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
Two participants solved a puzzle cube during timed competitions. The lists give the times, to the nearest second, that participant A and participant B took to solve the cube at various competitions.
Participant A: 23, 23, 25, 27
Participant B: 23, 23, 25, 27, 38
Which statement correctly compares the mean solve times, to the nearest second, of participant A and participant B at these competitions?
原库参考解析
均值・中位数与集中趋势挑战
Two participants solved a puzzle cube during timed competitions. The lists give the times, to the nearest second, that participant A and participant B took to solve the cube at various competitions.
Participant A: 23, 23, 24, 27
Participant B: 23, 23, 24, 27, 39
Which statement correctly compares the mean solve times, to the nearest second, of participant A and participant B at these competitions?
原库参考解析
均值・中位数与集中趋势挑战
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 77.9°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
均值・中位数与集中趋势进阶
Rattlesnakes do not lay eggs but give birth to live young. For the last three years, a rattlesnake at a zoo gave birth to one litter per year. The table shows the number of rattlesnakes in each of these three litters.
What is the mean number of rattlesnakes per litter for these three litters?
原库参考解析
均值・中位数与集中趋势挑战
Data set X: 13, 16, 19, 21, 24, 25, 25, 26, 38
Data set Y: 13, 16, 19, 21, 24, 25, 25, 26, 33
Data set Y is created by replacing the number 38 in data set X with the number 33.
Which of the following statements is true about the means and medians of data set X and data set Y?
原库参考解析
均值・中位数与集中趋势进阶
74, 76, 78, 81, 84, 87, 90, 91, 95
What is the median of the data shown?
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 14 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?
参考答案
均值・中位数与集中趋势进阶
a, 26, 28, 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 35, the median is 28, and the range is 71. What is the value of c?
参考答案
均值・中位数与集中趋势挑战
Data set A and data set B each consist of 24 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?
参考答案
均值・中位数与集中趋势挑战
Two data sets of 14 integers each are summarized in the histograms shown. For each of the histograms, the first interval represents the frequency of integers greater than or equal to 10, but less than 20. The second interval represents the frequency of integers greater than or equal to 20, but less than 30, and so on. What is the smallest possible difference between the mean of data set A and the mean of data set B?
参考答案
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?
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?
参考答案
均值・中位数与集中趋势挑战
The histogram summarizes the distribution of an original data set that represents the number of points per game a basketball team has scored in the last 50 games played. If the team scores 18 points in the next game and this game is added to the original data set to create a new data set of 51 values, which of the following must be true?
I. The median number of points per game for the new data set is less than the median number of points per game for the original data set.
II. The mean number of points per game for the new data set is less than the mean number of points per game for the original data set.
参考答案
