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
Ari examined a set of 86 plants. The lightest plant had a mass of 2.8 kilograms (kg) and the heaviest plant had a mass of 6.3 kg. Chihiro examined the same set of 86 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 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
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
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 251 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
均值・中位数与集中趋势进阶
12, 22, 24, 33, 39, 42, 44
The given list shows the 7 values in a data set. Which of the following data sets has the same median as the given list?
原库参考解析
均值・中位数与集中趋势挑战
At a nature preserve, a wildlife biologist counted ducks from an observation deck at the same time each day for 41 days. The table summarizes the resulting data set, data set A.
The data value 13 was recorded in error and is removed from data set A to create data set B, which consists of the remaining 40 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・中位数与集中趋势挑战
At a nature preserve, a wildlife biologist counted bald eagles from an observation deck at the same time each day for 21 days. The table summarizes the resulting data set, data set A.
The data value 19 was recorded in error and is removed from data set A to create data set B, which consists of the remaining 20 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
For a study, a research team measured the heights of 380 king penguins from the Kerguelen Islands and 190 king penguins from the Falkland Islands. The research team determined that the mean height of the 380 king penguins from the Kerguelen Islands was 87 centimeters and the mean height of the 190 king penguins from the Falkland Islands was 93 centimeters. What was the mean height, in centimeters, of all 570 king penguins the research team measured for this study?
原库参考解析
均值・中位数与集中趋势进阶
a, 26, 29, b, 31, 47, c
For the given data set, the data values are listed in ascending order, where a, b, and c are constants. For this data set, the mean is 36, the median is 29, and the range is 72. What is the value of c?
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 13 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・中位数与集中趋势挑战
a, 26, 29, b, 33, 49, c
For the given data set, the data values are listed in ascending order, where a, b, and c are constants. For this data set, the mean is 36, the median is 29, and the range is 72. What is the value of c ?
原库参考解析
