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
Each of the following frequency tables represents a data set. Which data set has the greatest mean?
原库参考解析
均值・中位数与集中趋势基础
The dot plot represents a data set of the number of bursts for 13 eruptions of a steam vent. If an additional eruption with 11 bursts is added to this data set to create a new data set of 14 eruptions, which of the following measures will be greater for the new data set than for the original data set?
Ⅰ. The median number of bursts
Ⅱ. The mean number of bursts
原库参考解析
均值・中位数与集中趋势挑战
The dot plot represents a data set of the number of bursts for 13 eruptions of a steam vent. If an additional eruption with 11 bursts is added to this data set to create a new data set of 14 eruptions, which of the following measures will be greater for the new data set than for the original data set?
Ⅰ.The median number of bursts
Ⅱ.The mean number of bursts
原库参考解析
均值・中位数与集中趋势进阶
6, 6, 8, 8, 8, 10, 21
Which of the following lists represents a data set that has the same median as the data set shown?
原库参考解析
均值・中位数与集中趋势基础
The box plot summarizes 15 data values. What is the median of this data set?
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 alpine marmots.
4,010; 4,010; 3,030; 4,050; 3,050
What is the mean mass, in grams, of these 5 alpine marmots?
原库参考解析
均值・中位数与集中趋势挑战
A list of 10 data values is shown.
6, 8, 16, 4, 17, 26, 8, 5, 5, 5What is the mean of these data?
原库参考解析
