Statistics 8th Edition Jia Junping Chapter 4 Summary and after class exercise answers

1. Examination site induction

2. Test site tips

（1） Concentration trend 、 A measure of discrete trends , Including the meaning of each indicator 、 Calculation formula 、 characteristic 、 significance 、 Scope of application （ choice question 、 Short answer 、 Calculation test site ）;
（2） The number of 、 The characteristics and application occasions of the three indicators of median and average , The relationship between the three indicators under skew distribution （ choice question 、 Short answer 、 Calculation test site ）;
（3） A measure of the shape of the distribution ： Numerical meaning of skewness coefficient and kurtosis coefficient （ choice question 、 Short answer test site ）.
（4） Calculation formula and application of standard score （ choice question 、 Short answer 、 Calculation test site ）;
（5） Rule of thumb 、 The concrete application of Chebyshev inequality （ Multiple choice test site ）.

3. Core of examination site

（1）： The measurement of concentration trends
 

【 Be careful 】 The relationship between centralized trend measurement indicators in the distribution of different degrees of skewness ：① In symmetric distribution , The number of 、 The median is equal to the average ;② Left biased distribution , The data has a minimum , Pull the average to the minimum , The mode and median are not affected by extreme values , Yes x＜Me＜Mo;③ Right biased distribution , The data has a maximum , It must pull the average to the maximum , therefore Mo＜Me＜x.

(2) A measure of the degree of dispersion
The degree of dispersion of data reflects the degree to which the value of each variable is far away from its central value , The smaller the degree of dispersion , The better the representativeness .

(3) Measure of distribution shape
Skewness is a measure of the symmetry of data distribution , Kurtosis is a measure of the degree of flattening or peaking of data distribution .
 

(4) A measure of relative position
（1） Standard score （ Normalized value or z fraction ）

① Calculation formula ：zi＝（xi－x）/s.

② characteristic ： The average is 0、 The standard deviation is 1; The data is linearly transformed , Do not change the position of the data in the Group , Nor does it change the shape of the distribution of this group of data .

③ purpose ：
a． Determine the relative position of a certain data in the group of data , It can be used to judge whether a group of data has outliers ;
b． When dealing with multiple variables with different dimensions , Standard scores can be used to standardize variables .
（2） Rule of thumb （3σ principle ）
When a set of data is symmetrically distributed , The rule of thumb shows ：

① There are about 68% The data is on average ±1 Within the range of standard deviation ;

② There are about 95% The data is on average ±2 Within the range of standard deviation ;

③ There are about 99% The data is on average ±3 Within the range of standard deviation .
therefore , On average ±3 The range of standard deviation contains almost all the data , And in the ±3 Data outside of the standard deviation is called outliers .
（3） Chebyshev inequality

① Concept ： For data with arbitrary distribution patterns , According to Chebyshev inequality , There are at least （1－1/k2） The data of this project falls in ±k Within standard deviation . among k It is greater than 1 Any value of , It doesn't have to be an integer . about k＝2,3,4, The meaning of this inequality is ：
a． There are at least 75% The data of falls on the average ±2 Within the range of standard deviation ;
b． There are at least 89% The data of falls on the average ±3 Within the range of standard deviation ;
c． There are at least 94% The data of falls on the average ±4 Within the range of standard deviation .

② characteristic ： Chebyshev inequality is applicable to data with any distribution shape .

Two exercises

1 A car retail store 10 Sales people 5 The number of cars sold in the month （ Company ： platform ） The order is as follows ：
2　4　7　10　10　10　12　12　14　15
requirement ：
（1） Calculate the mode of car sales 、 Median and average .
（2） Calculate the quartile according to the defined formula .
（3） Calculate the standard deviation of sales .
（4） Explain the characteristics of the distribution of automobile sales .

Explain ：（1）10 Sales people 5 Of the number of cars sold in the month , sales 10 There are the most people in this car , by 3 people , So mode M0＝10.
The median position ＝（n＋1）/2＝（10＋1）/2＝5.5, therefore Me＝（10＋10）/2＝10（ platform ）.
The average
（2） From the data in the question ：

①QL Location ＝n/4＝10/4＝2.5, namely QL In the 2 A numerical （4） And the 3 A numerical （7） Between 0.5 Location . therefore
QL＝（4＋7）/2＝5.5（ platform ）

②QU Location ＝3n/4＝3×10÷4＝7.5, namely QU In the 7 A numerical （12） And the 8 A numerical （12） Between 0.5 Location , therefore
QU＝（12＋12）/2＝12（ platform ）

（3） By average x＝9.6 Available ：

（4） Because the average is less than the median and mode , Therefore, the distribution of car sales is left biased .

2 Random sampling 25 Network users , Get their age data as shown in the table 4-4 Shown .

requirement ：
（1） Calculate the mode 、 Median .
（2） Calculate the quartile .

（3） Calculate the mean and standard deviation .
（4） Calculate skewness coefficient and kurtosis coefficient .
（5） Comprehensively analyze the distribution characteristics of netizens' age .

Explain ：（1） Counter table 4-4 The data in is arranged from small to large ：

According to the sorting data , The most frequent occurrence of age is 19 and 23, All appear 3 Time , So there are two modes , namely Mo＝19 and Mo＝23.
Due to the median position ＝（n＋1）/n＝（25＋1）/2＝13, therefore Me＝23（ year ）
（2） The quartile can be calculated from the data in the question ：

①QL Location ＝n/4＝25/4＝6.25, namely QL In the 6 A numerical （19） And the 7 A numerical （19） Between 0.25 Location , therefore

QL＝19＋0.25×（19－19）＝19（ year ）

② because QU Location ＝3×25/4＝25/4＝18.75, namely QU In the 18 A numerical （25） And the 19 A numerical （27） Between 0.75 Location , therefore
QU＝25＋0.75×（27－25）＝26.5（ year ）
（3） The average

By average x＝24 Available ：

（4） The coefficient of skewness ：

Peak state coefficient ：

（5） The results of a comprehensive analysis of the age distribution of netizens are as follows ： From mode 、 Median and average , Netizens are aged 23 People around the age of are in the majority ; The standard deviation is large , It shows that there is a large age difference between netizens ; The skewness coefficient is greater than 1, It shows that the age distribution of netizens is right , And the degree of deflection is very large . The kurtosis coefficient is positive , It is peak distribution , It shows that the age distribution of netizens is relatively concentrated .

3 An e-commerce 6 Sales data for each day of the month

requirement ：
（1） Calculate the average and median daily sales of the department store .
（2） Calculate the quartile .
（3） Calculate the standard deviation of daily sales .

Explain ：（1） The average of daily sales is ：

Will table 4-5 Data sorting in ：

The median position ＝（30＋1）/2＝15.5, therefore Me＝（272＋273）/2＝272.5
（2） The quartile can be calculated from the data in the question ：

① because QL Location ＝30/4＝7.5, namely QL In the 7 A numerical （258） And the 8 A numerical （261） Between 0.5 Location . therefore

QL＝（258＋261）/2＝259.5（ Ten thousand yuan ）②QU Location ＝3×30÷4＝22.5, namely QL In the 22 A numerical （284） And the 23 A numerical （291） Between 0.5 Location . therefore
QU＝（284＋291）/2＝287.5（ Ten thousand yuan ）
（3） By the average of daily sales x＝274.1 Available ：

4 Draw in an area 120 companies , Group by profit amount .

requirement ：

（1） Calculation 120 The average and standard deviation of the profits of enterprises .
（2） Calculate the skewness coefficient and kurtosis coefficient of the distribution .

Explain ：（1） The average calculation process is shown in table 4-7 Shown .
surface 4-7　 A region 120 Calculation process table of average profit of enterprises

By the table 4-7 The average can be obtained from the data in ：

The calculation process of standard deviation is shown in table 4-8 Shown .

surface 4-8　 A region 120 Calculation process table of standard deviation of profits of enterprises

By the table 4-8 Data available in ：

（2） The calculation process of skewness coefficient and kurtosis coefficient is shown in table 4-9 Shown .
surface 4-9　 Calculation process of skewness coefficient and kurtosis coefficient

From the data in the table, the skewness coefficient is ：

The kurtosis coefficient can be obtained from the data in the table ：

5 The average daily output of a product production line is 3700 Pieces of , The standard deviation is 50 Pieces of . If the yield on a given day is lower or higher than the average yield , And fall into ±2 Out of the range of standard deviation , Think of this production line “ Out of control ”. surface 4-10 It is the output of each day of the week , Which days did the production line lose control ？
surface 4-10　 Production line output data

Explain ： because x＝3700,s＝50, Use the formula zi＝（xi－x）/s You can calculate the standard score every day , As shown in the table 4-11 Shown .

surface 4-11　 Standard score table of production line output
By the table 4-11 According to the data in ： Monday 、 The absolute value of the standard score on Saturday is greater than 2, So I lost control on Monday and Saturday .

6 A product requires manual assembly , There are three alternative assembly methods . To test which method is better , Random sampling 15 One worker , Let them assemble in three ways . surface 4-12 yes 15 The number of products assembled by workers in three ways at the same time .
surface 4-12　 The number of products assembled by workers in the same time in three ways （ Company ： individual ）

requirement ：
（1） What method are you going to use to evaluate the advantages and disadvantages of the assembly method ？

（2） If you choose a method , What choice would you make ？ Try to explain why . Explain ：（1） It should be evaluated in terms of average and standard deviation . When comparing the discreteness of various methods , Discrete coefficients should be used .
（2） surface 4-13 Given with Excel Calculate some main descriptive statistics .
surface 4-13　 Description statistics

From the centralized trend of the three methods , Method A The average output of is the highest , The median and mode are also higher than the other two methods . In terms of dispersion , The discrete coefficients of the three methods are ：
vA＝sA/xA＝2.13/165.6＝0.013

vB＝sB/xB＝1.75/128.73＝0.014
vA＝sC/xC＝2.77/125.53＝0.022
Method A The dispersion coefficient of is the smallest , That is, the degree of dispersion is the smallest , Therefore, the method should be selected A.