节: 插盒和口述口语笔 | 13.12 口述盒和口述口语笔

章节大纲

Box-and-Whisker Plots
::插盒和口述口语笔

Consider the following list of numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
::考虑以下数字清单:1、2、3、4、5、6、7、8、9、10。

The is the $\begin{aligned} (\frac{n + 1}{2}) \end{aligned}$ th value. There are 10 values, so the median lies halfway between the $\begin{aligned} 5^{t h} \end{aligned}$ and the $\begin{aligned} 6^{t h} \end{aligned}$ value. The median is therefore 5.5. This splits the list cleanly into two halves.
::值为 (n+12) 值。有 10 值, 所以中位值介于 5 和 6 值之间。因此中位值为 5.5 。这可以将列表分割为两半。

The lower list is: 1, 2, 3, 4, 5
::较低的清单是: 1, 2, 3, 4, 5

And the upper list is: 6, 7, 8, 9, 10
::最高名单是: 6,7,8,8,9,10

The median of the lower half is 3. The median of the upper half is 8. These numbers, together with the median, cut the list into four quarters. We call the division between the lower two quarters the first quartile . The division between the upper two quarters is the third quartile (the second quartile is, of course, the median).
::下半部的中位数是3; 上半部的中位数是8; 上半部的中位数是8; 这些数字连同中位数一起将清单切成四个季度。我们称下两个季度之间的划分为第一个四分位数。上两个季度之间的划分是第三个四分位数( 当然第二个四分位数是中位数 )。

A box-and-whisker plot is formed by placing vertical lines at five positions, corresponding to the smallest value, the first quartile, the median, the third quartile and the greatest value. (These five numbers are often referred to as the .) A box is drawn between the position of the first and third quartiles, and horizontal line segments (the whiskers ) connect the box with the two extreme values.
::通过将垂直线设置在五个位置形成一个圆形图案,这些位置相当于最小值、第一个四分位、中位、第三个四分位和最大值。 (这五个数字通常称为.)在第一个四分位和第三个四分位的位置之间绘制一个框,将方框与两个极端值连接到水平线段(斜线)之间。

The box-and-whisker plot for the integers 1 through 10 is shown below.
::整数 1 至 10 的框和口令图如下。

With a box-and-whisker plot, a simple measure of dispersion can be gained from the distance from the first quartile to the third quartile. This inter- quartile range is a measure of the spread of the middle half of the data .
::通过一个盒子和口述图块,可以从第一个四分位数到第三个四分位数的距离中获取一个简单的分散度。这个赤道间距是衡量数据中间半部的分布的尺度。

Interpreting a Box-and-Whisker Plot
::解释一盒和口述口述口语笔

Forty students took a college algebra entrance test and the results are summarized in the box-and-whisker plot below. How many students would be allowed to enroll in the class if the pass mark was set at 65%? 60%?
::40名学生参加了大学代数入学考试,结果总结在下面的盒子和口述图中。如果通过率定在65%,那么有多少学生可以报名入学? 60%?

From the plot, we can see the following information:
::从图中,我们可以看到以下信息:

Lowest score = 50%
::最低得分=50%

First quartile = 60%
::第一个四分位数=60%

Median score = 65%
::中位得分=65%

Third quartile = 77%
::第三四分位=77%

Highest score = 97%
::最高得分=97%

Since the pass marks given in the question correspond with the median and the first quartile, the question is really asking how many students there are in: a) the upper half and b) the upper 3 quartiles.
::由于问题中给出的通过分数与中位数和第一个四分位数相同,问题确实在于学生人数:a) 上半数和b) 上四分位数。

Since there are 40 students, there are 20 in the upper half; that is, 20 students scored above 65%. Similarly, there are 30 students in the upper 3 quartiles, so 30 students scored above 60%.
::有40名学生,上半部有20名,即20名得分超过65%,同样,上四分位有30名学生,上四分位有30名学生,上四分位有30名得分超过60%。

Creating a Box-and-Whisker Plot
::创建纸箱和口述口语笔

Harika is rolling 3 dice and adding the numbers together. She records the total score for each of 50 rolls, and the scores she gets are shown below. Display the data in a box-and-whisker plot, and find both the range and the inter-quartile range .
::Harika 正在滚动 3 骰子, 并且将数字加在一起。她记录了每50卷的总得分, 她得到的得分在下面显示。显示一个盒子和脚印图中的数据, 并同时找到范围和孔径范围。

9, 10, 12, 13, 10, 14, 8, 10, 12, 6, 8, 11, 12, 12, 9, 11, 10, 15, 10, 8, 8, 12, 10, 14, 10, 9, 7, 5, 11, 15, 8, 9, 17, 12, 12, 13, 7, 14, 6, 17, 11, 15, 10, 13, 9, 7, 12, 13, 10, 12

First we’ll put the list in order. Since there are 50 data points, $\begin{aligned} (\frac{n + 1}{2}) = 26.5 \end{aligned}$ , so the median will be the mean of the $\begin{aligned} 25^{t h} \end{aligned}$ and $\begin{aligned} 26^{t h} \end{aligned}$ values. The median will split the data into two lists of 25 values; we can write them as two distinct lists.
::首先我们按顺序排列列表。因为有50个数据点, (n+12)=26.5, 因此中位数将是第25和26个数值的平均值。中位数将把数据分成两个25个数值的列表; 我们可以把它们写成两个不同的列表。

\begin{aligned} 5, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, \\ 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 17, 17 \end{aligned}

Since each sub-list has 25 values, the first and third quartiles of the entire data set can be found from the median of each smaller list. For 25 values, $\begin{aligned} (\frac{n + 1}{2}) = 13 \end{aligned}$ , and so the quartiles are given by the $\begin{aligned} 13^{t h} \end{aligned}$ value from each smaller sub-list.
::由于每个子列表有25个值, 整个数据集的第一和第二个四分位数可以从每个较小列表的中位数中找到。对于25个值, (n+12)=13, 因此四分位数由每个较小子列表的第13个值给出。

From the ordered list we can see the five number summary:
::从订购名单中,我们可以看到五个数字摘要:

The lowest value is 5
::最低值为5

The first quartile is 9
::第一个四分位数是九

The median is 10.5
::中位数为10.5

The third quartile is 13
::第三个四分位数是13

The highest value is 17.
::最高值为17。

The box-and-whisker plot therefore looks like this:
::因此,这个盒子和耳机的图案看起来是这样的:

The range is given by subtracting the smallest value from the largest value: $\begin{aligned} 17 - 5 = 12 \end{aligned}$ .
::从最大值(17-5=12)中减去最小值以给定范围。

The inter-quartile range is given by subtracting the first quartile from the third quartile: $\begin{aligned} 13 - 9 = 4 \end{aligned}$ .
::从第三个四分位数( 13- 9= 4) 中减去第一个四分位数( 13- 9= 4) 。

Representing Outliers in a Box-and-Whisker Plot
::以盒子和口述口述口述口述口述口述口述口述口述口述口述口述口述口述口语笔

Box-and-whisker plots can be misleading if we don’t take outliers into account. An outlier is a data point that does not fit well with the other data in the list. For box-and-whisker plots, we can define which points are outliers by how far they are from the box part of the diagram. Defining which data are outliers is somewhat arbitrary, but many books use the norm that follows. Our basic measure of distance will be the inter-quartile range (IQR).
::如果我们不考虑外线, 外线是一个数据点, 与列表中的其他数据不匹配。对于箱外线块, 我们可以从图框部分的距离来定义哪些点是外线。确定哪些数据是外线, 有点武断, 但许多书籍使用以下的规范。我们的基本距离测量标准将是跨赤道范围( IQR ) 。

A mild outlier is a point that falls more than 1.5 times the IQR outside of the box.
::温和的离值是盒子外IQR的1.5倍以上的点。

An extreme outlier is a point that falls more than 3 times the IQR outside of the box.
::一个极端的外向点是一个点跌落超过3倍的IQR外的盒子外。

When we draw a box-and-whisker plot, we don’t include the outliers in the “whisker” part of the plot; instead, we draw them as separate points.
::当我们绘制一幅纸箱和史诗的图时, 我们不把外线纳入图中“耳机”部分; 相反, 我们把它们作为单独的点来画。

Drawing a Box-and-Whisker Plot
::绘制纸箱和口述口语笔画

Draw a box-and-whisker plot for the following ordered list of data:
::为以下订购的数据列表绘制一个框和划线图 :

\begin{aligned} 1, 2, 5, 9, 10, 10, 11, 12, 13, 13, 14, 19, 25, 30 \end{aligned}

From the ordered list we see:
::我们从订购的清单中看到:

The lowest value is 1.
::最低值为1。

The first quartile $\begin{aligned} (Q_{1}) \end{aligned}$ is 9.
::第一个四分位数( Q1) 是 9 。

The median is 11.5.
::中位数为11.5。

The third quartile $\begin{aligned} (Q_{3}) \end{aligned}$ is 14.
::第三四分位数(Q3)为14。

The highest value is 30.
::最高值为30。

Before we start to draw our box-and-whisker plot, we can determine the IQR:
::在开始绘制我们的盒子和口语图之前, 我们可以确定IQR:

\begin{aligned} I Q R = Q_{3} - Q_{1} = 14 - 9 = 5 \end{aligned}

::IQR31=14-9=5

Outliers are points that fall more than 1.5 times the IQR outside of the box—in other words, values that are more than 7.5 units less than 9 or greater than 14. So any values less than 1.5 or greater than 21.5 are outliers.
::外差值是框外IQR的1.5倍以上,也就是说,值超过7.5个单位小于9或大于14。因此,任何值小于1.5或大于21.5的值都是外差值。

Looking back at the data we see:
::回顾我们所看到的数据:

The value of 1 is less than 1.5, so it is a mild outlier .
::1 的值小于 1.5, 所以它是一个微小的外差。

The value 2 is the lowest value that falls within the included range .
::值2是包含范围内的最低值。

The value 30 is greater than 21.5. In fact, it’s not just more than 7.5 units outside the box, it’s more than twice that far outside the box. Since it falls more than 3 times the IQR above the third quartile, it’s an extreme outlier .
::30 值大于 21.5 。事实上,它不只是盒子外7.5 个单位,而是盒子外的两倍多。由于它比第三四分位数的IQR跌幅超过3倍以上,它是一个极端的外差。

The value 25 is also greater than 21.5, so it is a mild outlier .
::25值也大于21.5,因此是轻微的外差值。

The value 19 is the highest value that falls within the included range .
::值19是包含范围内的最高值。

So when we draw our box-and-whisker plot, the whiskers will only go out as far as 2 and 19 respectively. The points outside of that range are all outliers. Here is the plot:
::所以当我们绘制我们的盒子和口哨图时, 胡须将分别高达2和19。范围外的点都是外围点。下面是图案:

Making Box-and-Whisker Plots Using a Graphing Calculator
::使用图形计算计算器制作纸箱和口述口语图

Graphing calculators make analyzing large lists of data easy. They have built-in algorithms for finding the median and the quartiles, and can be used to display box-and-whisker plots.
::图形计算器使得分析大量数据清单变得容易。他们拥有找到中位数和四分位数的内置算法, 并可用于显示框和断层图。

The ages of all the passengers traveling in a train carriage are shown below.
::乘坐火车旅行的所有乘客的年龄如下所示。

35, 42, 38, 57, 2, 24, 27, 36, 45, 60, 38, 40, 40, 44, 1, 44, 48, 84, 38, 20, 4, 2, 48, 58, 3, 20, 6, 40, 22, 26, 17, 18, 40, 51, 62, 31, 27, 48, 35, 27, 37, 58, 21

Use a graphing calculator to: obtain the 5 number s
::使用图形计算计算器以: 获得 5 s 数字

Obtain the 5 number summary for the data.
::获取数据5号摘要。

Create a box-and-whisker plot.
::创建纸箱和口哨图。

Determine if any of the points are outliers.
::确定其中任何点数是否为异常点。

Enter the data in your calculator:
::在您的计算器中输入数据 :

Press [START] then choose [EDIT] .
::按[裁键,然后选择[EDIT]。

Enter all 43 data points in list $\begin{aligned} L_{1} \end{aligned}$ .
::在 L1 列表中输入所有 43 个数据点。

Find the 5 number summary:
::查找5号摘要:

Press [START] again. Use the right arrow to choose [CALU] .
::再次按 [START] 键。使用正确的箭头选择 [CALU] 。

Highlight the 1-Var Stats option. Press [EDIT] .
::突出显示 1 - Var Stats 选项。按 [ EDIT] 键。

The single variable statistics summary appears.
::单一可变统计摘要出现。

Note the mean ( $\begin{aligned} \bar{x} \end{aligned}$ ) is the first item given.
::注意中值 (x) 是给出的第一个项目。

Use the down arrow to bring up the data for the five number summary . $\begin{aligned} n \end{aligned}$ is the number of data points, and the final fie numbers in the screen are the numbers we require.
::使用向下箭头显示五个数字摘要的数据。 n 是数据点的数量, 屏幕中的最后一条线是我们需要的数字。

	Symbol	Value
Lowest Value	minX	1
First Quartile	$\begin{aligned} Q_{2} \end{aligned}$	21
Median	Med	37
Third Quartile	$\begin{aligned} Q_{3} \end{aligned}$	45
Highest Value	maxX	84

Display the box-and-whisker plot:
::显示框和断层图 :

Bring up the [STARTPLOT] option by pressing [2nd]. [Y=] .
::按住[第2次],提出[裁武巴选项。 [Y=]。

Highlight 1:Plot1 and press [ENTER] .
::亮点 1: 绘图1 和按 [ENTER] 。

There are two types of box-and-whisker plots available. The first automatically identifies outliers. Highlight it and press [ENTER] .
::有两种类型的纸箱和耳机地块可供使用。第一种是自动识别外部线。突出显示并按 [ 键。

Press [WINDOW] and ensure that Xmin and Xmax allow for all data points to be shown. In this example, $\begin{aligned} X m i n = 0 \end{aligned}$ and $\begin{aligned} X m a x = 100 \end{aligned}$ .
::按 [WINDOW] 并确保 Xmin 和 Xmax 允许显示所有数据点。在此示例中, Xmin=0 和 Xmax=100 。

Press [GRAPH] and the box-and-whisker plot should appear.
::按[GRAPH]键和盒子和口哨的图案应该出现。

The calculator will automatically identify outliers and plot them as such. You can use the [TRACE] function along with the arrows to identify outlier values. In this case there is one outlier: 84.
::计算器将自动识别外部线, 并以此绘制。您可以使用 [TRACE] 函数和箭头来识别外部值。在此情况下, 只有一个外部线: 84 。

Example
::示例示例示例示例

Example 1
::例1

The box-and-whisker plots below represent the times taken by a school class to complete an obstacle course. The times have been separated into boys and girls. The boys and the girls each think that they did best. Determine the five number summary for both the boys and the girls and give a convincing argument for each of them.
::下面的盒子和小费地块代表了学校班级完成障碍课程的时间,这些时间被分为男孩和女孩,男孩和女孩都认为他们做得最好,确定男孩和女孩的5个数字摘要,并给每个男孩和女孩一个令人信服的理由。

Comparing two sets of data with a box-and-whisker plot is relatively straightforward. For example, you can see that the data for the boys is more spread out, both in terms of the range and the inter-quartile range.
::将两组数据与盒式和口语图作比较相对简单。例如,你可以看到,男孩的数据在范围上和孔径间范围上都比较分散。

The five number summary for each is shown in the table below.
::5个数字摘要如下表所示。

	Boys	Girls
Lowest value	1:30	1:40
First Quartile	2:00	2:30
Median	2:30	2:55
Third Quartile	3:30	3:20
Highest value	5:10	4:10

Here are some points each side could use in their argument:
::以下是各方在论点中可以使用的一些要点:

Boys:
::男孩:

The boys had the fastest time (1 minute 30 seconds), so the fastest individual was a boy.
::男孩有最快的时间(1分30秒) 所以最快的人是男孩

The boys also had the smaller median (2 minutes 30 seconds), meaning half of the boys were finished when only one fourth of the girls were finished (since the girls’ first quartile is also 2:30). In other words, the boys’ average time was faster.
::男孩的中位数也较小(2分30秒),这意味着一半的男孩只完成了四分之一的女孩(因为女孩的第一个四分位数也是2:30)就完成了。换句话说,男孩的平均时间更快。

Girls:
::女童:

The boys had the slowest time (5 minutes 10 seconds), so by the time all the girls were finished there was still at least one boy completing the course.
::男孩的学习时间最慢(5分钟10秒),因此,所有女孩完成学业时,至少有一名男孩完成了课程。

The girls had the smaller third quartile (3 min 20 seconds), meaning that even without taking the slowest fourth of each group into account, the girls were still quickest.
::女孩的第三个四分位数较小(3分20秒),这意味着即使没有考虑到每个群体中最慢的四分之一,女孩的速度仍然最快。

Review
::回顾

Draw a box-and-whisker plot for the following unordered data: 49, 57, 53, 54, 57, 49, 67, 51, 57, 56, 59, 57, 50, 49, 52, 53, 50, 58
::为下列未经排序的数据绘制盒子和脚图: 49、57、53、54、57、49、67、51、57、56、59、57、57、50、49、52、53、50、58

A simulation of a large number of runs of rolling 3 dice and adding the numbers results in the following 5-number summary: 3, 8, 10.5, 13, 18 . Make a box-and-whisker plot for the data and comment on the differences between it and the plot in example B.
::模拟大量滚动三骰子的运行并添加数字,得出以下5个数字摘要:3、8、10.5、13、18。制作一个插盒式图,以图解数据,并评论数据与图象B中的图象之间的差异。

The box-and-whisker plots below represent the percentage of people living below the poverty line by county in both Texas and California. Determine the 5-number summary for each state, and comment on the spread of each distribution.
::下面的方框和小费地块代表得克萨斯州和加利福尼亚州按县分列的生活在贫困线以下的人口比例。确定每个州的5个数字摘要,并评论每个分布的分布。

The 5-number summary for the average daily temperature in Atlantic City, $\begin{aligned} N J^{1} \end{aligned}$ (given in $\begin{aligned} ^{\circ} F \end{aligned}$ ) is: 31, 39, 52, 68, 76 . Draw the box-and-whisker plot for this data and use it to determine which of the following, if any, would be considered outliers if they were included in the data:
1. January’s record high temperature of $\begin{aligned} 78^{\circ} \end{aligned}$
  ::1月创纪录的高温78
2. January’s record low temperature of $\begin{aligned} - 8^{\circ} \end{aligned}$
  ::1月创纪录的低温为-8
3. April’s record high temperature of $\begin{aligned} 94^{\circ} \end{aligned}$
  ::4月创纪录的高温94
4. The all time record high of $\begin{aligned} 106^{\circ} \end{aligned}$
  ::创历史最高纪录的106
::大西洋城,NJ1(以F为主)平均日温度5号摘要:31、39、52、68、76。为这些数据绘制盒子和口述图,并用来确定如果数据中包括下列哪些(如果有的话)可被视为外部值:1月份创纪录的高温为781月创纪录的低温-84月创记录的94度。

In 1887 Albert Michelson and Edward Morley conducted an experiment to determine the speed of light. The data for the first 10 runs (5 results in each run) is given below. Each value represents how many kilometers per second over 299,000 km/s was measured. Create a box-and-whisker plot of the data. Be sure to identify outliers and plot them as such. 850, 740, 900, 1070, 930, 850, 950, 980, 980, 880, 960, 940, 960, 940, 880, 800, 850, 880, 900, 840, 880, 880, 800, 860, 720, 720, 620, 860, 970, 950, 890, 810, 810, 820, 800, 770, 760, 740, 750, 760, 890, 840, 780, 810, 760, 810, 790, 810, 820, 850
::1887年,Albert Michelson和Edward Morley进行了确定光速的实验,下文提供了前10次运行的数据(每次运行5个结果),每个数值代表299,000公里/秒每秒多少公里,超过299,000公里/秒,每个数值代表每秒多少公里。建立数据盒式和口式图块。一定要找出和绘制出局者。850、740、900、1070、930、850、950、950、950、980、880、880、960、940、940、880、800、800、850、880、900、840、880、800、860、720、720、660、870、970、950、890、810、810、810、820、820、820、760、760、740、750、760、890、840、780、880、810、760、810、810、820、850。

Is it possible to have outliers on both ends of a data set? Explain.
::数据集两端是否都可能有外部线? 解释一下。

Is it possible for more than half the values in a data set to be outliers? Explain.
::数据集中一半以上的值是否可能是外线?解释。

Is it possible for more than a quarter of the values in a data set to be outliers? Explain.
::数据集中超过四分之一的值是否可能是外部值?解释。

Is it possible for either of the whiskers in a box-and-whisker plot to be of zero length? Explain.
::盒子和口述图中的任何一只胡须都可能为零长度吗?解释一下。

Is it possible for either of the whiskers in a box-and-whisker plot to be longer than the box? Explain.
::盒子和口哨图中的任何一只胡须能否比盒子长?解释一下。

Is it possible for either of the whiskers in a box-and-whisker plot to be twice as long as the box? Explain.
::盒子和口哨图中的任何一只胡须能否比盒子长一倍?解释一下。

$\begin{aligned} ^{1} \end{aligned}$ Information taken from data published by Rutgers University Climate Lab.
::1 资料取自Rutgers大学气候实验室公布的数据。

Review
::回顾

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。

章节大纲

Box-and-Whisker Plots ::插盒和口述口语笔

Interpreting a Box-and-Whisker Plot ::解释一盒和口述口述口语笔

Creating a Box-and-Whisker Plot ::创建纸箱和口述口语笔

Representing Outliers in a Box-and-Whisker Plot ::以盒子和口述口述口述口述口述口述口述口述口述口述口述口述口述口述口语笔

Drawing a Box-and-Whisker Plot ::绘制纸箱和口述口语笔画

Making Box-and-Whisker Plots Using a Graphing Calculator ::使用图形计算计算器制作纸箱和口述口语图

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review ::回顾

Box-and-Whisker Plots
::插盒和口述口语笔

Interpreting a Box-and-Whisker Plot
::解释一盒和口述口述口语笔

Creating a Box-and-Whisker Plot
::创建纸箱和口述口语笔

Representing Outliers in a Box-and-Whisker Plot
::以盒子和口述口述口述口述口述口述口述口述口述口述口述口述口述口述口语笔

Drawing a Box-and-Whisker Plot
::绘制纸箱和口述口语笔画

Making Box-and-Whisker Plots Using a Graphing Calculator
::使用图形计算计算器制作纸箱和口述口语图

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review
::回顾