4) Mengolah Data Kelompok a) Ukuran Pemusatan 1) Mean (Nilai Rata-rata)
Statistika
A. Pengertian Statistika
Definisi
Data adalah seluruh keterangan, informasi, atau fakta tentang sesuatu atau permasalahan. Statistika adalah ilmu yang berhubungan dengan cara-cara mengumpulkan, mengurutkan, menyajikan, dan mengolah data.
1) Mengumpulkan Data
79, 80, 70, 68, 92, 48, 90, 92, 85, 76, 48, 90, 92, 85, 76, 88, 78, 74, 70, 38, 80, 63, 76, 49, 84, 61, 83, 88, 81, 82, 61, 83, 88, 81, 82, 51, 71, 72, 82, 70, 81, 91, 56, 65, 63, 74, 89, 73, 90, 97, 60, 66, 98, 93, 81, 93 72, 91, 67, 88, 75, 83, 79, 86
2) Mengurutkan Data
38, 48, 49, 51, 56, 60, 60, 61, 63, 63, 63, 65, 66, 67, 67, 68, 70, 70, 70, 71, 71, 72, 72, 72, 73, 74,74, 75, 75, 76, 76, 78, 79, 79, 80, 80, 80, 81, 81, 81, 82, 82, 83, 83, 84, 85, 86, 87, 88, 88, 88, 89, 90, 90, 90, 91, 91, 91, 92, 93, 93, 93, 97, 98
3) Menyajikan Data
a) Tabel
Banyak data: = 64
(1) Nilai Terendah
Nilai Terendah =
terkecil
Contoh diatas: Nilai Terendah = 98
(2) Nilai Tertinggi
Nilai Tertinggi =
terbesar
Contoh diatas: Nilai Tertinggi = 38
(3) Jangkauan
Jangkauan = −
terbesar terkecil
Contoh diatas: Jangkauan = 98 − 38 = 60
(4) Banyak Kelas
Banyak Kelas = 1 + 3,3 ⋅ log Contoh diatas: Banyak Kelas = 1 + 3,3 ⋅ log
= 1 + 3,3 ⋅ log 64 = 1 + 3,3 ⋅ 1,81 = 1 + 5,97 = 6,97
≈ 7
1
96 6 576 Jumlah 64 4875
87 16 1392 7. 92-100
78 17 1326 6. 83-91
69 14 966 5. 74-82
60 7 420 4. 65-73
51 3 153 3. 56-64
42 2. 47-55
42
(5) Panjang Kelas
) Nilai Tengah ( ) Frekuensi ( ) 1. 38-46
No ( ) Panjang Kelas (
= 8,57 ≈ 9
7
60
=
Banyak Kelas Contoh diatas:
= Jangkauan
b) Diagram (1) Diagram Garis (2) Diagram Batang
(3) Diagram Lingkaran
1 Nilai 42 = = 5,625 64 × 360
3 Nilai 51 = = 16,875 64 × 360
7 Nilai 60 = = 39,375 64 × 360
14 Nilai 69 = = 78,75 64 × 360
17 Nilai 78 = = 95,625 64 × 360
16 Nilai 87 = = 90 64 × 360
6 Nilai 96 = = 33,75 64 × 360
4) Mengolah Data Kelompok
a) Ukuran Pemusatan 1) Mean (Nilai Rata-rata)
( )+( )+( )+⋯+( ) 1 ⋅
1 2 ⋅
2 3 ⋅ 3 ⋅
̅ =
1
2 + + 3 +⋯+
Contoh: Diketahui: = 64
No ( ) Nilai Tengah ( ) Frekuensi ( ) ) Panjang Kelas (
1. 38-46
42
1
42 2. 47-55
51 3 153 3. 56-64
60 7 420 4. 65-73
69 14 966 5. 74-82
78 17 1326 6. 83-91
87 16 1392 7. 92-100
96 6 576 Jumlah 64 4875
- 2
- 3
- 4
- 5
- 6
7
14
= 74 − 0,5 = 73,5 = 9
= + (
1
2 −
) = 73,5 + (
1
2 (64)−25
14
) (9) = 73,5 + (
32−25
) (9) = 73,5 + (
96 6 576 Jumlah 64 4875
7
14
) (9) = 73,5 + (
1
2
) (9) = 73,5 + 4,5 = 78
3) Modus (Nilai Sering Muncul atau Frekuensi Terbesar)
= + (
1
1
= 1 + 3 + 7 + 14 = 25 = 17
87 16 1392 7. 92-100
)
Keterangan: = Median = Frekuensi kumulatif sebelum kelas median
̅ =
( 1 ⋅ 1 )+( 2 ⋅ 2 )+( 3 ⋅ 3 )+( 4 ⋅ 4 )+( 5 ⋅ 5 )+( 6 ⋅ 6 )+( 7 ⋅ 7 )
1
̅ =
4875
64
= 76,17
2) Median (Nilai Tengah)
= + (
1 2 − )
= Frekuensi kelas median = Tepi bawah
6. 83-91
= Panjang kelas Contoh: Jika
= 32, maka Letak data median = 32 No (
) Panjang Kelas ( ) Nilai Tengah ( ) Frekuensi ( )
1. 38-46
42
1
42 2. 47-55
51 3 153 3. 56-64
60 7 420 4. 65-73
69 14 966 5. 74-82
78 17 1326
- 2
- 2
3
1
96 6 576 Jumlah 64 4875
1
= 17 − 14 = 3
2
= 17 − 16 = 1 = 74 − 0,5 = 73,5
= 9 = + (
1
6. 83-91
) = 73,5 + (
3 3+1
) (9) = 73,5 + (
= 73,5 + 6,75 = 80,25
4
) (9) = 73,5 +
87 16 1392 7. 92-100
78 17 1326
4
17, maka No (
Keterangan: = Modus
1
= Selisih frekuensi kelas modus dengan frekuensi kelas sebelumnya
2
= Selisih frekuensi kelas modus dengan frekuensi kelas sesudahnya = Tepi bawah
= Panjang kelas Contoh: Jika frekuensi terbesar adalah
) Panjang Kelas ( ) Nilai Tengah ( ) Frekuensi ( )
69 14 966 5. 74-82
1. 38-46
42
1
42 2. 47-55
51 3 153 3. 56-64
60 7 420 4. 65-73
27
b) Ukuran Letak (1) Kuartil
1
1
4
( + 1) = + (
4 −
) , dimana
= 1,2,3; dan = banyak data
(a) Kuartil Bawah ( )
Letak data =
=
4 −
- (
1
1
1
)
(b) Kuartil Tengah ( ) atau Median
2
=
2
1
2 −
2
- (
- (
2
)
= 64,5 + 3,86 = 68,36
51 3 153 3. 56-64
96 6 576 Jumlah 64 4875
7. 92-100
87
16 13926. 83-91
78
17 13265. 74-82
69
14 96660 7 420 4. 65-73
42 2. 47-55
=
1
42
) Nilai Tengah ( ) Frekuensi ( ) 1. 38-46
No ( ) Panjang Kelas (
4 = 48,75
195
4 (64 + 1) =
3
4
( + 1) =
1
1
3 =
) (9) = 64,5 + (
7
27
) (9) = 64,5 +
7
3
) (9) = 64,5 + (
14
6
14
1
16−10
) (9) = 64,5 + (
14
4 (64)−10
1
) = 64,5 + (
1
1
4 −
3
Letak data
(c) Kuartil Atas ( )
2
42 2. 47-55
1
42
1. 38-46
) Panjang Kelas ( ) Nilai Tengah ( ) Frekuensi ( )
dari data: No (
3
, dan
,
60 7 420 4. 65-73
1
) Contoh: Tentukan
3
3
4 −
3
3
=
3
51 3 153 3. 56-64
69 14 966 5. 74-82
2 = 32,5
65
65
2 (64 + 1) =
1
2
( + 1) =
1
2 =
Letak data
4 = 16,25
4 (64 + 1) =
78 17 1326 6. 83-91
1
4
( + 1) =
1
1 =
( + 1) Letak data
4
Jawab: Letak data =
96 6 576 Jumlah 64 4875
87 16 1392 7. 92-100
- (
1 −
2
2
= + ( )
2
2
1
1 (64)−25
2
= 73,5 + ( ) (9)
17 32−25
= 73,5 + ( ) (9)
17
7
= 73,5 + ( ) (9)
17
63
= 73,5 +
17
= 73,5 + 3,71 = 77,21
3 −
3
4
= + ( )
3
3
1
3 (64)−42
4
= 82,5 + ( ) (9)
16 48−42
= 82,5 + ( ) (9)
16
6
= 82,5 + ( ) (9)
16
54
= 82,5 +
16
= 82,5 + 3,38 = 85,88
(2) Desil
Letak data = ( + 1)
10 −
10
, dimana = + ( ) = 1,2,3, ⋯ ,9; dan = banyak data
Contoh: Tentukan , , dan dari data:
4
7
9
No ( ) Nilai Tengah ( ) Frekuensi ( )) Panjang Kelas ( 1. 38-46
42
1
42 2. 47-55
51 3 153 3. 56-64
60 7 420 4. 65-73
69 14 966 5. 74-82
78 17 1326 6. 83-91
87 16 1392 7. 92-100
96 6 576 Jumlah 64 4875
Jawab: Letak data =
( + 1)
10
4 2 130
Letak data
= = 26 ( + 1) = (64 + 1) =
4
10
5
5
7 7 455
Letak data
= ( + 1) = (64 + 1) = = 45,5
7
10
10
10
9 9 585
Letak data
= = 58,5 ( + 1) = (64 + 1) =
9
10
10
10 No ( ) Nilai Tengah ( ) Frekuensi ( ) ) Panjang Kelas (
1. 38-46
42
1
42 2. 47-55
51 3 153 3. 56-64
60 7 420 4. 65-73
69 14 966 5. 74-82
78
17 13266. 83-91
87
16 13927. 92-100
96
6 576Jumlah 64 4875
4 −
4
10
= + ( )
4
4
4
4 (64)−25
10
= 73,5 + ( ) (9)
17 25,6−25
= 73,5 + ( ) (9)
17 0,6
= 73,5 + ( ) (9)
17 0,54
= 73,5 +
17
= 73,5 + 0,03 = 73,53
7 −
7
10
= + ( )
7
7
7
7 (64)−42
10
= 82,5 + ( ) (9)
16 44,8−42
= 82,5 + ( ) (9)
16 2,8
= 82,5 + ( ) (9)
16 25,2
= 82,5 +
16
= 82,5 + 1,58 = 84,08
9 −
9
10
= + ( )
9
9
9
9 (64)−58
10
= 91,5 + ( ) (9)
16 57,6−58
= 91,5 + ( ) (9)
16 −0,4
= 91,5 + ( ) (9)
16 3,6
= 82,5 −
16
= 82,5 − 0,23 = 82,27
c) Ukuran Penyebaran (1) Jangkauan
4 = 48,75
51 3 153 3. 56-64
42 2. 47-55
1
42
) Nilai Tengah ( ) Frekuensi ( ) 1. 38-46
No ( ) Panjang Kelas (
195
69 14 966
4 (64 + 1) =
3
4 ( + 1) =
3
3 =
Letak data
2 = 32,5
60 7 420 4. 65-73
5. 74-82
2 (64 + 1) =
4 −
) (9)
14
4 (64)−10
1
) = 64,5 + (
1
1
1
78 17 1326
1
=
1
96 6 576 Jumlah 64 4875
7. 92-100
87 16 1392
6. 83-91
65
1
38, 48, 49, 51, 56, 60, 60, 61, 63, 63, 63, 65, 66, 67, 67, 68, 70, 70, 70, 71, 71, 72, 72, 72, 73, 74,74, 75, 75, 76, 76, 78, 79, 79, 80, 80, 80, 81, 81, 81, 82, 82, 83, 83, 84, 85, 86, 87, 88, 88, 88, 89, 90, 90, 90, 91, 91, 91, 92, 93, 93, 93, 97, 98 Contoh diatas: Jangkauan = 98 − 38 = 60
,
1
42
1. 38-46
) Panjang Kelas ( ) Nilai Tengah ( ) Frekuensi ( )
, dan Simpangan Kuartil dari data: No (
3
2
51 3 153 3. 56-64
,
1
1 Contoh: Tentukan
−
3
Simpangan Kuartil =
(2) Simpangan Kuartil
42 2. 47-55
60 7 420 4. 65-73
2 ( + 1) =
4 ( + 1) =
1
2 =
Letak data
4 = 16,25
65
4 (64 + 1) =
1
1
69 14 966 5. 74-82
1 =
( + 1) Letak data
4
Jawab: Letak data =
96 6 576 Jumlah 64 4875
87 16 1392 7. 92-100
78 17 1326 6. 83-91
- (
16−10
= 64,5 + ( ) (9)
14
6
= 64,5 + ( ) (9)
14
3
= 64,5 + ( ) (9)
7
27
= 64,5 +
7
= 64,5 + 3,86 = 68,36
1 −
2
2
= + ( )
2
2
1
1 (64)−25
2
= 73,5 + ( ) (9)
17 32−25
= 73,5 + ( ) (9)
17
7
= 73,5 + ( ) (9)
17
63
= 73,5 +
17
= 73,5 + 3,71 = 77,21
3 −
3
4
= + ( )
3
3
1
3 (64)−42
4
= 82,5 + ( ) (9)
16 48−42
= 82,5 + ( ) (9)
16
6
= 82,5 + ( ) (9)
16
54
= 82,5 +
16
= 82,5 + 3,38 = 85,88
Simpangan Kuartil = −
3
1 Simpangan Kuartil = 85,88 − 68,36
Simpangan Kuartil = 17,52
(3) Simpangan Rata-rata
− ̅|] + [ − ̅|] + [ − ̅|] + ⋯ + [ − ̅|] [ | | | |
1
1
2
2
3
3
=
(4) Simpangan Baku (Varian)
2
2
2
2 [ ( − ̅) ] + [ ( − ̅) ] + [ ( − ̅) ] + ⋯ + [ ( − ̅) ]
1
1
2
2
3
3 = √ Contoh: Diketahui: ̅ = 76,17; tentukan simpangan rata-rata dan sinpangan bakunya!
2
2
̅ − ̅ | − ̅| ( − ̅) − ̅| − ̅) | ( 42 76,17 −34,17 34,17 11167,59
1 34,17 11167,59 51 76,17 −25,17 25,17 633,53 3 75,51 1900,59 60 76,17 −16,17 16,17 261,47 7 113,19 1830,29 69 76,17 −7,17 7,17 51,41
14 100,38 719,74 78 76,17 1,83 1,83 3,34 17 31,11 56,78 87 76,17 10,83 10,83 117,29 16 173,28 173,28 96 76,17 13,83 13,83 191,27
6 82,98 1147,62 Jumlah
= 64 610,62 16995,89 [ | − ̅|] + [ | − ̅|] + [ | − ̅|] + ⋯ + [ | − ̅|]
1
1
2
2
3
3
= 610,62
=
64 = 9,54
16995,89
= √
64 = √2665,56 = 51,63