Course: 15.3 五号摘要 | The Way To Learn

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五号摘要

When given a long list of numbers, it is useful to summarize the data . One way to summarize the data is to give the lowest number, the highest number and the middle number. In addition to these three numbers it is also useful to give the of the lower half of the data and the median of the upper half of the data. These five numbers give a very concise summary of the data.
::当给出一个长长的数字列表时,总结数据是有用的。总结数据的一种方法是提供最低的数字、最高的数字和中间数字。除了这三个数字外,还提供数据下半部分和上半部分的中值。这五个数字提供了非常简洁的数据摘要。

What is the five number summary of the following data?
::下列数据的5个数字摘要是什么?

0, 0, 1, 2, 63, 61, 27, 13

The Five Number Summary
::五号数字摘要

Suppose you have ordered data with $\begin{aligned} m \end{aligned}$ observations. The rank of each observation is shown by its index. The rank of an observation is the number of observations that are less than or equal to the value of that observation.
::假设您已经用 m 观察来命令数据。每种观察的级别以其索引显示。观察的顺序是少于或等于该观察值的观察次数。

$\begin{aligned} y_{1} \leq y_{2} \leq y_{3} \leq \dots \leq y_{m} \end{aligned}$
::y"y"2"y"3" y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"3"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"%"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y"y""y"y""""""y"""""y"""y"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""

In data sets that are large enough, you can divide the numbers into four parts called quartiles . The quartiles of interest are the first quartile , $\begin{aligned} Q 1 \end{aligned}$ , the second quartile , $\begin{aligned} Q 2 \end{aligned}$ , and the third quartile $\begin{aligned} Q 3 \end{aligned}$ . The second quartile , $\begin{aligned} Q 2 \end{aligned}$ , is defined to be the median of the data. The first quartile , $\begin{aligned} Q 1 \end{aligned}$ , is defined to be the median of the lower half of the data. The third quartile , $\begin{aligned} Q 3 \end{aligned}$ , is similarly defined to be the median of the upper half of the data.
::在足够大的数据组中,您可以将数字分为四部分,称为四分位数。感兴趣的四分位数是第一个四分位数,Q1、第二个四分位数、Q2和第三个四分位数 Q3。第二个四分位数,Q2的定义是数据的中位数。第一个四分位数,Q1的定义是数据下半数的中位数。第三个四分位数,Q3的定义同样是数据上半数的中位数。

These three numbers in addition to the minimum and maximum values are the five number summary. Note that there are variations of the five number summary that you can study in a statistics course.
::除了最低值和最高值外,这三个数字还有五个数字摘要。请注意,在统计课程中,可以学习的五个数字摘要有不同之处。

Take the following data:
::获取以下数据:

2, 7, 17, 19, 25, 26, 26, 32

There are 8 observations total in this set of data.
::这组数据中共有8个观测数据。

Lowest value (minimum) : 2
::最低价值(最低):2

$\begin{aligned} Q 1 : \frac{7 + 17}{2} = 12 \end{aligned}$ (Note that this is the median of the first half of the data - 2, 7, 17, 19)
::问题1:7+172=12(注意这是数据前半部分的中位数-2、7、17、19)

$\begin{aligned} Q 2 : \frac{19 + 25}{2} = 22 \end{aligned}$ (Note that this is the median of the full set of data)
::问题2:19+252=22(注意这是全套数据的中位数)

$\begin{aligned} Q 3 : 26 \end{aligned}$ (Note that this is the median of the second half of the data - 25, 26, 26, 32)
::问题3:26 (注意这是数据后半部分的中位数 -- -- 25、26、26、32)

Upper value (maximum) : 32
::上值(最大) : 32

The 5 number summary is 2, 12, 22, 26, 32.
::5号摘要是2号,12号,22号,26号,32号。

Examples
::实例

Example 1
::例1

Earlier you were asked to compute the five number summary for 0, 0, 1, 2, 63, 61, 27, 13. It helps to order the data.
::早些时候,你被要求计算5个数字摘要的 0, 0, 0, 1, 2, 63, 61, 27, 13。它有助于订购数据。

0, 0, 1, 2, 13, 27, 61, 63

Since there are 8 observations, the median is the average of the $\begin{aligned} 4^{t h} \end{aligned}$ and $\begin{aligned} 5^{t h} \end{aligned}$ observations: $\begin{aligned} \frac{2 + 13}{2} = 7.5 \end{aligned}$
::由于有8项观测,中位数是第四和第五次观测的平均值:2+132=7.5。

The lowest observation is 0.
::最低的观察是0。

The highest observation is 63.
::最高为63岁。

The middle of the lower half is $\begin{aligned} \frac{0 + 1}{2} = 0.5 \end{aligned}$
::下半部的中间部分是 0+12=0.5

The middle of the upper half is $\begin{aligned} \frac{27 + 61}{2} = 44 \end{aligned}$
::上半为27+612=44

The five number summary is 0, 0.5, 7.5, 44, 63
::五个数字摘要为0、0.5、7.5、44、63。

Example 2
::例2

Create a set of data that meets the following five number summary:
::创建一套符合以下五个数字摘要的数据:

{2, 5, 9, 18, 20}

Suppose there are 8 data points. The lowest point must be 2 and the highest point must be 20. The middle two points must average to be 9 so they could be 8 and 10. The second and third points must average to be 5 so they could be 4 and 6. The sixth and seventh points need to average to be 18 so they could be 18 and 18. Here is one possible answer:
::假设有8个数据点,最低点必须是2个,最高点必须是20个。中点两点必须平均为9个,这样可以达到8个和10个。第二点和第三点必须平均为5个,这样可以达到4个和6个。第六点和第七点平均必须达到18个,这样可以达到18个和18个。

2, 4, 6, 8, 10, 18, 18, 20

Example 3
::例3

Compute the five number summary for the following data:
::为下列数据计算五个数字摘要:

1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 8, 9, 10, 15

There are 20 observations.
::有20条意见。

Lower : 1
::下级:1

$\begin{aligned} Q 1 : \frac{2 + 3}{2} = 2.5 \end{aligned}$
::问题1:2+32=2.5

$\begin{aligned} Q 2 : \frac{4 + 5}{2} = 4.5 \end{aligned}$
::问题2:4+52=4.5

$\begin{aligned} Q 3 : \frac{6 + 7}{2} = 6.5 \end{aligned}$
::问题3:6+72=6.5

Upper : 15
::上上:15

Example 4
::例4

Compute the five number summary for the following data:
::为下列数据计算五个数字摘要:

4, 8, 11, 11, 12, 14, 16, 20, 21, 25

There are 10 observations total in this set of data.
::这组数据中共有10个观测数据。

Lowest value (minimum) : 4
::最低价值(最低):4

$\begin{aligned} Q 1 : 11 \end{aligned}$ (Note that this is the median of the first half of the data - 4, 8, 11, 11, 12)
::问题1:11 (注意这是数据前半部分的中位数 -- -- 4、8、11、11、11、12)

$\begin{aligned} Q 2 : \frac{12 + 14}{2} = 13 \end{aligned}$ (Note that this is the median of the full set of data)
::问题2:12+142=13(注意这是全套数据的中位数)

$\begin{aligned} Q 3 : 20 \end{aligned}$ (Note that this is the median of the second half of the data - 14, 16, 20, 21, 25)
::问题3:20(注意这是数据后半部分的中位数----14、16、20、21、25)

Upper value (maximum) : 25
::上值(最大) : 25

T he five number summary is 4, 11, 13, 20, 25.
::五个数字摘要为4、11、13、20、25。

Example 5
::例5

Compute the five number summary for the following data:
::为下列数据计算五个数字摘要:

3, 7, 10, 14, 19, 19, 23, 27, 29

There are 9 observations total. To calculate $\begin{aligned} Q 1 \end{aligned}$ and $\begin{aligned} Q 3 \end{aligned}$ , you should include the median in both the lower half and upper half calculations.
::共有9个观测组。要计算 Q1 和 Q3, 您应该将中值包含在下半部和上半部的计算中。

Lowest value (minimum) : 3
::最低价值(最低):3

$\begin{aligned} Q 1 : 10 \end{aligned}$ (this is the median of 3, 7, 10, 14, 19)
::问题1:10(中位数为3、7、10、14、19)

$\begin{aligned} Q 2 : 19 \end{aligned}$
::问题2:19

$\begin{aligned} Q 3 : 23 \end{aligned}$ (this is the median of 19, 19, 23, 27, 29)
::问题3:23(这是19、19、23、27、29的中位数)

Upper value (maximum) : 29
::上值(最大) : 29

T he five number summary is 3, 10, 19, 23, 29.
::五个数字摘要是3、10、19、23、29。

Summary

The five number summary consists of the lowest value (minimum), first quartile (Q1), second quartile (median or Q2), third quartile (Q3), and the highest value (maximum).
::五个数字摘要包括最低值(最低值)、第一个四分位位数(Q1)、第二个四分位数(中位数或Q2)、第三个四分位数(Q3)和最高值(最高值)。

Quartiles divide the data into four parts:
- Q1 is the median of the lower half.
  ::问题1是下半部分的中位数。
- Q2 is the median of the entire dataset.
  ::Q2是整个数据集的中位数。
- Q3 is the median of the upper half.
  ::问题3是上半部分的中位数。
::将数据分成四个部分:Q1是下半部分的中位数。Q2是整个数据集的中位数。Q3是上半部分的中位数。

Review
::回顾

Compute the five number summary for each of the following sets of data:
::计算下列每组数据的5个数字摘要:

0.16, 0.08, 0.27, 0.20, 0.22, 0.32, 0.25, 0.18, 0.28, 0.27
77, 79, 80, 86, 87, 87, 94, 99
79, 53, 82, 91, 87, 98, 80, 93
91, 85, 76, 86, 96, 51, 68, 92, 85, 72, 66, 88, 93, 82, 84
335, 233, 185, 392, 235, 518, 281, 208, 318
38, 33, 41, 37, 54, 39, 38, 71, 49, 48, 42, 38
3, 7, 8, 5, 12, 14, 21, 13, 18
6, 22, 11, 25, 16, 26, 28, 37, 37, 38, 33, 40, 34, 39, 23, 11, 48, 49, 8, 26, 18, 17, 27, 14
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
49, 57, 53, 54, 49, 67, 51, 57, 56, 59, 57, 50, 49, 52, 53, 50, 58
18, 20, 24, 21, 5, 23, 19, 22
900, 840, 880, 880, 800, 860, 720, 720, 620, 860, 970, 950, 890, 810, 810, 820, 800, 770, 850, 740, 900, 1070, 930, 850, 950, 980, 980, 880, 960, 940, 960, 940, 880, 800, 850, 880, 760, 740, 750, 760, 890, 840, 780, 810, 760, 810, 790, 810, 820, 850
13, 15, 19, 14, 26, 17, 12, 42, 18
25, 33, 55, 32, 17, 19, 15, 18, 21
149, 123, 126, 122, 129, 120

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。

Section outline

General

五号摘要

The Five Number Summary ::五号数字摘要

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

The Five Number Summary
::五号数字摘要

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)