Section outline

  • Measures of Central Tendency 
    ::中央时期措施

    The word “average” is often used to describe the general characteristics of a group of unequal objects. Mathematically, an average is a single number which can be used to summarize a collection of numerical values. In mathematics, there are several types of “averages” with the most common being the mean , the and the mode .
    ::“平均”一词常常用来描述一组不平等对象的一般特征。从数学角度讲,平均是一个单一的数字,可用于总结数字值的集合。在数学中,有几种“平均”类型,最常见的是平均值、平均值和模式。

     

     

    Mean
    ::平均平均

    The arithmetic mean of a group of numbers is found by dividing the sum of the numbers by the number of values in the group. In other words, we add all the numbers together and divide by the number of numbers.
    ::一组数字的算术平均值通过将数字的和数除以该组中的数值数而发现。换句话说,我们把所有数字加在一起,除以数字数。

    Finding the Mean
    ::寻找平均值

    Find the mean of the numbers 11, 16, 9, 15, 5, 18.
    ::找出数字11、16、9、15、5、18的平均值。

    There are six separate numbers, so the mean = 11 + 16 + 9 + 15 + 5 + 18 6 = 74 6 = 12 1 3 .
    ::有六个独立的数字, 所以平均值=11+16+9+15+5+186=746=1213。

    The arithmetic mean is what most people automatically think of when the word average is used with numbers. It’s generally a good way to take an average, but it can be misleading when a small number of the values lie very far away from the rest. A classic example would be when calculating average income. If one person (such as former Microsoft Corporation chairman Bill Gates) earns a great deal more than everyone else who is surveyed, then that one value can sway the mean significantly away from what the majority of people earn.
    ::算术平均值是用数字来使用单词平均值时大多数人自动想到的。 它一般是使用平均值的好方法,但当一小部分值远远远离其他值时,它可能会产生误导。 典型的例子就是计算平均收入。 如果一个人(如前微软公司董事长比尔·盖茨)的收入远远高于被调查的其他人,那么,一个值可以大大偏离大多数人的收入。

    Real-World Application: Annual Income 
    ::实际世界应用:年度收入

    The annual incomes for 8 professions are shown below. Form the data , calculate the mean annual income of the 8 professions.
    ::以下列出8个职业的年收入,并列出数据,计算8个职业的年平均收入。

    Profession Annual Income
    Farming, Fishing, and Forestry $19,630
    Sales and Related $28,920
    Architecture and Engineering $56,330
    Healthcare Practitioners $49,930
    Legal $69,030
    Teaching & Education $39,130
    Construction $35,460
    Professional Baseball Player* $2,476,590

    ( Source: Bureau of Labor Statistics , except (*)-The Baseball Players' Association (playbpa.com)).
    :sad资料来源:劳工统计局,但(*) -- -- 棒球运动员协会(playbpa.com))。

    There are 8 values listed, so the mean is
    ::列出8个值,平均值是

    19630 + 28920 + 56330 + 49930 + 69030 + 39130 + 35460 + 2476590 8 = $ 346 , 877.50

    As you can see, the mean annual income is substantially larger than the income of 7 out of the 8 professions. The effect of the single outlier (the baseball player) has a dramatic effect on the mean, so the mean is not a good method for representing the ‘average’ salary in this case.
    ::正如你所看到的,平均年收入大大高于8个职业中7个的收入。 单点(棒球运动员)的影响对平均值产生了巨大影响,因此,在这种情况下,这一平均值不是代表“平均”工资的好方法。

    Median
    ::中中

    The median is another type of average. It is defined as the value in the middle of a group of numbers. To find the median, we must first list all the numbers in order from least to greatest .
    ::中位数是另一个平均类型。 它被定义为一组数字中间的值。 要找到中位数, 我们首先必须列出所有数字, 从最小到最大 。

    Finding the Median 
    ::查找中位数

    Find the median of the numbers 11, 21, 6, 17, 9.
    ::找出11号 21号 6号 17号 9号的中位数

    We first list the numbers in ascending order: 6, 9, 11 , 17, 21.
    ::我们首先列出按上升顺序排列的数字:6、9、11、17、21。

    The median is the value in the middle of the set (in bold).
    ::中位数是集中间的值(以粗体表示)。

    The median is 11 . There are two values higher than 11 and two values lower than 11.
    ::中位数为11,有两个值高于11,两个值低于11。

    If there is an even number of values, then the median is the arithmetic mean of the two numbers in the middle (in other words, the number halfway between them).
    ::如果数值是偶数,则中位数是中间两个数字的算术平均值(换句话说,数字是中间两个数字之间的一半)。

    The median is a useful measure of average when the data set is highly skewed by a small number of points that are extremely large or extremely small. Such outliers will have a large effect on the mean, but will leave the median relatively unchanged.
    ::当数据集被极大或极小的少数点高度偏斜时,中位数是衡量平均值的一个有用尺度。 这种离值将对中位数产生很大影响,但中位数将保持相对不变。

    Mode
    ::模式模式

    The mode can be a useful measure of data when that data falls into a small number of categories. It is simply a measure of the most common number, or sometimes the most popular choice. The mode is an especially useful concept for data sets that contains non-numerical information, such as surveys of eye color or favorite ice-cream flavor.
    ::当数据属于少数类别时,该模式可以是一种有用的数据计量方法。它只是衡量最常见的数字,有时是最受欢迎的选择。对于包含非数字信息的数据集来说,该模式是一个特别有用的概念,例如眼色调查或最喜爱的冰淇淋口味。

    Of course, a data set can contain more than one mode; when it does, it is called multimodal . In fact, every value in a data set could be a mode, if every value appears an equal number of times. However, this situation is quite rare. You might encounter data sets with two or even three modes, but more than that would be unlikely unless you are working with very small sample sets.
    ::当然,数据集可以包含不止一种模式;当它存在时,它被称为多式联运。事实上,如果每个值显示的数值相同次数,则数据集中的每个值都可能是一种模式。然而,这种情况非常罕见。你可能遇到两个甚至三个模式的数据集,但是,除非你用非常小的样本组工作,否则数据组就不可能有比这更多的数据组。

    Real-World Application: Age of Customers 
    ::现实世界应用:客户年龄

    Jim is helping to raise money at his church bake sale by doing face painting for children. He collects the ages of his customers, and displays the data in the graph below. Find the mean, median and mode for the ages represented.
    ::Jim通过为儿童做面部绘画来帮助在教堂面包销售中筹集资金。 他收集客户的年龄,并在下图中显示数据。 找到代表年龄的平均、中位数和模式。

    By reading the graph we can see that there was one 2-year-old, three 3-year-olds, four 4-year-olds, etc. In total, there were 1 + 3 + 4 + 5 + 6 + 7 + 3 + 1 = 30 customers.
    ::通过阅读图表可以看出,有1名2岁、3名3岁、4名4岁等,总共有1名+3+4+5+6+7+3+1=30名客户。

    The mean age is found by adding up all the ages multiplied by the number of times each age appears, and then dividing by 30:
    ::平均年龄是通过将所有年龄乘以每个年龄的出现次数而得出,然后除以30而得出的:

    2 ( 1 ) + 3 ( 3 ) + 4 ( 4 ) + 5 ( 5 ) + 6 ( 6 ) + 7 ( 7 ) + 8 ( 3 ) + 9 ( 1 ) 30 = 170 30 = 5 2 3

    Since there are 30 children, the median is half way between the 15 t h and 16 t h oldest (that way there will be 15 younger and 15 older than the median age). Both the 15 t h and 16 t h oldest fall in the 6-year-old range , therefore the median is 6.
    ::由于有30名儿童,中位数介于第15和第16个最大年龄之间(即15岁比中位年龄小,15岁比中位年龄大),中位数介于第15和第16个最大年龄之间,因此中位数介于6岁之间。

    The mode is given by the age group with the highest frequency. Reading directly from the graph, we see that the mode is 7; there are more 7-year-olds than any other age.
    ::该模式由使用频率最高的年龄组提供。直接从图表中读取,我们看到该模式为7岁;7岁的人比任何其他年龄的人都多。

    Watch this video for assistance with the example above:
    ::观看这段影片,

     

     

    Example
    ::示例示例示例示例

    Example 1
    ::例1

    Find the mean, median and mode of the numbers 2, 17, 1, -3, 12, 8, 12, 16.
    ::找出数字2、17、1、3、12、8、12、16的平均值、中位数和模式。

    Mean = 2 + 17 + 1 + ( − 3 ) + 12 + 8 + 12 + 16 9 = 7. 22 ¯
    ::平均值=2+17+1+(-3)+12+8+12+169=7.22

    We first list the numbers in ascending order: -3, 1, 2, 8, 12, 12, 16, 17.
    ::我们首先列出按上升顺序排列的数字: -3, 1, 2, 8, 8, 12, 12, 16, 17。

    The median is the value in the middle of the set, so the median lies between 8 and 12. Halfway between 8 and 12 is 10, so 10 is the median .
    ::中位数是集的中间值, 所以中位数在8到12之间, 8到12之间的中间线是10, 中位数是10。

    The mode is the most frequent number or numbers. The only number that repeats is 12, so 12 is the mode.
    ::模式是最常见的数字或数字。重复的数字只有12个,所以12个是模式。

    Review 
    ::回顾

    1. Find the median and mode of the numbers given in Example A, specifically: 11, 16, 9, 15, 5, and 18.
      ::查找例A中给出的数字的中位数和模式,具体如下:11、16、9、15、5和18。
    2. Find the median and mode of the salaries given in Example B, specifically: $19,630, $28,920, $56,330, $49,930, $69,030, $39,130, $35,460, and $2,476,590.
      ::具体来说,找到例B中给出的薪金的中位数和模式,具体如下:19 630美元、28 920美元、56 330美元、49 930美元、69 030美元、39 130美元、35 460美元和2 476 590美元。
    3. Find the mean, median and mode of the data set: 14, 9, 3, 14, 2, 7, 13, 6.
      ::找出数据集的平均值、中位数和模式:14、9、3、14、2、7、13、6。
    4. Find the mean, median and mode of the data set: 5, 3, 5, 0, 1, 5, 3, 4, 4, 4.
      ::找出数据集的平均值、中位数和模式:5、3、5、5、0、1、5、3、4、4、4。
    5. Find the mean, median and mode of the data set: 8, 5, 10, 4, 4, 10, 6, 4, 7, 8, 2, 8, 10, 9, 2, 1, 6, 10, 5, 3.
      ::找出数据集的平均值、中位数和模式:8、5、10、4、4、10、6、4、7、8、2、8、8、10、9、2、1、6、10、5、3。
    6. Find the mean, median and mode of the following numbers. Which of these will give the best average ? 15, 19, 15, 16, 11, 11, 18, 21, 165, 9, 11, 20, 16, 8, 17, 10, 12, 11, 16, 14
      ::找到下列数字的平均值、中位数和模式,其中哪一个是最佳平均数? 15, 19, 15, 16, 11, 11, 18, 21, 165, 9, 11, 20, 16, 8, 8, 17, 10, 12, 11, 16, 14
    7. Ten house sales in Encinitas, California are shown in the table below. Find the mean, median and mode of the sale prices. Explain, using the data, why the median house price is most often used as a measure of the house prices in an area.
      ::加利福尼亚州Encinitas的10个房屋销售情况见下表:找到销售价格的中位数、中位数和模式;用数据解释为什么住宅价格中位数最常用来衡量某个地区的房屋价格。
    Address Sale Price Date Of Sale
    643 3RD ST $1,137,000 6/5/2007
    911 CORNISH DR $879,000 6/5/2007
    911 ARDEN DR $950,000 6/13/2007
    715 S VULCAN AVE $875,000 4/30/2007
    510 4TH ST $1,499,000 4/26/2007
    415 ARDEN DR $875,000 5/11/2007
    226 5TH ST $4,000,000 5/3/2007
    710 3RD ST $975,000 3/13/2007
    68 LA VETA AVE $796,793 2/8/2007
    207 WEST D ST $2,100,000 3/15/2007

    For 8-10, determine which measure of central tendency (mean, median, or mode) would be most appropriate for the following.
    ::对于8-10,确定对以下各点最合适的衡量中心趋势(平均值、中位数或模式)的尺度。

    1. The life expectancy of store-bought goldfish.
      ::商店买的金鱼的预期寿命。
    2. The age in years of audience for a kids TV program.
      ::儿童电视节目的听众年龄。
    3. The average actual weight of sacks of potatoes labeled as 5-pound bags.
      ::土豆袋的平均实际重量,标记为5磅袋。

    Review (Answers)
    ::回顾(答复)

    Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
    ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。