Pilih variabel untuk PCA
nilai <- StudentsPerformance %>% 
  select(math_score = math_score,
         reading_score = reading_score,
         writing_score = writing_score)

# 3. Standarisasi data
nilai_scaled <- scale(nilai)

# 4. Analisis PCA
pca_result <- prcomp(nilai_scaled, center = TRUE, scale. = TRUE)

# 5. Output ringkasan PCA
summary(pca_result)
## Importance of components:
##                           PC1     PC2    PC3
## Standard deviation     1.6488 0.48640 0.2121
## Proportion of Variance 0.9061 0.07886 0.0150
## Cumulative Proportion  0.9061 0.98500 1.0000
# 6. Melihat loading tiap komponen
pca_result$rotation
##                      PC1        PC2         PC3
## math_score    -0.5547249 -0.8310995  0.03941888
## reading_score -0.5898847  0.3594283 -0.72308186
## writing_score -0.5867847  0.4243641  0.68963669
# 7. Scree Plot
plot(pca_result, type = "l", col = "blue", main = "Scree Plot PCA")

# 8. Biplot
biplot(pca_result, col = c("red", "blue"))