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・极差与标准差挑战
The histograms summarize the distributions of scores for team A and team B. Which statement best compares the standard deviations of scores for these teams?
原库参考解析
分布图表・极差与标准差挑战
Each of the following frequency tables represents a data set. Which of these frequency tables represents the data set with the smallest standard deviation?
原库参考解析
分布图表・极差与标准差挑战
The histogram summarizes a data set of the weights, in pounds, of 14 salmon. If an additional weight of 68 pounds is added to the original data set to create a new data set of 15 weights of salmon, which of the following measures must be greater for the new data set than for the original data set?
Ⅰ. The median
Ⅱ. The mean
原库参考解析
分布图表・极差与标准差进阶
The bar graph shows the frequency of each data value in a data set.
What is the frequency of data value 2 in this data set?
原库参考解析
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.
17, 4, 20, 17, 18, 6
What is the range of the numbers of individuals for the 6 groups of ring-tailed lemurs?
原库参考解析
分布图表・极差与标准差挑战
Each of the following frequency tables represents a data set. Which of these frequency tables represents the data set with the smallest standard deviation?
原库参考解析
分布图表・极差与标准差挑战
A data set of the orbital periods, rounded to the nearest whole number of Earth days, for 13 of Jupiter's moons is represented in the dot plot. An additional moon with an orbital period of 621 days is added to the original data set to create a new data set of 14 orbital periods. Which statement best compares the mean and median of the new data set to the mean and median of the original data set?
原库参考解析
分布图表・极差与标准差挑战
The bar graph shows the number of trees planted for 4 separate plots of land on a tree farm.
What is the maximum number of trees planted on one of these 4 plots of land?
原库参考解析
分布图表・极差与标准差进阶
The average (arithmetic mean) of a list of 1,000 numbers is 32 , and the standard deviation is d, where d > 0. Which of the following two numbers, when added to the list, will result in a list of 1,002 numbers with a standard deviation of less than d ?
原库参考解析
分布图表・极差与标准差进阶
The table shows the frequency of values in a data set.
What is the minimum value of the data set?
原库参考解析
分布图表・极差与标准差挑战
Data set A: 8, 11, 12, 15, 21, 27
Data set B: 9, 12, 13, 16, 22, h
Data sets A and B each consist of 6 values as shown, where h is a constant. If the standard deviation of data set A is greater than the standard deviation of data set B, which of the following could be the value of h?
I. 23
II. 28
III. 29
原库参考解析
分布图表・极差与标准差挑战
Data set A: 8, 11, 13 ,16, 21, 26
Data set B: 9, 12, 14, 17, 22, h
Data sets A and B each consist of 6 values as shown, where h is a constant. If the standard deviation of data set A is greater than the standard deviation of data set B, which of the following could be the value of h?
I.23
II. 27
III. 28
原库参考解析
