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
A science class studied the feeding behavior of the birds that visited a bird feeder each school day for a period of time. The table summarizes the number of birds the class observed visiting the bird feeder each school day during this period of time. What is the median number of birds the class observed visiting the bird feeder each school day during this period of time?
原库参考解析
均值・中位数与集中趋势挑战
The histagram summarizes a data set of the weights, in pounds, of 14 salmon. If an additional weight of 21 pounds is added to the original data set to create a new data set of 15 weights, which of the following measures must be less for the new data set than for the original data set?
Ⅰ. The median
Ⅱ. The mean
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・中位数与集中趋势进阶
\( 5,\ 8,\ 11,\ 17,\ 29 \)
What is the mean of the data shown?
原库参考解析
均值・中位数与集中趋势挑战
Data set F consists of 55 integers, where the value of each integer is between 150 and 260. Data set G consists of the same 55 integers in data set F as well as the integer 10. Which of the following must be less for data set G than for data set F?
Ⅰ. The mean
Ⅱ. The median
原库参考解析
均值・中位数与集中趋势进阶
Data set A: 2, 2, 2, 5, 5, 8, 11, 11, 14, 14, 14
Data set B: 2, 5, 5, 8, 8, 8, 8, 8, 11, 11, 14
Data set A and data set B each have 11 values. Which of the following statements 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 researcher created a data set by recording the daily low temperature, to the nearest integer, for each of the 31 days in December 2022 in a town in Illinois. The frequency table summarizes this data set.
Based on the frequency table, what could be the median of the recorded daily low temperatures, in degrees Fahrenheit?
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・中位数与集中趋势挑战
12 , 12 , 18 , 24 , 29
What is the median of the data set shown?
原库参考解析
均值・中位数与集中趋势进阶
The weights of the four rock samples are 160 grams, 250 grams, 240 grams, and 330 grams, respectively. The weight of the fifth rock sample is x grams. If the average (arithmetic mean) weight of the first four rock samples is the same as the average weight of all five rock samples, what is the value of x ?
原库参考解析
均值・中位数与集中趋势挑战
Data set A and data set B each consist of 25 values. The given 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・中位数与集中趋势进阶
The list gives the mass, in grams, of 5 ruffed lemurs.
3,910; 3,910; 3,130; 3,950; 3,150
What is the mean mass, in grams, of these 5 ruffed lemurs?
原库参考解析
