# Statistik Data Majemuk

### Ukuran penyebaran data

###### 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

$$\bbox[5px, border: 2px solid magenta] {\sigma = \sqrt{8,46} = 2,91}$$

##### SOAL LATIHAN

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