Ukuran Pemusatan Data
Ukuran pemusatan data terdiri atas:
1. Rata-rata (Mean)
\(\bar x = \dfrac {\Sigma \: f_i \:.\: x_i}{\Sigma \: f} \)
\(\bar x = \bar x_s + \dfrac {\Sigma \: f_i \:.\: d_i}{\Sigma \: f} \)
\(\bar x = \bar x_s + \left(\dfrac {\Sigma \: f_i \:.\: u_i}{\Sigma \: f}\right) \:.\: c \)
\(\bar x_s\) = rata-rata sementara
\(d_i = x_i - \bar x_s\)
\(u_i = \dfrac {d_i}{c}\)
\(c\) = panjang kelas
2. Nilai tengah (Median)
\(Me = T_b + \left(\dfrac {\dfrac 12 n - \Sigma \: f_k}{f_k} \right) \:.\: c \)
\(T_b\) = tepi bawah kelas
\(f_k\) = frekuensi kelas
\(\Sigma \: f_k\) = jumlah frekuensi sebelum kelas
3. Nilai yang paling sering muncul (Modus)
\(Mo = T_b + \left(\dfrac {d_1}{d_1 + d_2} \right) \:.\: c \)
\(d_1\) = selisih antara frekuensi kelas dan frekuensi kelas sebelumnya
\(d_2\) = selisih antara frekuensi kelas dan frekuensi kelas sesudahnya
Contoh
Tentukan nilai Mean, Median dan Modus dari data di bawah ini:
| Nilai | Frekuensi |
| 56 - 60 | 3 |
| 61 - 65 | 8 |
| 66 - 70 | 22 |
| 71 - 75 | 40 |
| 76 - 80 | 10 |
| 81 - 85 | 9 |
| 86 - 90 | 8 |
Menentukan Mean dengan rumus dasar
| Nilai | \(x_i\) | \(f_i\) | \(f_i \:.\: x_i\) |
| 56 - 60 | 58 | 3 | 174 |
| 61 - 65 | 63 | 8 | 504 |
| 66 - 70 | 68 | 22 | 1496 |
| 71 - 75 | 73 | 40 | 2920 |
| 76 - 80 | 78 | 10 | 780 |
| 81 - 85 | 83 | 9 | 747 |
| 86 - 90 | 88 | 8 | 704 |
| \(\Sigma \: f = 100\) | \(\Sigma \: f_i \:.\: x_i = 7325\) |
\(\bbox[5px, border: 2px solid magenta] {\bar x = \dfrac {\Sigma \: f_i \:.\: x_i}{\Sigma \: f} = \dfrac {7325}{100} = 73,25}\)
Menentukan Mean dengan metode rata-rata sementara (\(\bar x_s = 73\))
| Nilai | \(x_i\) | \(d_i = x_i - x_s\) | \(f_i\) | \(f_i \:.\: d_i\) |
| 56 - 60 | 58 | −15 | 3 | −45 |
| 61 - 65 | 63 | −10 | 8 | −80 |
| 66 - 70 | 68 | −5 | 22 | −110 |
| 71 - 75 | \(\bbox[5px, border: 2px solid red] {73}\) | 0 | 40 | 0 |
| 76 - 80 | 78 | 5 | 10 | 50 |
| 81 - 85 | 83 | 10 | 9 | 90 |
| 86 - 90 | 88 | 15 | 8 | 120 |
| \(\Sigma \: f = 100\) | \(\Sigma \: f_i \:.\: d_i = 25\) |
\(\bbox[5px, border: 2px solid magenta] {\bar x = \bar x_s + \dfrac {\Sigma \: f_i \:.\: d_i}{\Sigma \: f} = 73 + \dfrac {25}{100} = 73,25}\)
Menentukan Mean dengan metode coding (\(\bar x_s = 73\))
| Nilai | \(x_i\) | \(d_i = x_i - x_s\) | \(u_i = \dfrac {d_i}{c}\) | \(f_i\) | \(f_i \:.\: u_i\) |
| 56 - 60 | 58 | −15 | −3 | 3 | −9 |
| 61 - 65 | 63 | −10 | −2 | 8 | −16 |
| 66 - 70 | 68 | −5 | −1 | 22 | −22 |
| 71 - 75 | \(\bbox[5px, border: 2px solid red] {73}\) | 0 | 0 | 40 | 0 |
| 76 - 80 | 78 | 5 | 1 | 10 | 10 |
| 81 - 85 | 83 | 10 | 2 | 9 | 18 |
| 86 - 90 | 88 | 15 | 3 | 8 | 24 |
| \(\Sigma \: f = 100\) | \(\Sigma \: f_i \:.\: u_i = 5\) |
\(\bbox[5px, border: 2px solid magenta] {\bar x = \bar x_s + \left(\dfrac {\Sigma \: f_i \:.\: u_i}{\Sigma \: f}\right) \:.\: c = 73 + \left(\dfrac {5}{100}\right) \:.\: 5 = 73,25}\)
Menentukan Median atau \(Q_2\)
Jumlah data = 100
Letak \(Q_2\) = \(\dfrac 12 \:.\: 100 = 50\)
Median terletak di antara data ke 50 dan 51, yang terletak pada kelas 71 - 75
| Nilai | \(x_i\) | \(f_i\) | |
| 56 - 60 | 58 | 3 | \({\color {red} \Sigma \: f_k}\)
\({ \color {red} 3 + 8 + 22 = 33 }\) |
| 61 - 65 | 63 | 8 | |
| 66 - 70 | 68 | 22 | |
| 71 - 75 | 73 | 40 | \({\color {red} f_k = 40}\) |
| 76 - 80 | 78 | 10 | |
| 81 - 85 | 83 | 9 | |
| 86 - 90 | 88 | 8 | |
| \(\Sigma \: f = 100\) |
\(\bbox[5px, border: 2px solid magenta] { Me = T_b + \left(\dfrac {\dfrac 12 n - \Sigma \: f_k}{f_k} \right) \:.\: c = 70,5 + \left(\dfrac {\dfrac 12 (100) - 33}{40} \right) \:.\: 5 = 70,5 + 2,125 = 72,625}\)
Menentukan Modus
Jumlah data = 100
Median terletak di antara data ke 50 dan 51, yang terletak pada kelas 71 - 75
| Nilai | \(x_i\) | \(f_i\) | |
| 56 - 60 | 58 | 3 | |
| 61 - 65 | 63 | 8 | |
| 66 - 70 | 68 | 22 | \({\color {red} d_1 = 40 - 22 = 18}\) |
| 71 - 75 | 73 | 40 | ⇐ Kelas modus |
| 76 - 80 | 78 | 10 | \({\color {red} d_2 = 40 - 10 = 30}\) |
| 81 - 85 | 83 | 9 | |
| 86 - 90 | 88 | 8 | |
| \(\Sigma \: f = 100\) |
\(\bbox[5px, border: 2px solid magenta] { Mo = T_b + \left(\dfrac {d_1}{d_1 + d_2} \right) \:.\: c = 70,5 + \left(\dfrac {18}{18 + 30} \right) \:.\: 5 = 70,5 + 1,875 = 72.375}\)
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