a, b, and c are distinct non-zero integers, and the standard deviation of {a, b, c} is greater than the standard deviation of {|a|, |b|, |c|}

题目

a, b, and c are distinct non-zero integers, and the standard deviation of {a, b, c} is greater than the standard deviation of {|a|, |b|, |c|}. If the range of {a, b, c} is 4, which of the following could be true?

I. The median of {a, b, c} is -2

II. The product of a, b, and c is a prime number

III. The mode of {|a|, |b|, |c|} is 1

选项

I only

II only

III only

I and III only

I, II, and III

解析

因为集合\(\{a, b, c\}\)的标准差大于\(\{|a|, |b|, |c|\}\)的标准差，所以\(\{a, b, c\}\)中一定同时包含正整数和负整数。这是因为如果所有数都是正数或都是负数，\(\{a, b, c\}\)和\(\{|a|, |b|, |c|\}\)的标准差会相等。结合“极差为4”这一条件，可知集合中最小和最大的整数可能是\((-3, 1)\)、\((-2, 2)\)或\((-1, 3)\)。因此可能的集合有： \(\{-3, -2, 1\}\) \(\{-3, -1, 1\}\) \(\{-2, -1, 2\}\) \(\{-2, 1, 2\}\) \(\{-1, 1, 3\}\) \(\{-1, 2, 3\}\) 我们来分析选项： I. \(\{a, b, c\}\)的中位数是-2 若集合为\(\{-3, -2, 1\}\)，该说法成立。 II. \(a\)、\(b\)、\(c\)的乘积是质数若集合为\(\{-3, -1, 1\}\)，该说法成立。 III. \(\{|a|, |b|, |c|\}\)的众数是1 若集合为\(\{-3, -1, 1\}\)或\(\{-1, 1, 3\}\)，该说法成立。因此，这三个选项都**有可能**成立。答案：E

在线客服

官方微信

公众号