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
The dot plot represents an original data set of the ages of the 20 cards in a baseball card collection. A card that is 12 years old is added to the collection, resulting in a new data set of the ages of 21 cards. Which statement best compares the median of the original data set and the median of the new data set?
参考答案
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?
参考答案
均值・中位数与集中趋势进阶
A farmer gave two groups of chickens different types of chicken feed for a month to measure the effect on egg production. The lists give the number of eggs collected from each chicken in each of the two groups.
Group G: 8, 15, 18, 18, 24
Group H: 15, 18, 18, 24
Which statement correctly compares the median number of eggs collected for group G and the median number of eggs collected for group H?
Group G: 8, 15, 18, 18, 24
Group H: 15, 18, 18, 24
Which statement correctly compares the median number of eggs collected for group G and the median number of eggs collected for group H?
参考答案
均值・中位数与集中趋势进阶
12, 21, 23, 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?
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?
参考答案
均值・中位数与集中趋势进阶
A farmer gave two groups of chickens different types of chicken feed for a month to measure the effect on egg production. The lists give the number of eggs collected from each chicken in each of the two groups.
Group G: 8, 15, 18, 18, 23
Group H: 15, 18, 18, 23
Which statement correctly compares the median number of eggs collected for group G and the median number of eggs collected for group H?
Group G: 8, 15, 18, 18, 23
Group H: 15, 18, 18, 23
Which statement correctly compares the median number of eggs collected for group G and the median number of eggs collected for group H?
参考答案
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
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
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.
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 Robert to complete each of the 5 tasks in a project. What was the mean time, in minutes, for Robert to complete a task for this project?
原库参考解析
均值・中位数与集中趋势挑战
The histogram summarizes data set A, which represents the number of points per player earned by 50 players of a game. A new player earns 18 points playing the game, and this number of points is added to data set A to create data set B with 51 values. Which of the following must be true?
Ⅰ. The median number of points per player for data set B is less than the median number of points per player for data set A.
Ⅱ. The mean number of points per player for data set B is less than the mean number of points per player for data set A.
原库参考解析
均值・中位数与集中趋势基础
71,72,73,76,77,79,83,87,93
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
Data set A consists of 10 positive integers less than 60.The list shown gives 9 of the integers from data set A.
43,45,44,43,38,39,40,46,40
The mean of these 9 integers is 42.If the mean of data set A is an integer that is greater than 42, what is the value of the largest integer from data set A?
原库参考解析
