Soalan 3:
Hitung varians dan sisihan piawai bagi set data yang berikut.
(a) 7, 9, 11, 8, 3, 7
(b) 50, 72, 63, 58, 55, 50, 70, 62, 66, 64
(c) 3.2, 4.4, 3.9, 4.1, 5.2, 4.8, 5.2
(d) 20, 27, 32, 47, 50, 38, 42, 40, 33, 37, 30
Penyelesaian:
(a)
$$ \begin{aligned} \min , \bar{x} & =\frac{7+9+11+8+3+7}{6} \\ & =7.5 \end{aligned} $$
$$ \begin{aligned} &\text { Varians, } \sigma^2=\frac{\sum x^2}{N}-(\bar{x})^2\\ &\begin{aligned} & =\frac{7^2+9^2+11^2+8^2+3^2+7^2}{6}-(7.5)^2 \\ & =5.917 \end{aligned} \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, } \sigma=\sqrt{5.917}\\ &=2.432 \end{aligned} $$
(b)
$$ \begin{aligned} & \min , \bar{x} \\ & =\frac{50+72+63+58+55+50+70+62+66+64}{10} \\ & =61 \end{aligned} $$
$$ \begin{aligned} & \text { Varians, } \sigma^2 \\ & =\frac{50^2+72^2+63^2+58^2+55^2+50^2+70^2+62^2+66^2+64^2}{10}-(61)^2 \\ & =52.8 \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, } \sigma=\sqrt{52.8}\\ &=7.266 \end{aligned} $$
(c)
$$ \begin{aligned} \min , \bar{x} & =\frac{3.2+4.4+3.9+4.1+5.2+4.8+5.2}{7} \\ & =4.4 \end{aligned} $$
$$ \begin{aligned} &\text { Varians, } \sigma^2\\ &\begin{aligned} & =\frac{3.2^2+4.4^2+3.9^2+4.1^2+5.2^2+4.8^2+5.2^2}{7}-(4.4)^2 \\ & =0.46 \end{aligned} \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, } \sigma=\sqrt{0.46}\\ &=0.6782 \end{aligned} $$
(d)
$$ \begin{aligned} & \min , \bar{x} \\ & =\frac{20+27+32+47+50+38+42+40+33+37+30}{11} \\ & =36 \end{aligned} $$
$$ \begin{aligned} &\text { Varians, } \sigma^2\\ &\begin{aligned} & =\frac{20^2+27^2+32^2+47^2+50^2+38^2+42^2+40^2+33^2+37^2+30^2}{11}-(36)^2 \\ & =70.18 \end{aligned} \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, } \sigma=\sqrt{70.18}\\ &=8.377 \end{aligned} $$
Hitung varians dan sisihan piawai bagi set data yang berikut.
(a) 7, 9, 11, 8, 3, 7
(b) 50, 72, 63, 58, 55, 50, 70, 62, 66, 64
(c) 3.2, 4.4, 3.9, 4.1, 5.2, 4.8, 5.2
(d) 20, 27, 32, 47, 50, 38, 42, 40, 33, 37, 30
Penyelesaian:
(a)
$$ \begin{aligned} \min , \bar{x} & =\frac{7+9+11+8+3+7}{6} \\ & =7.5 \end{aligned} $$
$$ \begin{aligned} &\text { Varians, } \sigma^2=\frac{\sum x^2}{N}-(\bar{x})^2\\ &\begin{aligned} & =\frac{7^2+9^2+11^2+8^2+3^2+7^2}{6}-(7.5)^2 \\ & =5.917 \end{aligned} \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, } \sigma=\sqrt{5.917}\\ &=2.432 \end{aligned} $$
(b)
$$ \begin{aligned} & \min , \bar{x} \\ & =\frac{50+72+63+58+55+50+70+62+66+64}{10} \\ & =61 \end{aligned} $$
$$ \begin{aligned} & \text { Varians, } \sigma^2 \\ & =\frac{50^2+72^2+63^2+58^2+55^2+50^2+70^2+62^2+66^2+64^2}{10}-(61)^2 \\ & =52.8 \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, } \sigma=\sqrt{52.8}\\ &=7.266 \end{aligned} $$
(c)
$$ \begin{aligned} \min , \bar{x} & =\frac{3.2+4.4+3.9+4.1+5.2+4.8+5.2}{7} \\ & =4.4 \end{aligned} $$
$$ \begin{aligned} &\text { Varians, } \sigma^2\\ &\begin{aligned} & =\frac{3.2^2+4.4^2+3.9^2+4.1^2+5.2^2+4.8^2+5.2^2}{7}-(4.4)^2 \\ & =0.46 \end{aligned} \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, } \sigma=\sqrt{0.46}\\ &=0.6782 \end{aligned} $$
(d)
$$ \begin{aligned} & \min , \bar{x} \\ & =\frac{20+27+32+47+50+38+42+40+33+37+30}{11} \\ & =36 \end{aligned} $$
$$ \begin{aligned} &\text { Varians, } \sigma^2\\ &\begin{aligned} & =\frac{20^2+27^2+32^2+47^2+50^2+38^2+42^2+40^2+33^2+37^2+30^2}{11}-(36)^2 \\ & =70.18 \end{aligned} \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, } \sigma=\sqrt{70.18}\\ &=8.377 \end{aligned} $$
Soalan 4:
Suatu set data mengandungi tujuh nombor. Hasil tambah tujuh nombor ini ialah 84 dan hasil tambah kuasa dua bagi nombor-nombor ini ialah 1 920. Hitung varians dan sisihan piawai bagi set data ini.
Penyelesaian:
$$ \text { Diberi } \sum x=84, \sum x^2=1920 $$
$$ \min , \begin{aligned} \bar{x} & =\frac{\sum x}{N} \\ & =\frac{84}{7} \\ & =12 \end{aligned} $$
$$ \begin{aligned} \text { Varians } & =\frac{\sum x^2}{N}-(\bar{x})^2 \\ & =\frac{1920}{7}-(12)^2 \\ & =130.3 \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, }\\ &\begin{aligned} \sigma & =\sqrt{130.3} \\ & =11.41 \end{aligned} \end{aligned} $$
Suatu set data mengandungi tujuh nombor. Hasil tambah tujuh nombor ini ialah 84 dan hasil tambah kuasa dua bagi nombor-nombor ini ialah 1 920. Hitung varians dan sisihan piawai bagi set data ini.
Penyelesaian:
$$ \text { Diberi } \sum x=84, \sum x^2=1920 $$
$$ \min , \begin{aligned} \bar{x} & =\frac{\sum x}{N} \\ & =\frac{84}{7} \\ & =12 \end{aligned} $$
$$ \begin{aligned} \text { Varians } & =\frac{\sum x^2}{N}-(\bar{x})^2 \\ & =\frac{1920}{7}-(12)^2 \\ & =130.3 \end{aligned} $$
$$ \begin{aligned} &\text { Sisihan piawai, }\\ &\begin{aligned} \sigma & =\sqrt{130.3} \\ & =11.41 \end{aligned} \end{aligned} $$
Soalan 5:
Julat dan sisihan piawai suatu set nombor x1, x2, x3, ..., x10 ialah 10 dan 5.2 masing-masing.
Hitung
(a) julat dan sisihan piawai bagi set nombor
2x1, 2x2, 2x3, ..., 2x10,
$$ \begin{aligned} &\text { (b) julat dan sisihan piawai bagi set nombor }\\ &\frac{x_1-1}{4}, \frac{x_2-1}{4}, \frac{x_3-1}{4}, \ldots, \frac{x_{10}-1}{4} \end{aligned} $$
Penyelesaian:
(a)
$$ \begin{aligned} \text { Julat baharu } & =2\left(x_1, x_2, x_3, \ldots, x_{10}\right) \\ & =2(10) \\ & =20 \end{aligned} $$
$$ \begin{aligned} \text { Sisihan piawai baharu } & =2(5.2) \\ & =10.4 \end{aligned} $$
(b)
$$ \begin{aligned} \text { Julat baharu } & =1 / 4\left(x_1, x_2, x_3, \ldots, x_{10}\right) \\ & =1 / 4(10) \\ & =2.5 \end{aligned} $$
$$ \begin{aligned} \text { Sisihan piawai baharu } & =1 / 4(5.2) \\ & =1.3 \end{aligned} $$
Julat dan sisihan piawai suatu set nombor x1, x2, x3, ..., x10 ialah 10 dan 5.2 masing-masing.
Hitung
(a) julat dan sisihan piawai bagi set nombor
2x1, 2x2, 2x3, ..., 2x10,
$$ \begin{aligned} &\text { (b) julat dan sisihan piawai bagi set nombor }\\ &\frac{x_1-1}{4}, \frac{x_2-1}{4}, \frac{x_3-1}{4}, \ldots, \frac{x_{10}-1}{4} \end{aligned} $$
Penyelesaian:
(a)
$$ \begin{aligned} \text { Julat baharu } & =2\left(x_1, x_2, x_3, \ldots, x_{10}\right) \\ & =2(10) \\ & =20 \end{aligned} $$
$$ \begin{aligned} \text { Sisihan piawai baharu } & =2(5.2) \\ & =10.4 \end{aligned} $$
(b)
$$ \begin{aligned} \text { Julat baharu } & =1 / 4\left(x_1, x_2, x_3, \ldots, x_{10}\right) \\ & =1 / 4(10) \\ & =2.5 \end{aligned} $$
$$ \begin{aligned} \text { Sisihan piawai baharu } & =1 / 4(5.2) \\ & =1.3 \end{aligned} $$