6  Koefisien Korelasi

6.1 Koefisien Korelasi Sederhana

\[ r_{XY}=\frac{n\Sigma{XY}-\Sigma{X}\Sigma{Y}}{\sqrt{[n\Sigma{X^2}-(\Sigma{X^2})][n\Sigma{Y^2}-(\Sigma{Y^2})]}} \]

dimana

  • \(r_{XY}\) adalah koefisien korelasi antara \(X\) dan \(Y\)

  • \(n\) adalah ukuran sampel

  • \(X\) menunjukkan skor individu pada variabel \(X\)

  • \(Y\) menunjukkan skor individu pada variabel \(Y\)

  • \(XY\) adalah hasil kali setiap skor \(X\) dan skor \(Y\)

  • \(X^2\) adalah kuadrat skor \(X\) individu

  • \(Y^2\) adalah kuadrat skor \(Y\) individu

6.1.1 Menghitung korelasi ESCS dengan skor matematika

\(X\) = ESCS

\(Y\) = Skor matematika

Table 6.1: Skor ESCS dan matematika siswa
Matematika ESCS X^2 Y^2 XY
466.4 0.1
306.7 -2.2
496.6 -2.1
298.4 -1.4
349.8 -0.4
463.2 -1.4
442.0 -1.6
322.6 -2.2
327.4 -3.5
380.4 -1.4
Sumber: PISA Indonesia 2022
MATH ESCS X^2 Y^2 XY
466.4 0.1 0.01 217528.96 46.64
306.7 -2.2 4.84 94064.89 -674.74
496.6 -2.1 4.41 246611.56 -1042.86
298.4 -1.4 1.96 89042.56 -417.76
349.8 -0.4 0.16 122360.04 -139.92
463.2 -1.4 1.96 214554.24 -648.48
442.0 -1.6 2.56 195364.00 -707.20
322.6 -2.2 4.84 104070.76 -709.72
327.4 -3.5 12.25 107190.76 -1145.90
380.4 -1.4 1.96 144704.16 -532.56
Jumlah 3,854 −16 35 1,535,492 −5,972

  • \(\Sigma{X}\) = \(-16\)

  • \(\Sigma{Y}\) = \(3854\)

  • \(\Sigma{X^2}\) = \(35\)

  • \(\Sigma{Y^2}\) = \(1535492\)

  • \(\Sigma{XY}\) = \(-5972\)

\[ r_{XY}=\frac{n\Sigma{XY}-\Sigma{X}\Sigma{Y}}{\sqrt{[n\Sigma{X^2}-(\Sigma{X^2})][n\Sigma{Y^2}-(\Sigma{Y^2})]}} \]

\[ r_{XY}=\frac{(10 \times -5972.5)-(-16.1\times 3853.5)}{[(10\times 34.95)-(-16.1)^2][(10\times 1535492)-3853.5^2]} \]

\[ r_{XY}=\frac{2316.35}{6755.576}=0.343 \]

6.2 Scatterplot

ggplot(pisa, aes(x = ESCS, y = MATH)) + 
    geom_point()

6.3 Correlation Matrix

cor(pisa_22)
             ESCS        MATH        age          sex       growth
ESCS   1.00000000  0.28096883 0.08417389  0.017629678  0.129667961
MATH   0.28096883  1.00000000 0.04676718 -0.066554009  0.326085699
age    0.08417389  0.04676718 1.00000000  0.034307427  0.023934173
sex    0.01762968 -0.06655401 0.03430743  1.000000000 -0.004227732
growth 0.12966796  0.32608570 0.02393417 -0.004227732  1.000000000

Korelasi

Hubungan

0.8 - 1

Sangat kuat

0.6 - 0.8

Kuat

0.4 - 0.6

Moderat

0.2 - 0.4

Lemah

0 - 0.2

Sangat lemah atau tidak ada hubungan

6.4 Korelasi Menggunakan fungsi pcor()

library(ppcor)
Loading required package: MASS

Attaching package: 'MASS'
The following object is masked from 'package:dplyr':

    select
pcor(pisa_22)
$estimate
             ESCS        MATH         age         sex      growth
ESCS   1.00000000  0.25408411 0.072488957  0.03456075 0.040761670
MATH   0.25408411  1.00000000 0.023765420 -0.07689320 0.304522960
age    0.07248896  0.02376542 1.000000000  0.03475594 0.005523948
sex    0.03456075 -0.07689320 0.034755938  1.00000000 0.016759634
growth 0.04076167  0.30452296 0.005523948  0.01675963 1.000000000

$p.value
               ESCS         MATH         age         sex       growth
ESCS   0.000000e+00 1.633756e-20 0.009093995 0.214090418 1.427902e-01
MATH   1.633756e-20 0.000000e+00 0.393000181 0.005649785 3.596530e-29
age    9.093995e-03 3.930002e-01 0.000000000 0.211512538 8.426397e-01
sex    2.140904e-01 5.649785e-03 0.211512538 0.000000000 5.469473e-01
growth 1.427902e-01 3.596530e-29 0.842639722 0.546947341 0.000000e+00

$statistic
           ESCS       MATH       age        sex     growth
ESCS   0.000000  9.4427938 2.6124450  1.2430080  1.4663726
MATH   9.442794  0.0000000 0.8544751 -2.7720874 11.4916936
age    2.612445  0.8544751 0.0000000  1.2500366  0.1985580
sex    1.243008 -2.7720874 1.2500366  0.0000000  0.6024996
growth 1.466373 11.4916936 0.1985580  0.6024996  0.0000000

$n
[1] 1297

$gp
[1] 3

$method
[1] "pearson"