题目
Set Y contains elements {2, 4, 6, 8} which of the following sets have smaller standard deviation than set Y?
I. {22, 24, 26, 28}
II. {-4, -2, -8, -6, -10}
III. {0, 2, 4, 6}
解析
已知集合\(Y = \{2, 4, 6, 8\}\)
\(Y\)的均值 = \(\frac{4 + 6}{2} = 5\)
各项与均值的偏差 = \(\{2-5, 4-5, 6-5, 8-5\}\)
= \(\{-3, -1, 1, 3\}\)
### I. \(\{22, 24, 26, 28\}\)
均值 = \(\frac{24 + 26}{2} = 25\)
各项与均值的偏差 = \(\{22-25, 24-25, 26-25, 28-25\}\)
= \(\{-3, -1, 1, 3\}\)
因此,**标准差与\(Y\)相同**
### II. \(\{-4, -2, -8, -6, -10\}\)
按升序排列:\(\{-10, -8, -6, -4, -2\}\)
均值 = \(-6\)
各项与均值的偏差 = \(\{-4, -2, 0, 2, 4\}\)
因此,**标准差大于\(Y\)**
### III. \(\{0, 2, 4, 6\}\)
均值 = \(\frac{2 + 4}{2} = 3\)
各项与均值的偏差 = \(\{0-3, 2-3, 4-3, 6-3\}\)
= \(\{-3, -1, 1, 3\}\)
因此,**标准差与\(Y\)相同**
答案:选项E