这部分需要会什么？
- 计算均值、中位数与加权均值
- 根据均值反求总和或缺失数据
- 判断新增数据和极端值对集中趋势的影响
核心概念与公式
均值 \(\bar x=(x_1+\cdots+x_n)/n\)，因此总和等于均值乘个数。中位数须先按大小排序；奇数个数据取中间一个，偶数个取中间两个的平均值。
合并两组数据时，整体均值为 \((n_1\bar x_1+n_2\bar x_2)/(n_1+n_2)\)。只有两组人数相同，才能直接平均两组均值。均值会被极端数值明显影响，中位数通常更稳定。
做题时怎样使用
- 先确认数据个数、频数和是否已经排序
- 求均值用总和；求中位数看位置；合并组别先还原各组总和
- 若数据改动，分别分析总和和排序位置如何变化
容易混淆的地方
频数表中的一个数可能出现多次，不能把每种数值只算一次。中位数的位置变化不等于数据总和变化。
用一个例题串起方法
知识点教学例题4 个数的均值为 12，加入第五个数后均值为 14。第五个数是多少？
展开解题思路 ＋
原总和为 \(4\times12=48\)，新总和为 \(5\times14=70\)。加入的数为 \(70-48=22\)。
本知识点题库
先独立作答，再查看答案与解析。
均值・中位数与集中趋势挑战
An environmental scientist is investigating the volatility of 175 organic compounds by finding the boiling point of each compound. The mean boiling point of all 175 compounds is 167 degrees Celsius. The scientist classifies each of these compounds as either semivolatile or volatile. Of the 175 compounds, 50 compounds were classified as semivolatile, and these 50 compounds have a mean boiling point of 322 degrees Celsius. The remaining 125 compounds were classified as volatile. What is the mean boiling point, in degrees Celsius, of the 125 compounds classified as volatile compounds?
原库参考解析
均值・中位数与集中趋势挑战
Data set A: 5,8,11,11,14, 20
Data set B:5,8,11,11,14,20,41
Two data sets are shown. Which statement best compares the medians of the data sets?
原库参考解析
均值・中位数与集中趋势挑战
\( 6,\ 6,\ 12,\ 16,\ 35 \)
What is the median of the data set shown?
原库参考解析
均值・中位数与集中趋势进阶
Data set A and data set B each consist of 33 values. The table shows the frequencies of the values for each data set. Which of the following statements best compares the means of the two data sets?
原库参考解析
均值・中位数与集中趋势进阶
The list gives the mass, in grams, of 5 ruffed lemurs.
3,810; 3,810; 3,530; 3,850; 3,550
What is the mean mass, in grams, of these 5 ruffed lemurs?
原库参考解析
均值・中位数与集中趋势进阶
The list gives the mass, in grams, of 5 ruffed lemurs:
3,810; 3,810; 3,030; 3,850; 3,050
What is the mean mass, in grams, of these 5 ruffed lemurs?
原库参考解析
均值・中位数与集中趋势进阶
The list gives the mass, in grams, of 5 ruffed lemurs.
3,710; 3,710; 3,130; 3,750; 3,150
What is the mean mass, in grams, of these 5 ruffed lemurs?
原库参考解析
均值・中位数与集中趋势挑战
28, 28, 29, 30, 32, 33, 37
What is the mean of the data set shown?
原库参考解析
均值・中位数与集中趋势挑战
The histogram summarizes a data set of the weights, in pounds, of 14 salmon. If an additional weight of 32 pounds is added to the original data set to create a new data set of 15 weights. Which of the following measures must be less for the new data set than for the original data set?
Ⅰ. The median
Ⅱ.The mean
原库参考解析
均值・中位数与集中趋势进阶
The frequency table summarizes the mass, in grams, of 35 objects. What is the mean mass, in grams, of these objects?
原库参考解析
均值・中位数与集中趋势挑战
For data set \( A \), the table summarizes the distribution of the number of pieces of mail received by a business each day during a period of \( 11 \) days.
The data value \( 17 \) is removed from data set \( A \) to create data set \( B \), which consists of the remaining \( 10 \) data values. Which statement best compares the median of data set \( A \) and the median of data set \( B \)?
原库参考解析
均值・中位数与集中趋势进阶
A survey of the daily commuting times, in minutes, of seven different Chicago-based employees at a certain company yielded a data set consisting of the values 31, 13, 44, 32, 53, 22, and 44. A survey of the daily commuting times, in minutes, of seven different San Francisco-based employees at the same company yielded a data set consisting of the values 46, 31, 19, 33, 48, x, and 38. If the means of the two data sets are equal, what is the value of x?
原库参考解析
