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
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?
原库参考解析
分布图表・极差与标准差进阶
The weight of each object in two groups, A and B, was recorded. The histograms represent the distributions of weight, in grams, for the objects in groups A and B.
The mean weight of the objects in group A is equal to 2.9, and the mean weight of the objects in group B is equal to 7.9. Which of the following statements about the standard deviations of weight for the objects in these two groups is true?
原库参考解析
分布图表・极差与标准差基础
For a certain region, the table shows the average number of store employees in 2016 by type of store. Based on the table, how much greater was the average number of store employees in warehouse stores than in supermarkets?
原库参考解析
分布图表・极差与标准差挑战
The dot plots represent the distributions of values in data sets A and B.
Which of the following statements must be true?
I. The median of data set A is equal to the median of data set B.
II. The standard deviation of data set A is equal to the standard deviation of data set B.
原库参考解析
分布图表・极差与标准差基础
To study the characteristics of sea stars in a group of tide pools, researchers measured the diameter of the sea stars within the tide pools. The dot plot gives the diameter, to the nearest inch, of each of the sea stars in these tide pools.
Based on the dot plot, how many sea stars had a diameter, to the nearest inch, of 16 inches?
原库参考解析
分布图表・极差与标准差进阶
4,4,4,4,8,8,8,13,13
Which frequency table correctly represents the data listed?
原库参考解析
分布图表・极差与标准差进阶
Each of the dot plots shown represents the number of glue sticks brought in by each student for two classes,class A and class B. Which statement best compares the standard deviations of the numbers of glue sticks brought in by each student for these two classes?
原库参考解析
分布图表・极差与标准差基础
A group of students voted on five after-school activities. The bar graph shows the number of students who voted for each of the five activities. How many students chose activity 3?
原库参考解析
分布图表・极差与标准差基础
The dot plot represents the 15 values in data set A. Data set B is created by adding 56 to each of the values in data set A. Which of the following correctly compares the medians and the ranges of data sets A and B?
原库参考解析
分布图表・极差与标准差基础
The bar graph shows the distribution of 419 cans collected by 10 different groups for a food drive. How many cans were collected by group 6?
原库参考解析
分布图表・极差与标准差挑战
The dot plot represents the 15 values in data set A. Data set B is created by adding 56 to each of the values in data set A. Which of the following correctly compares the medians and the ranges of data sets A and B?
原库参考解析
分布图表・极差与标准差挑战
23, 27, 27, 32, 35, 36, 52
What is the range of the 7 scores shown?
原库参考解析
