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・中位数与集中趋势进阶
The list gives the mass, in grams, of 5 ruffed lemurs.
3,910, 3,910, 3,430, 3,950, 3,450
What is the mean mass, in grams, of these 5 ruffed lemurs?
原库参考解析
均值・中位数与集中趋势基础
Scientists collected acorns that each housed a colony of a particular ant species and then analyzed each colony's structure. For these colonies, the table summarizes the number of queen ants per colony. For these colonies, what is the median number of queen ants per colony?
原库参考解析
均值・中位数与集中趋势进阶
A list of 10 data values is shown.
10,14,22,6,24,26,14,8,8,8
What is the mean of these data?
原库参考解析
均值・中位数与集中趋势挑战
Data set A: 2, 3, 4, 5,5
Data set B: 2, 3, 4, 10
Which of the following statements best compares the means of data set A and data set B?
原库参考解析
均值・中位数与集中趋势挑战
The table shows the amount of time it took a participant in a study to complete each of 5 tasks. What was the mean time, in minutes, for this participant to complete a task?
原库参考解析
均值・中位数与集中趋势挑战
The table shows the amount of time it took a participant in a study to complete each of 5 tasks. What was the mean time, in minutes, for this participant to complete a task?
原库参考解析
均值・中位数与集中趋势进阶
The list gives the speeds, in miles per hour, of 7 serves made by a professional tennis player.
107, 109, 109, 111, 151, 154, 155
What is the mean speed, in miles per hour, of these 7 serves?
原库参考解析
均值・中位数与集中趋势进阶
Data sets X and Y are summarized in the tables shown, where a, b, and c are positive, consecutive integers and a < b < c. Which statement comparing the means of the data sets is true?
原库参考解析
均值・中位数与集中趋势挑战
The table shows the frequencies of the values in a data set. Which of the following frequency tables represents a data set with the same mean as this data set?
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 23 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・中位数与集中趋势挑战
Ari examined a set of 85 plants. The lightest plant had a mass of 2.8 kilograms kg and the heaviest plant had a mass of 6.2 kg. Chihiro examined the same set of 85 plants and also an additional plant with a mass of 11.6 kg. Which of the following must be true about the set of plants that Ari examined and the set of plants that Chihiro examined?
原库参考解析
均值・中位数与集中趋势进阶
The table shows the frequencies of the data values for four data sets. Which data set has the greatest mean?
原库参考解析
