Ukuran Penyebaran Data
Ukuran penyebaran data terdiri atas:
1. Simpangan rata-rata
\(SR = \dfrac {\Sigma \: |x_i - \bar x |}{n}\)
\(SR = \dfrac {\Sigma \: \left(f \:.\: |x_i - \bar x |\right)}{n}\)
2. Ragam (Varians)
\(\sigma^2 = \dfrac {\Sigma \: |x_i - \bar x |^2}{n}\)
\(\sigma^2 = \dfrac {\Sigma \: \left( f_i \:.\: |x_i - \bar x |^2\right)}{n}\)
\(\sigma^2 = \dfrac {\Sigma \: x_i^2}{n} - \left(\dfrac {\Sigma \: x_i}{n} \right)^2\)
\(\sigma^2 = \dfrac {\Sigma \: f_i \:.\: x_i^2}{n} - \left(\dfrac {\Sigma \: f \:.\: x_i}{n} \right)^2\)
3. Simpangan baku (standar deviasi)
Simpangan baku (\(\sigma\)) adalah akar dari varians.
Contoh
Tentukan nilai simpangan rata-rata, varians dan simpangan baku dari data di bawah ini:
| Nilai | Frekuensi |
| 1 - 3 | 5 |
| 4 - 6 | 12 |
| 7 - 9 | 13 |
| 10 - 12 | 9 |
Pada tabel, tambahkan kolom \((x_i - \bar x)\) dan \((x_i - \bar x)^2\)
| Nilai | \(x_i\) | \((x_i - \bar x)\) | \((x_i - \bar x)^2\) | \(f_i\) | \(f_i \:.\: x_i\) | \(f_i \:.\: (x_i - \bar x)\) | \(f_i \:.\: (x_i - \bar x)^2\) |
| 1 - 3 | 2 | −5 | 25 | 5 | 10 | −25 | 125 |
| 4 - 6 | 5 | −2 | 4 | 12 | 60 | −24 | 48 |
| 7 - 9 | 8 | 1 | 1 | 13 | 104 | 13 | 13 |
| 10 - 12 | 11 | 4 | 16 | 9 | 99 | 36 | 144 |
| \(\Sigma = 39\) | \(\Sigma = 273\) | \(\Sigma = 0\) | \(\Sigma = 330\) |
Rata-rata
\(\bbox[5px, border: 2px solid magenta] {\bar x = \dfrac {\Sigma \: f_i \:.\: x_i}{\Sigma \: f} = \dfrac {273}{39} = 7}\)
Simpangan rata-rata
\(\bbox[5px, border: 2px solid magenta] {SR = \dfrac {\Sigma \: \left(f_i \:.\: |x_i - \bar x |\right)}{\Sigma \: f} = \dfrac {0}{39} = 0}\)
Varians
\(\bbox[5px, border: 2px solid magenta] {\sigma^2 = \dfrac {\Sigma \: \left( f_i \:.\: |x_i - \bar x |^2\right)}{\Sigma \: f} = \dfrac {330}{39} = 8,46}\)
Simpangan baku
\(\bbox[5px, border: 2px solid magenta] {\sigma = \sqrt{8,46} = 2,91}\)
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