这部分需要会什么？
- 从频数表、直方图和箱线图读取分布
- 区分极差、四分位距与标准差
- 判断整体平移或缩放对离散程度的影响
核心概念与公式
极差为最大值减最小值；四分位距 \(IQR=Q_3-Q_1\)，描述中间 50% 数据的跨度。标准差衡量数据围绕均值的离散程度，越集中通常越小；不能只凭极差相同就断定标准差相同。
全部数据加同一个常数时，均值和中位数同幅移动，但极差、四分位距和标准差不变。全部数据乘 \(c\) 时，标准差、极差和四分位距都乘 \(|c|\)。直方图通常只给出区间频数，未必能确定每个数据的精确值。
做题时怎样使用
- 先看坐标轴、组距、频数和总样本量
- 根据所问指标选用端点、四分位数或整体集中程度
- 比较变换前后时，判断是平移还是缩放，而不是重新猜测分布
容易混淆的地方
箱线图的箱体长度表示四分位距，不表示样本量。标准差不会为负；所有数据相同才有标准差为零。
用一个例题串起方法
知识点教学例题一组数据的均值为 10，标准差为 3。每个值先乘 2，再加 5，新的均值和标准差分别是多少？
展开解题思路 ＋
新均值为 \(2\times10+5=25\)。乘 2 使标准差变为 \(2\times3=6\)，再加 5 不改变离散程度，所以标准差为 6。
本知识点题库
先独立作答，再查看答案与解析。
分布图表・极差与标准差进阶
4, 4, 12, 25, 25
What is the range of the data set shown?
原库参考解析
分布图表・极差与标准差进阶
The values in data sets X and Y are shown in the table.
The standard deviation of data set X is q, and the standard deviation of data set Y is s. Which of the following statements about the standard deviation of the data sets is true?
原库参考解析
分布图表・极差与标准差基础
The table summarizes the distribution of average pitching speed, in miles per hour (mph), for a sample of 24 pitchers in a youth softball league.
According to the table, how many pitchers in the sample have an average pitching speed of at least 30 mph but less than 35 mph?
原库参考解析
分布图表・极差与标准差挑战
A zoologist observed the nests of wood ducks in an area. Each year, the zoologist recorded the number of eggs in each of the first 35 nests that were observed and created a frequency table for the data set. Which of the following frequency tables represents the data set with the smallest standard deviation?
原库参考解析
分布图表・极差与标准差进阶
In each of the following data sets of 5 values, p is a constant. Which of these data sets has the largest standard deviation?
原库参考解析
分布图表・极差与标准差挑战
A zoologist observed the nests of wood ducks in an area. Each year, the zoologist recorded the number of eggs in each of the first 50 nests that were observed and created a frequency table for the data set. Which of the following frequency tables represents the data set with the smallest standard deviation?
原库参考解析
分布图表・极差与标准差进阶
In each of the following data sets of 5 values, p is a constant. Which of these data sets has the largest standard deviation?
参考答案
分布图表・极差与标准差进阶
A marine biologist is studying a habitat that contains shoal bass, a species of freshwater fish. The box plot shown summarizes the lengths, in centimeters (cm), of a sample of 75 shoal bass from this habitat. What is the range, in cm, of the lengths of the shoal bass in the sample?
参考答案
分布图表・极差与标准差进阶
The list gives the number of individuals in each of 6 groups of ring-tailed lemurs.
14, 5, 17, 14, 15, 7
What is the range of the numbers of individuals for the 6 groups of ring-tailed lemurs?
参考答案
分布图表・极差与标准差挑战
A marine biologist is studying a habitat that contains shoal bass, a species of freshwater fish. The box plot shown summarizes the lengths, in centimeters (cm), of a sample of 75 shoal bass from this habitat. What is the range, in cm, of the lengths of the shoal bass in the sample?
参考答案
分布图表・极差与标准差挑战
The table shown summarizes the distribution of 65 employees based on the number of complete years they have worked at a certain company.
How many of the 65 employees have worked at the company for 21 or more complete years?
How many of the 65 employees have worked at the company for 21 or more complete years?
参考答案
分布图表・极差与标准差挑战
Which of the following lists represents a data set with the smallest standard deviation?
参考答案
