Modul 8: Review - R untuk Sains Data

Rangkuman Materi R

Author

Tim Asisten Lab Matematika UI

Published

November 10, 2025

Kembali ke Arsip Praktikum PSD 2025

Tujuan Pembelajaran

Setelah menyelesaikan modul ini, mahasiswa diharapkan mampu:

  • 1. Menguasai Operasi Dasar dan Manajemen Data di R
    • Melakukan operasi aritmatika dan manipulasi variabel di R
    • Membuat dan memanipulasi vector serta melakukan sampling
    • Mengimport data dari berbagai format (.csv, .xlsx)
    • Menghitung statistik deskriptif (mean, median, modus, variansi, standar deviasi)
  • 2. Memvisualisasikan Data dengan Efektif
    • Membuat visualisasi frekuensi (bar chart, pie chart, histogram)
    • Membuat visualisasi dua variabel (scatter plot, boxplot, line chart)
    • Menggunakan package ggplot2 untuk visualisasi yang lebih fleksibel
    • Memvisualisasikan data time series
  • 3. Bekerja dengan Distribusi Probabilitas
    • Menghitung PMF/PDF, CDF, quantile, dan generate random numbers
    • Menggunakan distribusi diskrit (Bernoulli, Binomial, Poisson, Uniform diskrit)
    • Menggunakan distribusi kontinu (Normal, Uniform kontinu)
    • Memahami package distr dan distrEx untuk manipulasi distribusi
  • 4. Melakukan Analisis Statistika Inferensia
    • Membuat dan menggunakan loop (for loop dan replicate) untuk simulasi
    • Memahami dan menerapkan distribusi sampling (sample mean, distribusi-t)
    • Melakukan taksiran titik dan taksiran interval (confidence intervals)
    • Melakukan pengujian hipotesis (uji-z, uji-t, ANOVA, uji Chi-squared)
  • 5. Menganalisis Hubungan Antar Variabel
    • Menghitung korelasi antar variabel (Pearson, Spearman, Kendall)
    • Memvisualisasikan korelasi dengan heatmap
    • Melakukan regresi linear sederhana
    • Menginterpretasi hasil analisis regresi

Skenario: Menjadi Data Analyst Profesional 📊

Selamat! Anda telah menyelesaikan hampir seluruh perjalanan pembelajaran di Pengantar Sains Data. Kini saatnya untuk mengkonsolidasikan semua ilmu yang telah Anda pelajari.

Bayangkan Anda adalah seorang Data Analyst di perusahaan konsultan yang diminta untuk melakukan analisis komprehensif terhadap berbagai dataset klien. Untuk itu, Anda perlu menguasai:

  1. Fundamental Skills: Kemampuan dasar R untuk manipulasi data dan perhitungan statistik
  2. Visualization Skills: Kemampuan membuat visualisasi yang informatif dan menarik
  3. Statistical Knowledge: Pemahaman distribusi probabilitas dan aplikasinya
  4. Inferential Skills: Kemampuan membuat kesimpulan dari sampel ke populasi
  5. Analytical Skills: Kemampuan menganalisis hubungan antar variabel

Modul ini dirancang sebagai review komprehensif yang akan membantu Anda mengingat kembali dan mengintegrasikan semua keterampilan tersebut.

1. Dasar-Dasar R dan Manajemen Data

1.1 Operasi Dasar dan Variabel

R adalah bahasa pemrograman yang powerful untuk analisis data. Mari kita mulai dengan operasi dasar:

R
# Operasi dasar
2 + 3      # Penjumlahan
[1] 5
7 - 5      # Pengurangan
[1] 2
3 * 5      # Perkalian
[1] 15
3 / 4      # Pembagian
[1] 0.75
2^3        # Pangkat
[1] 8
2:4        # Sequence
[1] 2 3 4
R
# Menggunakan <- (lebih umum) atau =
apel <- 4
jeruk = 7

# Memanggil variabel
print(apel)
[1] 4
# Operasi dengan variabel
apel * jeruk
[1] 28
banyak_buah <- apel + jeruk
R
# Melihat semua variabel
ls()

# Menghapus variabel
rm(banyak_buah)

# Menghapus semua variabel
rm(list = ls())

1.2 Vector dan Sampling

Vector adalah kumpulan data dengan tipe yang sama, merupakan struktur data fundamental di R.

R
# Membuat vector numerik
nilai_ujian <- c(85, 92, 78, 88, 95)
nilai_ujian
[1] 85 92 78 88 95
# Membuat vector karakter
nama_buah <- c("apel", "jeruk", "pisang")
nama_buah
[1] "apel"   "jeruk"  "pisang"
# Mengakses elemen vector
nama_buah[1]        # Elemen pertama
[1] "apel"
nilai_ujian[2:4]    # Elemen ke-2 sampai ke-4
[1] 92 78 88

Enam Tipe Dasar Vector

Ada enam tipe utama, namun empat di antaranya akan Anda gunakan 99% setiap saat.

Ini adalah tipe default untuk angka di R. Tipe ini bisa menampung angka desimal.

R
ipk <- c(3.55, 4.0, 2.89)
ipk
[1] 3.55 4.00 2.89
class(ipk) # class() memberitahu kita tipe objeknya
[1] "numeric"

Jika Anda yakin hanya akan bekerja dengan bilangan bulat, Anda bisa membuatnya secara eksplisit dengan menambahkan L di belakang angka. Ini menghemat memori.

R
jumlah_sks <- c(24L, 21L, 18L)
jumlah_sks
[1] 24 21 18
class(jumlah_sks)
[1] "integer"

Untuk menyimpan data teks. Selalu diapit oleh tanda kutip " atau '.

R
mata_kuliah <- c("Kalkulus", "Aljabar Linear", "Statistika")
mata_kuliah
[1] "Kalkulus"       "Aljabar Linear" "Statistika"    
class(mata_kuliah)
[1] "character"

Hanya bisa berisi tiga nilai: TRUE, FALSE, atau NA (Not Available/Missing). Sangat penting untuk operasi perbandingan dan filter.

R
lulus_ujian <- c(TRUE, TRUE, FALSE)
lulus_ujian
[1]  TRUE  TRUE FALSE
class(lulus_ujian)
[1] "logical"

Sampling sangat penting untuk analisis statistik:

R
# Sampling tanpa pengembalian (without replacement)
sample(nama_buah, 2)
[1] "jeruk"  "pisang"
# Sampling dengan pengembalian (with replacement)
sample(nama_buah, 5, replace = TRUE)
[1] "jeruk"  "pisang" "pisang" "apel"   "pisang"

1.3 Import Data dan Package

Package memperluas fungsionalitas R. Untuk import data, kita memerlukan package khusus.

R
# Instal package (hanya sekali)
install.packages("readr")
install.packages("readxl")
R
# Aktifkan package (setiap sesi)
library(readr)
library(readxl)

Import Data dari Berbagai Format:

R
# Import CSV
df1 <- read.csv("./data.csv")

# Import Excel
df2 <- read_excel("./data.xlsx")

# Melihat data
View(df1)
head(df1)  # 6 baris pertama
tail(df1)  # 6 baris terakhir

1.4 Eksplorasi Data dan Statistik Deskriptif

Mari kita gunakan dataset iris yang sudah tersedia di R:

R
# Mengaktifkan dataset built-in
data("iris")

# Melihat 6 baris pertama
head(iris)
  Sepal.Length Sepal.Width Petal.Length Petal.Width Species
1          5.1         3.5          1.4         0.2  setosa
2          4.9         3.0          1.4         0.2  setosa
3          4.7         3.2          1.3         0.2  setosa
4          4.6         3.1          1.5         0.2  setosa
5          5.0         3.6          1.4         0.2  setosa
6          5.4         3.9          1.7         0.4  setosa

Informasi Struktur Data:

R
# Melihat dimensi (baris x kolom)
dim(iris)
[1] 150   5
# Melihat nama kolom
names(iris)
[1] "Sepal.Length" "Sepal.Width"  "Petal.Length" "Petal.Width"  "Species"     
# Struktur data
str(iris)
'data.frame':   150 obs. of  5 variables:
 $ Sepal.Length: num  5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 ...
 $ Sepal.Width : num  3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 ...
 $ Petal.Length: num  1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 ...
 $ Petal.Width : num  0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 ...
 $ Species     : Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 1 1 1 ...

Ringkasan Statistik:

R
# Ringkasan lengkap
summary(iris)
  Sepal.Length    Sepal.Width     Petal.Length    Petal.Width   
 Min.   :4.300   Min.   :2.000   Min.   :1.000   Min.   :0.100  
 1st Qu.:5.100   1st Qu.:2.800   1st Qu.:1.600   1st Qu.:0.300  
 Median :5.800   Median :3.000   Median :4.350   Median :1.300  
 Mean   :5.843   Mean   :3.057   Mean   :3.758   Mean   :1.199  
 3rd Qu.:6.400   3rd Qu.:3.300   3rd Qu.:5.100   3rd Qu.:1.800  
 Max.   :7.900   Max.   :4.400   Max.   :6.900   Max.   :2.500  
       Species  
 setosa    :50  
 versicolor:50  
 virginica :50

Statistik Deskriptif Spesifik:

R
# Mean (rata-rata)
mean(iris$Sepal.Length)
[1] 5.843333
# Median (nilai tengah)
median(iris$Sepal.Length)
[1] 5.8
# Modus
library(DescTools)
Mode(iris$Sepal.Length)
[1] 5
attr(,"freq")
[1] 10
R
# Variansi
var(iris$Sepal.Length)
[1] 0.6856935
# Standar deviasi
sd(iris$Sepal.Length)
[1] 0.8280661
# Range (rentang)
range(iris$Sepal.Length)
[1] 4.3 7.9
R
# Minimum
min(iris$Sepal.Length)
[1] 4.3
# Maksimum
max(iris$Sepal.Length)
[1] 7.9
# Banyak data
length(iris$Sepal.Length)
[1] 150
# Total/jumlah
sum(iris$Sepal.Length)
[1] 876.5
# Standard error
se <- sd(iris$Sepal.Length) / sqrt(length(iris$Sepal.Length))
se
[1] 0.06761132

Subsetting/Filter Data:

R
# Menggunakan subset
iris_setosa <- subset(iris, Species == "setosa")
iris_longsepal <- subset(iris, Sepal.Length >= 5.5)

# Menggunakan dplyr
library(dplyr)

Attaching package: 'dplyr'
The following objects are masked from 'package:stats':

    filter, lag
The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union
iris_setosa <- filter(iris, Species == "setosa")

2. Visualisasi Data

2.1 Visualisasi Frekuensi (Base R)

Visualisasi membantu kita memahami distribusi dan pola dalam data.

Membuat Tabel Frekuensi:

R
# Tabel frekuensi untuk Species
freq_species <- data.frame(table(iris$Species, dnn = "Species"))
freq_species
     Species Freq
1     setosa   50
2 versicolor   50
3  virginica   50
R
barplot(freq_species$Freq, 
        names.arg = freq_species$Species,
        main = "Persebaran Spesies Iris",
        xlab = "Spesies", 
        ylab = "Frekuensi",
        col = c("red", "green", "blue"),
        border = "black")

R
pie(freq_species$Freq,
    labels = freq_species$Species,
    main = "Persebaran Spesies Iris",
    col = c("red", "green", "blue"))

R
hist(iris$Sepal.Length,
     main = "Histogram Sepal Length",
     xlab = "Sepal Length",
     ylab = "Frekuensi",
     col = "lightblue",
     border = "black",
     breaks = 15)

2.2 Visualisasi Dua Variabel (Base R)

R
# Boxplot berdasarkan grup
boxplot(Sepal.Length ~ Species, data = iris,
        main = "Sepal Length by Species",
        xlab = "Species",
        ylab = "Sepal Length",
        col = c("red", "green", "blue"))

R
plot(iris$Sepal.Length, iris$Sepal.Width,
     main = "Sepal Length vs Width",
     xlab = "Sepal Length",
     ylab = "Sepal Width",
     col = iris$Species,
     pch = 19)
legend("topright", legend = levels(iris$Species),
       col = 1:3, pch = 19)

R
# Menggunakan time series data
data("nhtemp")
plot(nhtemp, 
     main = "Average Yearly Temperature in New Haven",
     ylab = "Temperature (F)",
     xlab = "Year",
     col = "blue",
     lwd = 2)

2.3 Visualisasi dengan ggplot2

ggplot2 adalah package yang sangat powerful untuk visualisasi di R.

R
install.packages("ggplot2")
R
library(ggplot2)
R
ggplot(iris, aes(x = Sepal.Length, y = Sepal.Width, color = Species)) +
  geom_point(size = 3) +
  labs(title = "Sepal Length vs Width by Species",
       x = "Sepal Length (cm)",
       y = "Sepal Width (cm)") +
  theme_minimal()

R
# Menggunakan dataset mtcars
data(mtcars)
ggplot(mtcars, aes(x = factor(cyl))) +
  geom_bar(fill = "steelblue", width = 0.7) +
  labs(title = "Distribution of Cylinders",
       x = "Number of Cylinders",
       y = "Count") +
  theme_minimal()

R
ggplot(iris, aes(x = Sepal.Length)) +
  geom_histogram(binwidth = 0.3, fill = "lightblue", color = "black") +
  labs(title = "Histogram of Sepal Length",
       x = "Sepal Length (cm)",
       y = "Frequency") +
  theme_minimal()

R
ggplot(iris, aes(x = Species, y = Sepal.Length, fill = Species)) +
  geom_boxplot() +
  labs(title = "Sepal Length by Species",
       x = "Species",
       y = "Sepal Length (cm)") +
  theme_minimal() +
  theme(legend.position = "none")

R
# Menggunakan dataset economics
ggplot(economics, aes(x = date, y = unemploy)) +
  geom_line(color = "blue", linewidth = 1) +
  labs(title = "Unemployment Over Time",
       x = "Date",
       y = "Number of Unemployed (thousands)") +
  theme_minimal()

3. Distribusi Probabilitas

3.1 Konsep Dasar Distribusi

Untuk setiap distribusi di R, tersedia 4 fungsi utama:

  • d...(): Density function (PMF untuk diskrit, PDF untuk kontinu)
  • p...(): Probability function (CDF - Cumulative Distribution Function)
  • q...(): Quantile function (inverse CDF)
  • r...(): Random number generator (RNG)

3.2 Distribusi Diskrit

R
install.packages("Rlab")
R
library(Rlab)
R
# X ~ Bernoulli(p=0.7)
p <- 0.7

# PMF: P(X = x)
dbern(0, prob = p)  # P(X=0)
[1] 0.3
dbern(1, prob = p)  # P(X=1)
[1] 0.7
# CDF: P(X ≤ x)
pbern(0, prob = p)
[1] 0.3
pbern(1, prob = p)
[1] 1
# Quantile
qbern(0.3, prob = p)
[1] 0
# Random generation
rbern(10, prob = p)
 [1] 0 1 0 0 1 0 0 1 1 0
# Visualisasi PMF
support <- c(0, 1)
plot(support, dbern(support, prob = p),
     type = "h", lwd = 5, col = "blue",
     ylim = c(0, 1),
     main = "PMF Bernoulli(p=0.7)",
     xlab = "x", ylab = "P(X = x)")
points(support, dbern(support, prob = p), pch = 19, cex = 2, col = "blue")

R
# X ~ Binomial(n=10, p=0.7)
n <- 10
p <- 0.7

# PMF: P(X = x)
dbinom(2, size = n, prob = p)
[1] 0.001446701
# CDF: P(X ≤ x)
pbinom(2, size = n, prob = p)
[1] 0.001590386
# Quantile
qbinom(0.05, size = n, prob = p)
[1] 5
# Random generation
random_binom <- rbinom(100, size = n, prob = p)

# Visualisasi PMF
x <- 0:n
plot(x, dbinom(x, size = n, prob = p),
     type = "h", lwd = 5, col = "darkgreen",
     main = "PMF Binomial(10, 0.7)",
     xlab = "x", ylab = "P(X = x)")
points(x, dbinom(x, size = n, prob = p), pch = 19, cex = 2, col = "darkgreen")

# Histogram dari random generation
hist(random_binom,
     breaks = 10,
     main = "Histogram 100 sampel dari Binom(10, 0.7)",
     xlab = "Nilai X",
     ylab = "Frekuensi",
     col = "lightgreen")

R
# X ~ Poisson(λ=3)
lambda <- 3

# PMF: P(X = x)
dpois(2, lambda = lambda)
[1] 0.2240418
# CDF: P(X ≤ x)
ppois(2, lambda = lambda)
[1] 0.4231901
# Quantile
qpois(0.5, lambda = lambda)
[1] 3
# Random generation
rpois(100, lambda = lambda)
  [1] 5 7 4 3 4 1 2 3 5 4 1 0 4 1 4 4 2 3 5 5 5 1 7 5 7 2 1 1 2 3 2 2 6 3 1 7 6
 [38] 3 3 2 7 4 1 7 3 6 1 0 3 1 2 2 1 2 3 5 3 1 1 3 3 5 5 2 3 4 1 2 3 3 8 5 3 1
 [75] 4 3 4 3 2 4 2 2 3 3 5 3 4 1 5 4 3 2 2 3 3 2 2 3 3 2
# Visualisasi PMF
x <- 0:15
plot(x, dpois(x, lambda = lambda),
     type = "h", lwd = 5, col = "purple",
     main = "PMF Poisson(λ=3)",
     xlab = "x", ylab = "P(X = x)")
points(x, dpois(x, lambda = lambda), pch = 19, cex = 2, col = "purple")

R
library(distr)
library(distrEx)
R
# Membuat distribusi uniform diskrit pada {1,2,3,4,5,6} (dadu)
support <- c(1, 2, 3, 4, 5, 6)
k <- length(support)
prob <- rep(1/k, k)
dist_dadu <- DiscreteDistribution(support, prob)

# Plot
plot(dist_dadu, main = "Distribusi Uniform Diskrit (Dadu)")

# PMF
d(dist_dadu)(4)
[1] 0.1666667
# CDF
p(dist_dadu)(4)
[1] 0.6666667
# Quantile
q(dist_dadu)(0.5)
[1] 3
# Random generation
r(dist_dadu)(10)
 [1] 2 1 5 1 5 4 1 3 3 1
# Ekspektasi dan variansi
E(dist_dadu)
[1] 3.5
var(dist_dadu)
[1] 2.916667

3.3 Distribusi Kontinu

R
# X ~ N(μ=50, σ²=100)
mu <- 50
sigma <- 10

# Contoh: P(45 < X < 62)
prob <- pnorm(62, mean = mu, sd = sigma) - pnorm(45, mean = mu, sd = sigma)
cat(sprintf("P(45 < X < 62) = %.4f\n", prob))
P(45 < X < 62) = 0.5764
# Menggunakan Z-score
z1 <- (62 - mu) / sigma
z2 <- (45 - mu) / sigma
prob_z <- pnorm(z1) - pnorm(z2)
cat(sprintf("Dengan Z-score: %.4f\n", prob_z))
Dengan Z-score: 0.5764
# Visualisasi
x <- seq(20, 80, length = 200)
plot(x, dnorm(x, mean = mu, sd = sigma),
     type = "l", lwd = 2, col = "darkred",
     main = "PDF Normal N(50, 100)",
     xlab = "x", ylab = "f(x)")

# Area P(45 < X < 62)
x_shade <- seq(45, 62, length = 100)
y_shade <- dnorm(x_shade, mean = mu, sd = sigma)
polygon(c(45, x_shade, 62), c(0, y_shade, 0), col = rgb(1, 0, 0, 0.3), border = NA)
abline(v = mu, col = "blue", lty = 2)

R
# X ~ N(μ=0, σ²=1) - Distribusi Normal Standar

# PDF: f(x)
dnorm(0, mean = 0, sd = 1)
[1] 0.3989423
# CDF: P(X ≤ x)
pnorm(1.96, mean = 0, sd = 1)  # ≈ 0.975
[1] 0.9750021
# Quantile: P(X ≤ q) = p
qnorm(0.975, mean = 0, sd = 1)  # ≈ 1.96
[1] 1.959964
# Random generation
rnorm(100, mean = 0, sd = 1)
  [1] -0.980520682 -0.915321404 -0.242386558  2.029502355  0.783383026
  [6]  2.397292595  0.836565411  2.496647504 -1.194228162 -0.214326910
 [11] -0.644745295 -0.942863174  0.582820234  0.624942260 -0.064528109
 [16]  0.538575959  0.454537066  0.587575703  0.513498628 -0.648273294
 [21] -0.361535386 -0.208882508 -0.191647582 -1.515058032  1.202807428
 [26]  2.462176101 -0.054692734  0.928465006 -1.849197876 -0.176551199
 [31] -0.076880383  1.318193516  2.718757185 -1.593345393  2.132974962
 [36] -0.535683237 -0.920330812 -0.930286111  0.280197305  0.917064954
 [41] -2.346834487 -0.626612005 -0.223786056  2.561655198  0.246609717
 [46] -0.084556006  1.088663428  0.062612465  0.165482326  1.282162313
 [51] -0.386215486  0.384398038 -0.212153146 -0.398740606  0.979794148
 [56] -0.457217645  1.398951416 -0.614087144 -0.511371504 -1.234483691
 [61] -0.379607236  0.190716622  0.424068282 -0.816954468 -0.235109421
 [66] -0.555044910 -0.468561578  1.307533652  1.926788129  0.980300805
 [71]  0.009675772  0.921451351  0.742992046  0.507275510 -0.619532895
 [76] -0.384915337  0.164747569  0.633441643 -1.258560231 -0.887734437
 [81]  1.064301904 -0.117037256  0.899000183 -1.691872858 -0.433265642
 [86]  0.651411703 -0.183497059 -0.572777184  0.296999871  0.572448351
 [91]  1.554619793  0.132949022  1.112457114  1.544183698  1.246944848
 [96] -0.073408988  0.047217060  0.464199343 -1.421813362 -0.710557694
# Visualisasi PDF
x <- seq(-4, 4, length = 200)
plot(x, dnorm(x, mean = 0, sd = 1),
     type = "l", lwd = 2, col = "blue",
     main = "PDF Normal Standar N(0, 1)",
     xlab = "x", ylab = "f(x)")
abline(v = 0, col = "red", lty = 2)

R
# X ~ Uniform(a=20, b=105)
a <- 20
b <- 105

# PDF
dunif(50, min = a, max = b)
[1] 0.01176471
# CDF: P(X ≤ x)
punif(50, min = a, max = b)
[1] 0.3529412
# Quantile
qunif(0.5, min = a, max = b)
[1] 62.5
# Random generation
random_unif <- runif(1000, min = a, max = b)

# Visualisasi PDF
x <- seq(a-10, b+10, length = 200)
plot(x, dunif(x, min = a, max = b),
     type = "l", lwd = 3, col = "orange",
     main = "PDF Uniform(20, 105)",
     xlab = "x", ylab = "f(x)",
     ylim = c(0, 0.02))

# Histogram dari random generation
hist(random_unif,
     breaks = 30,
     main = "Histogram 1000 sampel dari Uniform(20, 105)",
     xlab = "Nilai X",
     col = "lightyellow",
     probability = TRUE)

4. Looping dan Distribusi Sampling

4.1 Looping di R

For Loop:

R
# For loop sederhana
for (i in 1:5) {
  print(i * 2)
}
[1] 2
[1] 4
[1] 6
[1] 8
[1] 10
# Menyimpan hasil
hasil <- numeric(5)
for (i in 1:5) {
  hasil[i] <- i^2
}
print(hasil)
[1]  1  4  9 16 25

Fungsi replicate:

Fungsi replicate sangat berguna untuk simulasi statistik.

R
# Mengulangi operasi 100 kali
hasil <- replicate(100, mean(rnorm(10)))
head(hasil)
[1] -0.39491921 -0.04959657  0.04368116 -0.66597750  0.30500491 -0.38296128
# Equivalent dengan for loop
hasil2 <- numeric(100)
for (i in 1:100) {
  hasil2[i] <- mean(rnorm(10))
}

4.2 Distribusi Sampling

Simulasi Distribusi Sample Mean:

R
# Populasi
populasi <- c(1, 2, 3, 4, 5, 6, 7, 8)

# Simulasi 1000 kali, sample size 30
set.seed(123)
sample_means <- replicate(1000, {
  sample_data <- sample(populasi, 30, replace = TRUE)
  mean(sample_data)
})

# Visualisasi
hist(sample_means,
     main = "Distribution of Sample Means (n=30)",
     xlab = "Sample Mean",
     col = "lightblue",
     breaks = 30,
     probability = TRUE)

# Overlay normal curve
curve(dnorm(x, mean = mean(sample_means), sd = sd(sample_means)), 
      add = TRUE, col = "red", lwd = 2)

# Mean dan SD dari distribusi sample mean
cat(sprintf("Mean of sample means: %.4f\n", mean(sample_means)))
Mean of sample means: 4.5013
cat(sprintf("SD of sample means: %.4f\n", sd(sample_means)))
SD of sample means: 0.4173

Perbandingan Berbagai Ukuran Sampel:

R
# Simulasi dengan berbagai ukuran sampel
par(mfrow = c(2, 2))

for (n in c(5, 10, 30, 50)) {
  sample_means <- replicate(1000, mean(sample(populasi, n, replace = TRUE)))
  
  hist(sample_means,
       main = paste("Sample Size =", n),
       xlab = "Sample Mean",
       col = "lightgreen",
       breaks = 20,
       probability = TRUE)
  
  curve(dnorm(x, mean = mean(populasi), sd = sd(populasi)/sqrt(n)), 
        add = TRUE, col = "red", lwd = 2)
}

par(mfrow = c(1, 1))

4.3 Central Limit Theorem (CLT)

R
# Parameter populasi
mu <- 50
sigma <- 10
n <- 36

# Mean dan SD dari distribusi sample mean (berdasarkan CLT)
mu_xbar <- mu
sigma_xbar <- sigma / sqrt(n)

cat(sprintf("μ_X̄ = %.4f\n", mu_xbar))
μ_X̄ = 50.0000
cat(sprintf("σ_X̄ = %.4f\n", sigma_xbar))
σ_X̄ = 1.6667
# Contoh: P(45 < X̄ < 55)
prob <- pnorm(55, mean = mu_xbar, sd = sigma_xbar) - 
        pnorm(45, mean = mu_xbar, sd = sigma_xbar)

cat(sprintf("\nP(45 < X̄ < 55) = %.4f\n", prob))

P(45 < X̄ < 55) = 0.9973

4.4 Distribusi-t

R
# Distribusi-t dengan df = 14
df <- 14

# PDF
dt(2, df = df)
[1] 0.05950834
# CDF: P(T ≤ t)
pt(2, df = df)
[1] 0.967356
# Quantile (t-tabel)
qt(0.975, df = df)  # untuk α=0.025
[1] 2.144787
# Random generation
rt(100, df = df)
  [1] -2.62249002 -4.15406058  0.84861094  0.19334457  0.66940683  0.57401384
  [7]  0.56665056  0.05276854 -0.51014017 -0.81351881  1.08366796 -0.63582645
 [13] -0.30269643 -0.36414360  2.70317304 -0.71835290 -0.49096799  0.67749892
 [19] -0.12060391 -1.28350798  0.94712489 -0.76391289  1.97659297 -0.47178298
 [25] -0.23250629  0.55180275 -0.92795831  1.68531129  0.79484392 -0.21530552
 [31]  0.73522088  2.58144307  0.02351631 -1.45275040 -2.71785259 -0.64265257
 [37] -1.05466980 -0.48310054  2.13913209 -0.61807360 -0.14338389 -1.23879954
 [43] -1.16364309  0.79165505  0.95575103 -0.21311130 -2.09228500  0.99684543
 [49]  0.92379512  4.75640219 -0.15232457 -0.34448416 -0.28409469  0.14944620
 [55]  0.50741149 -1.41303840  0.15097845  0.55426210 -2.14923488 -0.78741090
 [61] -0.77821930  0.33839129 -2.33949419  0.69374910  0.06553566  0.17039155
 [67]  0.04655677  0.10147598 -0.68217637 -0.13113890  1.44278813 -1.08139378
 [73] -1.27419142  2.82360289  0.33074624  0.62801599 -0.58281145  0.07018520
 [79] -0.15619497  0.52467228 -0.81856676 -0.81316681 -1.69441437 -0.27579374
 [85] -0.24511727 -1.09968646  0.54965428 -0.12506035  0.12225031 -0.23304499
 [91]  0.43738080 -0.11189371 -1.87866487  0.31271523  1.06313345  2.65770332
 [97]  0.59661662  2.10665958 -0.66862829  1.20781391
# Visualisasi: Perbandingan t vs Normal
x <- seq(-4, 4, length = 200)
plot(x, dnorm(x), type = "l", lwd = 2, col = "blue",
     main = "Perbandingan Distribusi t vs Normal",
     xlab = "x", ylab = "Density")
lines(x, dt(x, df = 5), col = "red", lwd = 2)
lines(x, dt(x, df = 14), col = "green", lwd = 2)
legend("topright", 
       legend = c("Normal", "t(df=5)", "t(df=14)"),
       col = c("blue", "red", "green"), lwd = 2)

5. Inferensia Statistika

R
library(distrEx)

5.1 Taksiran Titik

R
# Load data
data(mtcars)

# Taksiran mean
mean_mpg <- mean(mtcars$mpg)
cat(sprintf("Taksiran mean mpg: %.4f\n", mean_mpg))
Taksiran mean mpg: 20.0906
# Taksiran variansi
var_mpg <- var(mtcars$mpg)
cat(sprintf("Taksiran variansi mpg: %.4f\n", var_mpg))
Taksiran variansi mpg: 36.3241
# Taksiran proporsi
prop_matic <- sum(mtcars$am == 0) / nrow(mtcars)
prop_manual <- sum(mtcars$am == 1) / nrow(mtcars)

cat(sprintf("Proporsi mobil matic: %.4f\n", prop_matic))
Proporsi mobil matic: 0.5938
cat(sprintf("Proporsi mobil manual: %.4f\n", prop_manual))
Proporsi mobil manual: 0.4062

5.2 Taksiran Interval (Confidence Intervals)

R
# Data
x_bar <- mean(mtcars$mpg)
sigma <- 6  # variansi diketahui
n <- nrow(mtcars)
alpha <- 0.05

# CI 95%
z_critical <- qnorm(1 - alpha/2)
margin_error <- z_critical * sigma / sqrt(n)

lower <- x_bar - margin_error
upper <- x_bar + margin_error

cat(sprintf("CI 95%% untuk mean (σ diketahui): (%.4f, %.4f)\n", lower, upper))
CI 95% untuk mean (σ diketahui): (18.0118, 22.1695)
R
# Data
x_bar <- mean(mtcars$mpg)
s <- sd(mtcars$mpg)
n <- nrow(mtcars)
alpha <- 0.05

# CI 95% menggunakan t-distribution
t_critical <- qt(1 - alpha/2, df = n-1)
margin_error <- t_critical * s / sqrt(n)

lower <- x_bar - margin_error
upper <- x_bar + margin_error

cat(sprintf("CI 95%% untuk mean (σ tidak diketahui): (%.4f, %.4f)\n", lower, upper))
CI 95% untuk mean (σ tidak diketahui): (17.9177, 22.2636)
R
data(iris)

# Data
x1 <- iris$Sepal.Length[iris$Species == "setosa"]
x2 <- iris$Sepal.Length[iris$Species == "versicolor"]

mean1 <- mean(x1)
mean2 <- mean(x2)
var1 <- var(x1)
var2 <- var(x2)
n1 <- length(x1)
n2 <- length(x2)

# Variansi pooled
var_pooled <- ((n1-1)*var1 + (n2-1)*var2) / (n1 + n2 - 2)
sd_pooled <- sqrt(var_pooled)

alpha <- 0.05

# CI 95%
diff <- mean2 - mean1
t_critical <- qt(1-alpha/2, df = n1+n2-2)
margin_error <- t_critical * sd_pooled * sqrt(1/n1 + 1/n2)

lower <- diff - margin_error
upper <- diff + margin_error

cat(sprintf("CI 95%% untuk beda mean: (%.4f, %.4f)\n", lower, upper))
CI 95% untuk beda mean: (0.7546, 1.1054)
R
data(sleep)

# Hitung perbedaan
diff <- with(sleep, extra[group == 2] - extra[group == 1])

mean_diff <- mean(diff)
sd_diff <- sd(diff)
n <- length(diff)
alpha <- 0.05

# CI 95%
t_critical <- qt(1-alpha/2, df = n-1)
margin_error <- t_critical * sd_diff / sqrt(n)

lower <- mean_diff - margin_error
upper <- mean_diff + margin_error

cat(sprintf("CI 95%% untuk beda mean (paired): (%.4f, %.4f)\n", lower, upper))
CI 95% untuk beda mean (paired): (0.7001, 2.4599)

5.3 Pengujian Hipotesis

R
# H0: μ = 250
# H1: μ ≠ 250
# α = 0.05

mu_0 <- 250
alpha <- 0.05

# Statistik uji
x_bar <- mean(mtcars$disp)
s <- sd(mtcars$disp)
n <- nrow(mtcars)

Z <- (x_bar - mu_0) / (s / sqrt(n))

# Daerah penolakan
Z_lower <- qnorm(alpha/2)
Z_upper <- qnorm(1 - alpha/2)

# P-value
p_value <- 2 * (1 - pnorm(abs(Z)))

# Keputusan
keputusan <- ifelse(Z < Z_lower | Z > Z_upper, "H0 ditolak", "H0 diterima")

cat(sprintf("Statistik uji Z = %.4f\n", Z))
Statistik uji Z = -0.8799
cat(sprintf("Daerah penolakan: Z < %.4f atau Z > %.4f\n", Z_lower, Z_upper))
Daerah penolakan: Z < -1.9600 atau Z > 1.9600
cat(sprintf("P-value = %.6f\n", p_value))
P-value = 0.378914
cat(sprintf("Keputusan: %s\n", keputusan))
Keputusan: H0 diterima
R
# H0: μ = 250
# H1: μ ≠ 250

result <- t.test(mtcars$disp, mu = 250, alternative = "two.sided")

cat("Hasil Uji-t:\n")
Hasil Uji-t:
print(result)

    One Sample t-test

data:  mtcars$disp
t = -0.8799, df = 31, p-value = 0.3857
alternative hypothesis: true mean is not equal to 250
95 percent confidence interval:
 186.0372 275.4065
sample estimates:
mean of x 
 230.7219 
R
# H0: μ1 = μ2
# H1: μ1 ≠ μ2

# Variansi sama
result_equal <- t.test(mtcars$disp[mtcars$am == 0],
                       mtcars$disp[mtcars$am == 1],
                       alternative = "two.sided",
                       var.equal = TRUE)

cat("Hasil Uji-t (Variansi Sama):\n")
Hasil Uji-t (Variansi Sama):
print(result_equal)

    Two Sample t-test

data:  mtcars$disp[mtcars$am == 0] and mtcars$disp[mtcars$am == 1]
t = 4.0152, df = 30, p-value = 0.0003662
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
  72.15611 221.54025
sample estimates:
mean of x mean of y 
 290.3789  143.5308 
# Variansi berbeda
result_unequal <- t.test(mtcars$disp[mtcars$am == 0],
                         mtcars$disp[mtcars$am == 1],
                         alternative = "two.sided",
                         var.equal = FALSE)

cat("\nHasil Uji-t (Variansi Berbeda):\n")

Hasil Uji-t (Variansi Berbeda):
print(result_unequal)

    Welch Two Sample t-test

data:  mtcars$disp[mtcars$am == 0] and mtcars$disp[mtcars$am == 1]
t = 4.1977, df = 29.258, p-value = 0.00023
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
  75.32779 218.36857
sample estimates:
mean of x mean of y 
 290.3789  143.5308 
R
# Membuat data simulasi
set.seed(123)
pops.1 <- rnorm(n = 10, mean = 1, sd = 1)
pops.2 <- rnorm(n = 10, mean = 0.9, sd = 1)
pops.3 <- rnorm(n = 10, mean = 1.1, sd = 1)

pop.source <- c(rep("pops.1", 10), rep("pops.2", 10), rep("pops.3", 10))
pops.value <- c(pops.1, pops.2, pops.3)
data_anova <- data.frame(pop.source, pops.value)

# Uji ANOVA
# H0: μ1 = μ2 = μ3
# H1: setidaknya satu mean berbeda

result_anova <- aov(pops.value ~ pop.source, data = data_anova)
summary(result_anova)
            Df Sum Sq Mean Sq F value Pr(>F)
pop.source   2   1.16  0.5802    0.61  0.551
Residuals   27  25.68  0.9512               
# F-tabel
k <- 3  # Banyak populasi
n <- 10  # Ukuran sampel per populasi
alpha <- 0.05
F_tabel <- qf(1-alpha, df1 = k-1, df2 = k*(n-1))

cat(sprintf("\nF-tabel = %.4f\n", F_tabel))

F-tabel = 3.3541
R
# Menguji apakah dadu seimbang
set.seed(123)
dadu <- sample(1:6, 120, replace = TRUE)

# Frekuensi observasi
obs <- as.numeric(table(dadu))

# Uji Chi-squared
result_chi <- chisq.test(obs, p = rep(1/6, 6))

cat("Hasil Uji Chi-Squared:\n")
Hasil Uji Chi-Squared:
print(result_chi)

    Chi-squared test for given probabilities

data:  obs
X-squared = 5.6, df = 5, p-value = 0.3471
# Chi-squared tabel
k <- 6
alpha <- 0.05
chi_sq_tabel <- qchisq(1-alpha, df = k-1)

cat(sprintf("\nChi-squared tabel = %.4f\n", chi_sq_tabel))

Chi-squared tabel = 11.0705
R
# Menguji independensi treatment dan improvement
treatment_url <- "https://raw.githubusercontent.com/selva86/datasets/master/treatment.csv"

treatment <- read.csv(treatment_url)

# Tabel kontingensi
tabel <- table(treatment$treatment, treatment$improved)

# Uji Chi-squared
result <- chisq.test(tabel)
print(result)

6. Korelasi dan Regresi

6.1 Korelasi

Menghitung Korelasi:

R
data(mtcars)

# Korelasi semua variabel (Pearson)
cor_matrix <- cor(mtcars)
round(cor_matrix, 2)
       mpg   cyl  disp    hp  drat    wt  qsec    vs    am  gear  carb
mpg   1.00 -0.85 -0.85 -0.78  0.68 -0.87  0.42  0.66  0.60  0.48 -0.55
cyl  -0.85  1.00  0.90  0.83 -0.70  0.78 -0.59 -0.81 -0.52 -0.49  0.53
disp -0.85  0.90  1.00  0.79 -0.71  0.89 -0.43 -0.71 -0.59 -0.56  0.39
hp   -0.78  0.83  0.79  1.00 -0.45  0.66 -0.71 -0.72 -0.24 -0.13  0.75
drat  0.68 -0.70 -0.71 -0.45  1.00 -0.71  0.09  0.44  0.71  0.70 -0.09
wt   -0.87  0.78  0.89  0.66 -0.71  1.00 -0.17 -0.55 -0.69 -0.58  0.43
qsec  0.42 -0.59 -0.43 -0.71  0.09 -0.17  1.00  0.74 -0.23 -0.21 -0.66
vs    0.66 -0.81 -0.71 -0.72  0.44 -0.55  0.74  1.00  0.17  0.21 -0.57
am    0.60 -0.52 -0.59 -0.24  0.71 -0.69 -0.23  0.17  1.00  0.79  0.06
gear  0.48 -0.49 -0.56 -0.13  0.70 -0.58 -0.21  0.21  0.79  1.00  0.27
carb -0.55  0.53  0.39  0.75 -0.09  0.43 -0.66 -0.57  0.06  0.27  1.00
# Korelasi antara dua variabel
cor_disp_wt <- cor(mtcars$disp, mtcars$wt)
cat(sprintf("Korelasi disp dan wt: %.4f\n", cor_disp_wt))
Korelasi disp dan wt: 0.8880

Visualisasi Korelasi (Heatmap):

R
install.packages("corrplot")
R
library(corrplot)
R
# Plot heatmap
corrplot(cor_matrix, method = "color", type = "upper",
         tl.col = "black", tl.srt = 45,
         addCoef.col = "black", number.cex = 0.7,
         title = "Correlation Heatmap",
         mar = c(0,0,1,0))

Uji Korelasi:

R
# H0: ρ = 0 (tidak ada korelasi)
# H1: ρ ≠ 0 (ada korelasi)

result_cor <- cor.test(mtcars$disp, mtcars$wt)

cat("Hasil Uji Korelasi:\n")
Hasil Uji Korelasi:
print(result_cor)

    Pearson's product-moment correlation

data:  mtcars$disp and mtcars$wt
t = 10.576, df = 30, p-value = 1.222e-11
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.7811586 0.9442902
sample estimates:
      cor 
0.8879799

Visualisasi dengan ggscatter:

R
install.packages("ggpubr")
R
library(ggpubr)
R
ggscatter(mtcars, x = "disp", y = "wt",
          add = "reg.line",
          cor.coef = TRUE, cor.method = "pearson",
          xlab = "Displacement", ylab = "Weight",
          title = "Correlation between Displacement and Weight")

6.2 Regresi Linear Sederhana

Membuat Model Regresi:

R
# Model: wt = β₀ + β₁(disp) + ε
model <- lm(wt ~ disp, data = mtcars)

# Melihat ringkasan
summary(model)

Call:
lm(formula = wt ~ disp, data = mtcars)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.89044 -0.29775 -0.00684  0.33428  0.66525 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 1.5998146  0.1729964   9.248 2.74e-10 ***
disp        0.0070103  0.0006629  10.576 1.22e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4574 on 30 degrees of freedom
Multiple R-squared:  0.7885,    Adjusted R-squared:  0.7815 
F-statistic: 111.8 on 1 and 30 DF,  p-value: 1.222e-11

Koefisien Regresi:

R
# Ekstrak koefisien
beta_0 <- coef(model)[1]
beta_1 <- coef(model)[2]

cat(sprintf("Persamaan regresi: wt = %.4f + %.6f × disp\n", beta_0, beta_1))
Persamaan regresi: wt = 1.5998 + 0.007010 × disp
cat(sprintf("\nInterpretasi:\n"))

Interpretasi:
cat(sprintf("- Intercept (β₀) = %.4f\n", beta_0))
- Intercept (β₀) = 1.5998
cat(sprintf("- Slope (β₁) = %.6f\n", beta_1))
- Slope (β₁) = 0.007010
cat(sprintf("- Setiap kenaikan 1 unit displacement, rata-rata weight naik %.6f unit\n", beta_1))
- Setiap kenaikan 1 unit displacement, rata-rata weight naik 0.007010 unit

Visualisasi Regresi:

R
# Base R
plot(mtcars$disp, mtcars$wt,
     main = "Weight vs Displacement",
     xlab = "Displacement",
     ylab = "Weight",
     pch = 19, col = "blue")
abline(model, col = "red", lwd = 2)

# Menambahkan equation ke plot
legend("topleft", 
       legend = sprintf("wt = %.2f + %.4f × disp", beta_0, beta_1),
       bty = "n", cex = 0.9)

Dengan ggplot2:

R
ggplot(mtcars, aes(x = disp, y = wt)) +
  geom_point(size = 3, color = "blue") +
  geom_smooth(method = "lm", se = TRUE, color = "red") +
  labs(title = "Linear Regression: Weight vs Displacement",
       x = "Displacement (cu.in.)",
       y = "Weight (1000 lbs)",
       subtitle = sprintf("wt = %.2f + %.4f × disp", beta_0, beta_1)) +
  theme_minimal()
`geom_smooth()` using formula = 'y ~ x'

Diagnostik Model:

R
# Plot diagnostik
par(mfrow = c(2, 2))
plot(model)

par(mfrow = c(1, 1))

# R-squared
r_squared <- summary(model)$r.squared
adj_r_squared <- summary(model)$adj.r.squared

cat(sprintf("\nGoodness of Fit:\n"))

Goodness of Fit:
cat(sprintf("R-squared: %.4f\n", r_squared))
R-squared: 0.7885
cat(sprintf("Adjusted R-squared: %.4f\n", adj_r_squared))
Adjusted R-squared: 0.7815
cat(sprintf("Interpretasi: %.2f%% variasi weight dijelaskan oleh displacement\n", 
            r_squared * 100))
Interpretasi: 78.85% variasi weight dijelaskan oleh displacement

Prediksi:

R
# Prediksi untuk nilai baru
new_data <- data.frame(disp = c(150, 200, 250))
predictions <- predict(model, newdata = new_data, interval = "prediction")

cat("\nPrediksi Weight untuk Displacement baru:\n")

Prediksi Weight untuk Displacement baru:
result_df <- cbind(new_data, predictions)
print(result_df)
  disp      fit      lwr      upr
1  150 2.651363 1.696444 3.606283
2  200 3.001880 2.052322 3.951437
3  250 3.352396 2.403390 4.301401

Kesimpulan

Luar biasa! Anda telah menyelesaikan review komprehensif materi R untuk Pengantar Sains Data. Mari kita rangkum perjalanan pembelajaran Anda:

Keterampilan yang Telah Dikuasai:

  1. Fundamental Programming 💻
    • Operasi dasar, variabel, dan vector di R
    • Import dan manipulasi data
    • Subsetting dan filtering data
    • Perhitungan statistik deskriptif
  2. Data Visualization 📊
    • Visualisasi univariat (histogram, bar chart, pie chart)
    • Visualisasi bivariat (scatter plot, boxplot, line chart)
    • Visualisasi advanced dengan ggplot2
    • Interpretasi visual patterns
  3. Probability & Distributions 🎲
    • Distribusi diskrit (Bernoulli, Binomial, Poisson)
    • Distribusi kontinu (Normal, Uniform)
    • PMF/PDF, CDF, quantile, dan random generation
    • Package distr untuk manipulasi distribusi
  4. Statistical Inference 📈
    • Looping dan simulasi distribusi sampling
    • Central Limit Theorem dan aplikasinya
    • Taksiran titik dan interval
    • Pengujian hipotesis (z-test, t-test, ANOVA, Chi-squared)
  5. Relationship Analysis 🔗
    • Korelasi (Pearson, Spearman, Kendall)
    • Visualisasi korelasi dengan heatmap
    • Regresi linear sederhana
    • Interpretasi dan evaluasi model

Selamat! Anda telah menguasai toolkit fundamental R untuk data science. Terus berlatih dan jangan takut untuk bereksperimen! 🎉

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