Tentukan simpangan rata-rata, simpangan baku, dan ragam

Simpangan Rata-rata ≈ 10,66; Simpangan Baku ≈ 13,35; Ragam ≈ 178,22.

Pembahasan

Untuk menentukan simpangan rata-rata, simpangan baku, dan ragam dari data hasil ujian matematika 30 siswa, kita perlu melakukan beberapa langkah perhitungan:

1.  **Menentukan Nilai Tengah (xi) setiap kelas interval:**
    *   21-30: (21+30)/2 = 25,5
    *   31-40: (31+40)/2 = 35,5
    *   41-50: (41+50)/2 = 45,5
    *   51-60: (51+60)/2 = 55,5
    *   61-70: (61+70)/2 = 65,5
    *   71-80: (71+80)/2 = 75,5
    *   81-90: (81+90)/2 = 85,5

2.  **Menghitung Rata-rata (Mean) (ar{x}):**
    *   Jumlah siswa (f) = 1 + 1 + 3 + 10 + 8 + 5 + 2 = 30
    *   Jumlah (f * xi) = (1*25,5) + (1*35,5) + (3*45,5) + (10*55,5) + (8*65,5) + (5*75,5) + (2*85,5)
        = 25,5 + 35,5 + 136,5 + 555 + 524 + 377,5 + 171 = 1825
    *   $ar{x} = \sum(f 	imes xi) / \sum f = 1825 / 30 \approx 60,83$

3.  **Menghitung Simpangan Rata-rata (SR):**
    *   SR = $\sum |xi - \bar{x}| f / \sum f$
    *   $|25,5 - 60,83| 	imes 1 = 35,33 	imes 1 = 35,33$
    *   $|35,5 - 60,83| 	imes 1 = 25,33 	imes 1 = 25,33$
    *   $|45,5 - 60,83| 	imes 3 = 15,33 	imes 3 = 45,99$
    *   $|55,5 - 60,83| 	imes 10 = 5,33 	imes 10 = 53,30$
    *   $|65,5 - 60,83| 	imes 8 = 4,67 	imes 8 = 37,36$
    *   $|75,5 - 60,83| 	imes 5 = 14,67 	imes 5 = 73,35$
    *   $|85,5 - 60,83| 	imes 2 = 24,67 	imes 2 = 49,34$
    *   $\sum |xi - \bar{x}| f = 35,33 + 25,33 + 45,99 + 53,30 + 37,36 + 73,35 + 49,34 = 319,9$
    *   SR = $319,9 / 30 \approx 10,66$

4.  **Menghitung Simpangan Baku (SB) (s):**
    *   SB = $\sqrt{\frac{\sum (xi - \bar{x})^2 f}{\sum f}}$
    *   $(25,5 - 60,83)^2 	imes 1 = (-35,33)^2 	imes 1 = 1248,2089 	imes 1 = 1248,2089$
    *   $(35,5 - 60,83)^2 	imes 1 = (-25,33)^2 	imes 1 = 641,6089 	imes 1 = 641,6089$
    *   $(45,5 - 60,83)^2 	imes 3 = (-15,33)^2 	imes 3 = 235,0089 	imes 3 = 705,0267$
    *   $(55,5 - 60,83)^2 	imes 10 = (-5,33)^2 	imes 10 = 28,4089 	imes 10 = 284,089$
    *   $(65,5 - 60,83)^2 	imes 8 = (4,67)^2 	imes 8 = 21,8089 	imes 8 = 174,4712$
    *   $(75,5 - 60,83)^2 	imes 5 = (14,67)^2 	imes 5 = 215,2089 	imes 5 = 1076,0445$
    *   $(85,5 - 60,83)^2 	imes 2 = (24,67)^2 	imes 2 = 608,6089 	imes 2 = 1217,2178$
    *   $\sum (xi - \bar{x})^2 f = 1248,2089 + 641,6089 + 705,0267 + 284,089 + 174,4712 + 1076,0445 + 1217,2178 = 5346,667
    *   Ragap = $\sum (xi - \bar{x})^2 f / \sum f = 5346,667 / 30 \approx 178,22$
    *   SB = $\sqrt{178,22} \approx 13,35$

5.  **Menghitung Ragam (Varians) ($s^2$):**
    *   Ragam adalah kuadrat dari simpangan baku.
    *   Ragam = $SB^2 \approx 13,35^2 \approx 178,22$

Jadi:
*   Simpangan Rata-rata: $\approx 10,66$
*   Simpangan Baku: $\approx 13,35$
*   Ragam: $\approx 178,22$