4.3 相对频率

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  相对变化

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  If you were evaluating a set of data describing the numbers of “A’s”, “B’s”, “C’s”, and “D’s” that students earned on a particular test, and needed to display the data on a relative frequency table , how would you go about it? 
  ::如果你在评估一套数据,描述学生在特定测试中赚取的 " A " 、 " B " 、 " C " 和 " D " 数字,并且需要将数据显示在相对频率表中,你将如何进行?

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  Relative Frequencies 
  ::相对变化

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  Frequency tables are closely related to histograms and stem-and-leaf plots. A relative frequency table is specifically designed to display the ratio of each individual frequency to the total frequency of the data. To begin building a relative frequency table, start by grouping values into categories, classes, or intervals, depending on the type of data. You should try to limit the number of intervals or classes to less than a dozen in most cases, and you can use the square root of the number of actual data points as a guide if you wish.
  ::频度表与直方图和干叶图密切相关。一个相对频度表是专门设计用来显示每个单个频率与数据总频率之比的。要开始构建一个相对频度表,首先根据数据类型,将数值分组为类别、类别或间隔。在大多数情况下,应尝试将间隔或类别数目限制在十几个以下,如果需要,可以使用实际数据点数的平方根作为指南。

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  Once you have all of your data separated into separate classes or categories, (often known as “ binning ”, since the data is divided up into multiple “ bins ,” one for each specified class, category, or interval), tally the number of values in each category and the total number of values all together.
  ::一旦您的所有数据被分解成不同的类别或类别(通常称为“宾馆”,因为数据分为多个“宾馆”,每个指定的类别、类别或间距各一个),将每个类别中的数值数和数值总数加起来。

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  To calculate the relative frequency of each category, divide the category, class, or interval frequency by the overall frequency. The decimal you get will represent the part of the entire sample that is represented by that category. Once you have calculated all of the relative frequencies for every category, add them up to make sure they total 1.0.
  ::要计算每类的相对频率, 将类别、 类别或间隔频率除以总频率。 您得到的小数点将代表该类别所代表的整个样本部分。 一旦您计算了每一类的所有相对频率, 将它们相加以确保总计为1.0 。

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  Note! If you are graphing the relative frequencies of a  continuous variable , you will need to specify how to handle any values that fall right on the boundary of a category (also commonly called a class). Here are a couple of ways to do this:
  ::注意 。 如果您正在绘制连续变量的相对频率图, 您需要指定如何处理属于某个类别( 通常也称为类) 边界上的任何值 。 以下有几种方法可以做到这一点 :

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  • You can specify on your table that values equal to lower class limits are included in a class, but values equal to upper class limits are not (this is the conventional method). This means that a value of 5 would be considered part of a 5-10 class, but not part of a 1-5 class.
    ::您可以在表格中指定,一个等级中包含的数值等于较低等级限值,但与最高等级限值相等的数值则不是(这是常规方法)。这意味着5值将被视为5-10级的一部分,但不属于1-5级的一部分。
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  • You can also define each category so that there are no overlapping values:
    ::您也可以定义每一类别,这样就不会出现重叠值:

  1-4.99 5-9.99 10-14.99 15-20

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  Determining Relative Frequencies 
  ::确定相对频率

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  1. You are given a bag of marbles in multiple colors, if there are 25 red, 22 yellow, 17 green, and 28 blue marbles, what are the relative frequencies of each color?
  ::1. 如果有25个红色、22个黄色、17个绿色和28个蓝色大理石,那么每种颜色的相对频率是多少?

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  Start by totaling the number of marbles:  $\begin{array}{r}25+22+17+28=92\end{array}$ total marbles
  ::从弹珠总数开始:25+22+17+28=92

  

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  Divide the number of each color by the total number of marbles:
  ::将每种颜色的数除以大理石的总数:

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  $\begin{array}{r}\frac{25redmarbles}{92totalmarbles}=.272\end{array}$
  ::25 红色大理石 92 共 25 红大理石= 272

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  $\begin{array}{r}\frac{22yellowmarbles}{92totalmarbles}=.239\end{array}$
  ::22个黄色大理石92

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  $\begin{array}{r}\frac{17greenmarbles}{92totalmarbles}=.185\end{array}$
  ::17个绿色大理石92 总计大理石=185

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  $\begin{array}{r}\frac{28bluemarbles}{92totalmarbles}=.304\end{array}$
  ::28个蓝色弹珠92 总计弹珠=304

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  Add your totals together to verify that they equal 1:
  ::将总数加在一起,以核实总数等于1:

  $\begin{array}{r}.272+.239+.185+.304=1.0\end{array}$

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  Note that each of the relative frequencies can also be understood as percentages:
  ::请注意,每一相对频率也可以被理解为百分比:

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  .272 = 27.2% red marbles
  ::272=27.2%红弹珠

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  .239 = 23.9% yellow marbles
  ::239 = 23.9%黄珠

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  .185 = 18.5% green marbles
  ::185=18.5%绿弹珠

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  .304 = 30.4% blue marbles
  ::304 = 30.4% 蓝球

  $\begin{array}{r}27.2\mathrm{\%}+23.9\mathrm{\%}+18.5\mathrm{\%}+30.4\mathrm{\%}=100\mathrm{\%}\end{array}$

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  2. A police officer is reviewing accident statistics for her city. She notes that there were a total of 23 incidents involving teen drivers between ages sixteen and twenty-one, 19 incidents involving drivers aged twenty-two through twenty-six, 19 involving twenty-seven to forty-year-olds, and 18 for ages above forty-one.
  ::2. 一名警官正在审查她所在城市的事故统计,她指出,共有23起事件涉及16至21岁的青少年驾驶员,19起事件涉及22至26岁的驾驶员,19起事件涉及27至40岁的驾驶员,18起事件涉及41岁以上的驾驶员。

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  What are the relative frequencies for each age range ?
  ::每个年龄组的相对频率是多少?

  

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  The total number of accidents is:
  ::事故总数如下:

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  $\begin{array}{r}23+19+19+18=79\text{total accidents}\end{array}$
  ::23+19+19+19+18=79

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  The relative frequencies are:
  ::相对频率是:

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  $\begin{array}{r}\frac{23inagerange16-21}{79total}=.291\end{array}$
  ::23岁,年龄在16-2179岁之间,总数=291

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  $\begin{array}{r}\frac{19inagerange22-26}{79total}=.241\end{array}$
  ::19岁,年龄在22-2679岁之间,总数=241

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  $\begin{array}{r}\frac{19inagerange27-40}{79total}=.241\end{array}$
  ::19岁,年龄在27-40.79岁之间,总数=241

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  $\begin{array}{r}\frac{18inagerange41+}{79total}=.228\end{array}$
  ::18岁,年龄在41+79之间,总计=228

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  Verify that the relative frequencies total 1.0:
  ::核实相对频率共1.0:

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  $\begin{array}{r}.291+.241+.241+.228=1.001(duetorounding)\end{array}$
  ::291+241+241+221+228=1.001(由于四舍五入)

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  Calculating Probability 
  ::计算概率

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  A local high school has 150 students who drive to school. Examining the parking lot, you note that there are 25 white cars, 35 red cars, 13 green cars, 19 blue cars, and 58 others.
  ::当地一所高中有150名学生开车去学校。在停车场的检查中,你注意到有25辆白色汽车、35辆红色汽车、13辆绿色汽车、19辆蓝色汽车和其他58辆汽车。

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  What are the chances, expressed as percentages that randomly chosen students have each of the different colored cars?
  ::随机挑选的学生拥有每种不同彩色汽车的几率是多少?

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  We are given the total number of cars in the question: 150
  ::我们得到的问题中的汽车总数是:150辆。

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  Divide each of the individual colors by the total and convert the decimal answers to percentages:
  ::将每个颜色除以总和,然后将小数点回答转换为百分比:

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  $\begin{array}{r}\frac{25whitecars}{150totalcars}=.167=16.7\mathrm{\%}\end{array}$
  ::总共25辆白色汽车150辆=167=16.7%

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  $\begin{array}{r}\frac{35redcars}{150totalcars}=.233=23.3\mathrm{\%}\end{array}$
  ::35辆红色轿车150,车总车数=233=23.3%

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  $\begin{array}{r}\frac{13greencars}{150totalcars}=.087=8.7\mathrm{\%}\end{array}$
  ::13辆绿色轿车150辆,共计0.877辆,占8.7%

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  $\begin{array}{r}\frac{19bluecars}{150totalcars}=.127=12.7\mathrm{\%}\end{array}$
  ::总共19辆蓝色轿车150辆=.1127=12.7%

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  $\begin{array}{r}\frac{58others}{150totalcars}.387=38.7\mathrm{\%}\end{array}$
  ::其他58 150辆汽车总数 387=38.7%

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  Earlier Problem Revisited
  ::重审先前的问题

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  If you were evaluating a set of data describing the numbers of “A’s”, “B’s”, “C’s”, and “D’s” that students earned on a particular test, and needed to display the data on a relative frequency table, how would you go about it?
  ::如果你在评估一套数据,描述学生在特定测试中赚取的 " A " 、 " B " 、 " C " 和 " D " 数字,并且需要将数据显示在相对频率表中,你将如何进行?

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  Add up the number of entries in each category, A, B, C, and D, to get the total number of data points. Divide the number of values in each category by the total to get the relative frequencies. Convert the decimal values to percentages if necessary.
  ::添加每个类别A、B、C和D的条目数,以获得数据点的总数。将每个类别中的数值除以总数,以获得相对频率。如有必要,将小数点值转换为百分比。

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  Examples
  ::实例

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  Example 1
  ::例1

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  The Sackmore and Headbut village football teams have played each other 50 times. Sackmore has won 10 times, Headbut has won 35 times, and the teams have drawn 5 times. Based on past performance, what is the probability that Sackmore will win the next match?
  ::萨克摩和赫布特村的足球队相互比赛了50次。 萨克莫尔赢了10次,赫布特赢了35次,球队也抽了5次。 根据过去的表现,萨克摩赢得下一场比赛的可能性有多大?

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  So far, Sackmore has won 10 out of the 50 matches. We can write this as a fraction, whic (reduced) is: $\begin{array}{r}\frac{1}{5}\end{array}$ . This fraction isn't really the probability of Sackmore winning, but it is an estimate of that probability. We say that the relative frequency of Sackmore winning  $\begin{array}{r}\frac{1}{5}\end{array}$ . 
  ::到目前为止,萨克摩尔在50场比赛中赢得了10场比赛。我们可以把它写成一个小分数,即:15。这个小分数并不是萨克摩尔赢得比赛的概率,而是这一概率的估计。我们说,萨克摩尔赢得15场比赛的相对频率是15次。

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  Example 2
  ::例2

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  Tony estimates that the probability that there will be an empty space in the car park when he arrives at work is $\begin{array}{r}\frac{4}{5}\end{array}$ . His estimate is based on 50 observations. On how many of these 50 days was he unable to find an empty space in the car park?
  ::Tony估计,他上班时在停车场内空空空间的概率是45。 他的估算基于50次观察。 在这50天中,有多少天他无法在停车场找到空空间?

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  If Tony has figured that he is able to find a space 4 of every 5 times he arrives, then he is not able to find a space 1 in every 5 times. if we set the ratio:  $\begin{array}{r}\frac{1}{5}=\frac{x}{50}\end{array}$ , we can solve for  $\begin{array}{r}x\end{array}$  to find that he did not have a space 10 times. 
  ::如果Tony发现他每到一次时能发现5个空位中的4个,那么他就不能每5个空位中找到1个。如果我们设定比例:15=x50,我们就能解决x发现他没有10个空位的问题。

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  Example 3
  ::例3

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  A pair of dice (one red, one green) is cast 30 times, and on 4 of these occasions, the sum of the numbers facing up is 7. What is the relative frequency that the sum is 7?Out of thirty throws, four of them were 7's. The relative frequency is  $\begin{array}{r}\frac{4}{30}\end{array}$  or  $\begin{array}{r}\frac{2}{15}\end{array}$ . 
  ::一对骰子(一个红色,一个绿色)投了30次,在其中4次中,面对的数字总和是7.7;在30次中,7次的相对频率是多少?30次中,4次是7次。相对频率是430次或215次。

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  Example 4
  ::例4

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  The students in a class were asked what kind of music they liked. 18 liked rock, 11 liked pop, 5 liked hip hop, and 8 liked country. Create a frequency and relative frequency table using this information.
  ::一个班级的学生被问及他们喜欢哪种音乐。 18人喜欢摇滚,11人喜欢流行,5人喜欢流行,8人喜欢国家。 使用这些信息创建频率和相对频率表。

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  To create the frequency table , we just need one column for each category. 
  ::为了创建频率表,我们只需要为每一类别设置一列。

  Rock	Pop	Hip Hop	Country
  18	11	5	8

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  To convert to a relative frequency table, just divide each frequency by the total: 
  ::转换为相对频率表时,将每个频率除以总数:

  Rock	Pop	Hip Hop	Country
  $\begin{array}{r}\frac{18}{42}=.43\end{array}$	$\begin{array}{r}\frac{11}{42}=.26\end{array}$	$\begin{array}{r}\frac{5}{42}=.12\end{array}$	$\begin{array}{r}\frac{8}{42}=.19\end{array}$

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  Example 5
  ::例5

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  In 1990, there were approximately 10,000 fast food outlets in the US that specialized in Mexican food. Of these, the largest were Taco Bell with 4809 outlets, Taco John's with 430 outlets and Del Taco with 275 outlets. The relative frequency that a fast food outlet that specializes in Mexican food is none of the above is:
  ::1990年,美国有大约10,000个专门从事墨西哥食品的快食品销售点。 其中最大的是塔科贝尔(Taco Bell)有4809个销售点,塔科约翰(Taco John)有430个销售点,德尔塔科(Del Taco)有275个销售点。 专门从事墨西哥食品的快食品销售点的相对频率与上述情况无关:

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  The likelihood that a restaurant is not one of the top three would equal the number of Mexican fast food restaurants that are not one of the three: $\begin{array}{r}10,000-4809-430-275=4486\end{array}$ , divided by the total number of Mexican fast food restaurants, 10,000 :
  ::餐馆不是前三位之一的可能性相当于墨西哥快餐餐厅的数量,而快餐餐馆不是前三位之一:10,000-4809-430-275-4475=4486,除以墨西哥快餐餐厅的总数,10,000:

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  $$\begin{array}{r}\frac{4,486}{10,000}=.4486or44.86\mathrm{\%}\end{array}$$

  
  ::4 48610,000=4486,即44.86%

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  Review 
  ::回顾

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  30 Students in a class surveyed each other to find out their favorite movie series, and recorded the results in a table like the one shown below.
  ::30名班级学生互相调查,以了解他们最喜欢的电影系列,并将结果记录在一张表格中,如下表所示。

  Movie Series	Number of Likes
  Twilight	7
  Lord of the Rings	5
  Pirates of the Caribbean	9
  Harry Potter	6
  Narnia	2
  High School Musical	1

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  1. What was the relative frequency for Narnia?
  ::1. 纳尔尼亚的相对频率是多少?

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  2. What was the relative frequency for Pirates of the Caribbean?
  ::2. 加勒比海盗的相对频率是多少?

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  3. 100 people were asked whether they were left-handed. 8 people answered yes. What is the relative frequency of left-handed people in the survey?
  ::3. 100人被问及他们是否是左撇子。 8人回答是。 调查中左撇子的相对频率是多少?

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  4. The relative frequency of getting a white candy from a particular bag is 0.3. If the bag contains 100 candies, estimate the number of whites.
  ::4. 从某一特定包袋中获取白糖的相对频率为0.3,如果包袋中含有100盒糖果,估计白糖的数量。

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  5. Kyle observed 80 cars as they drove by his bedroom window. 24 of them were red. What is the relative frequency of red cars?
  ::5. Kyle在经过卧室窗户时观察到80辆汽车,其中24辆是红色的,红色汽车的相对频率是多少?

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  6. The relative frequency of rain in April is .6. There are 30 days in April. Estimate the number of days of rain expected in April.
  ::6. 4月份的降雨相对频率为6.4天,4月份为30天,估计4月份预计的降雨天数。

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  Use the table below listing the heights of 100 male semiprofessional soccer players.
  ::使用下表列出100名男性半专业足球运动员的身高。

  HEIGHTS (INCHES)	FREQUENCY OF STUDENTS	RELATIVE FREQUENCY
  59.95-61.95	5
  61.95-63.95	3	3100 = 0.03
  63.95-65.95		15100 = 0.15
  65.95-67.95	40	40100 = 0.40
  67.95-69.95	17
  69.95-71.95	12	12100 = 0.12
  71.95-73.95		7100 = 0.07
  73.95-75.95	1	1100 = 0.01
  	Total = 100	Total =

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  7. Fill in the blanks and check your answers.
  ::7. 填空并检查你的答案。

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  8. The percentage of heights that are from 67.95 to 71.95 inches is:
  ::8. 从67.95至71.95英寸的高度百分比为:

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  9. The percentage of heights that are from 67.95 to 73.95 inches is:
  ::9. 从67.95至73.95英寸的高度百分比为:

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  10. The percentage of heights that are more than 65.95 inches is:
  ::10. 超过65.95英寸的高度百分比为:

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  11. The number of players in the sample who are between 61.95 and 71.95 inches tall is:
  ::11. 抽样中身高在61.95至71.95英寸之间的球员人数如下:

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  12. What kind of data does this chart highlight, qualitative or quantitative?
  ::12. 本图表着重说明何种数据,定性或定量数据?

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  13. What is the height interval for the players who fall under the frequency of .03?
  ::13. 处于0.03频率下的球员的高度间隔是多少?

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  Review (Answers)
  ::回顾(答复)

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  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 。