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 maximum temperature on April 1. in degrees Fahrenheit (°F), was recorded each year at a certain weather station for 11 years. The histogram summarizes the recorded data set.
The temperature of 76.2 °F is removed from this data set to create a new data set of 10 temperatures. Which of the following statements must be true?
I.The mean of the new data set is less than the mean of the original data set.
II.The median of the new data set is less than the median of the original data set.
原库参考解析
分布图表・极差与标准差进阶
The dot plot shows the distribution of capacity for a set of capacitors, set A, which a researcher used for a certain experiment. For another experiment, the researcher used a different set of capacitors, set B. Set B has the same number of capacitors as set A but the capacity of each capacitor of set B is 14 microfarads (\( μ \)F) greater than the capacity of each respective capacitor in set A. Which of the following is true about the capacities of the capacitors in set B?
原库参考解析
分布图表・极差与标准差进阶
The histogram shows the distribution of 20 lengths, in feet, in a data set. The first bar represents the lengths that are less than 50 feet, the second bar represents the lengths that are at least 50 feet but less than 100 feet, and so on. Which of the following could be the maximum length, in feet, in this data set?
原库参考解析
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?
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
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・极差与标准差进阶
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・极差与标准差进阶
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
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 2.1, and the mean weight of the objects in group B is equal to 11.1. 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・极差与标准差进阶
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・极差与标准差挑战
Which of the following lists represents a 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
The bar graph shows the distribution of the number of raspberries per carton for 18 cartons of raspberries at a grocery store.
How many of these cartons of raspberries contain exactly 78 raspberries?
原库参考解析
