第3章 数据的描述
一、单项选择题 20~30道题
1平均数是对()
A、总体单位数的平均
B、变量值的平均
C、标志的平均
D、变异的平均
参考答案:B
2、权数的最根本作用体现在()的变动上。
A次数
B标志值
C权数比重
D标志值和次数
参考答案:C
3、某单位职工的平均年龄为35岁,这是对()的平均
A变量
B变量值
C数量标志
D数量指标
参考答案:B
4、加权算术平均数的实质权数是
A各单位占总单位数的比重
B各组的次数
C各组的标志值
D各组的频数
参考答案A
5、,集中趋势的主要测度值是()
A算术平均数
B中位数
C众数
D几何平均数
参考答案A
1、 某副食品公司所属的三个商店,2004年计划规定销售额分别为500万元,600万元,800万元,执行的结果分别为104%,105%,105%,则该公司三个商店平均完成计划的计算方法为:
A
B
C
D
参考答案B
2、 在常用的 平均数中,易受极端值影响的是
A算术平均数和几何平均数
B调和平均数和几何平均数
C算术平均数和众数
D众数和中位数
参考答案D
9、在有组距数列计算平均数时,用组中值代表组内变量一般水平的假定条件是
A各组的次数必须相等
B组中值能取整数
C各组变量值在本组内呈均匀分布
D各组必须是封闭组
参考答案:C
10标志变异指标与平均数代表性之间存在
A正比关系
B反比关系
C恒等关系
D倒数关系
参考答案B
11、在掌握了各组单位成本和各组产量资料时,计算平均单位成本所使用的方法应是
A算术平均数
B调和平均数
C几何平均数
D中位数
参考答案A
12两个总体的平均数不等,但标准差相等,则
A平均数小的代表性大
B平均数大的代表性大
C两个平均数代表性相同
D无法判断代表性大小
参考答案A
13标志变异指标是说明变量分布的
A变动趋势
B集中趋势
C离散趋势
D一般趋势
参考答案:C
14、再对一般水平不同的总体进行分布离散程度的比较时,应使用
A极差
B平均差
C标准差
D变异系数
参考答案D
15、皮尔逊(pearson)偏态系数测定数据分布,当SK小于0时,数据分布是
A左偏
B正态
C右偏
D无法判断
参考答案A
16当峰度系数<0数据分布为
A扁平分布
B尖峰分布
C峰度适中
D无法判断
参考答案A
17、假定100名大学生的四级英语考试成绩数据是:平均数70分,标准差5分。那么有多少个学生的考试成绩在58——82分之间?
A 75%
B 82.6%
C 89%
D 95.45%
参考答案B
18、某高校学生参加英语四级考试的合格率为90%,则合格率的方差为
A 0.9
B 0.09
C 0.3
D 0.03
参考答案B
19、下面是抽样调查的10个家庭住房面积(单位:平方米):55,75,75,90,90,90,90,105,120,150,这10个家庭住房面积的众数为
A 90
B 75
C 55
D 150
参考答案A
20、数据之间的差异程度或频数分布的分散程度称为
A集中趋势
B离散程度
C偏态
D峰度
参考答案B
21、某生产小组有9名工人,日产零件数分别为:12,15,9,12,13,12,14,11,10。根据数据计算的结果为
A均值=中位数=众数
B均值>中位数>众数
C中位数>均值>众数
D众数>中位数>均值
参考答案A
二、多项选择题
1、作为一个完整的统计表,从结构上看,一般要由下面哪几个部分组成:
A表头
B行标题、列标题
C数据资料
D主词
E宾词
参考答案 ABC
2、按对主词是否分组及分组的程度,统计表可以分为:
A简单表
B时间数列表
C复合分组表
D空间数列表
E、简单分组表
参考答案ACE
3、在数据集中趋势的测度值中,不受极端值影响的测度值是
A均值
B几何平均数
C众数
D中位数
E四分位数
参考答案CDE
4、在对两组数据进行差异程度比较时,不能直接比较两组数据的方差,因为两组数据的
A均值不同
B方差不同
C数据个数不同
D计量单位不同
E离差之和不同
参考答案AD
5、在数据离散程度的度量值中,不受极端值影响的测度值是
A全距
B异众比率
C四分位差
D标准差
E离散系数
参考答案BC
6、标志变异指标的作用,主要有
A衡量平均数代表性大小
B反映社会经济活动过程的节奏性和均衡性
C反映总体单位分布的均匀性和稳定性
D科学地确定必要的抽样单位的因素
E分析社会经济现象某总体的变动趋势
参考答案ABC
7、当总体单位的标志值出现极大极小且差异巨大时,反映总体一般水平应采用
A算术平均数
B调和平均数
C几何平均数
D众数
E中位数
参考答案DE
8、甲班同学的平均身高为
,标准差为
,乙班同学的平均身高为
,标准差为
,如果( ),甲班身高的均值的代表性低。
A
>
,
>
B
<
,
>
C
=
,
>
D
<
,
<
E
<
,
=
参考答案BCE
9、某公司下属5个企业,共有2000名职工,已知每个企业某月的产值计划完成百分比及其实际产值,要计算公司平均产值计划完成程度,则()
A应采用算术平均数
B应采用调和平均数
C实际产值是权数
D工人数是权数
E企业数是权数
参考答案BC
10、几何平均数主要用途是
A确定分组组数
B确定组距
C对比率、指数等进行平均
D计算组中值
E计算平均发展速度
参考答案 CE
11、四分位距
A主要用于对顺序数据离散程度的测度。也用于数值型数据
B四分位距反映了中间50%数据的离散程度
C数值越小,说明中间的数据越分散
D可用于衡量中位数的代表性
E且不受极端值的影响。
参考答案ABDE
12、方差的计算公式
A
B
C
D
E
参考答案ABCD
13、比例(成数)的标准差计算公式为
A
B
C
D
E
参考答案ACD
14、对于偏态系数
A若偏态系数 SK =0 为对称分布
B 若偏态系数 SK> 0 为右偏或正偏分布;
C若偏态系数 SK> 0为左偏或负偏分布;
D偏态系数 SK< 0 为左偏或负偏分布
E偏态系数 SK< 0为右偏或正偏分布
参考答案ABD
15、Z分数
A是变量值与其平均数的离差除以标准差的值。
B主要用于对变量的标准化处理,不改变该组数据的分布形状。
C Z分数给出了一组数据中各数值的相对位置,它并没有改变一个数据在该组中的位置
D标准化后,不同
观测值的比较只有相对意义,没有绝对意义
E。Z分数较多用于在对多个具有不同量纲的指标进行处理时。
参考答案ABCDE
三、名词解释(选择出若干)
集中趋势
算术平均数
调和平均数
几何平均数
众数
中位数
四分位数
离中趋势
异众比率
四分位差
全距
平均差
方差和标准差
变异系数
偏态。
偏态系数
峰度
峰度系数
Z分数
统计表
统计图
[参考答案]
集中趋势是一组数据向其中心值靠拢的倾向和程度。
算术平均数也称均值,是集中趋势的主要测度值,也是最常用的一个测度值。
调和平均数也称倒数平均数,它是变量值倒数的算术平均数的倒数。是均值的另一种表现形式。
几何平均数是n 个比率或变量值连乘积的 n 次方根。是集中趋势的测度值之一。
众数是位置平均数,是集中趋势的测度值之一,是出现次数最多的变量值。
中位数是把一组数据,按大小顺序排列后,处在数列中点位置的数值。是典型的位置平均数,集中趋势的测度值之一。
将数据按大小顺序排序后,分割成四等分,得到三个分割点,每个分割点上的数值称为四分位数。它是集中趋势的测度值之一。
离中趋势反映的是一组数据中各观测值之间的差异或离散程度。即反映各变量值远离其中心值的程度(离散程度)。
异众比率是离散程度的测度值之一,指的是非众数组的频数占总频数的比率。
四分位距是第三个四分位数与第一个四分位数之差的二分之一
全距也称极差,是一组数据的最大值与最小值之差
平均差是一组数据值与其均值之差的绝对值的平均数
方差是一组数据中各数值与其算术平均数离差平方的平均数,标准差是方差的平方根。
变异系数也叫离散系数,是标准差(或平均差)与其均值的比值,用来反映总体分布数列中变量值差异程度的相对指标。
统计学将数据分布的不对称性称作偏态。
偏态系数就是对数据分布不对称性的的测度值
峰度是次数曲线的高峰形态。即数据分布的扁平或尖峰程度
峰度系数是数据分布峰度的度量值
Z分数也称标准分数(standard score),或标准化值,是变量值与其平均数的离差除以标准差的值。
统计表是系统显示统计数据的重要方式之一。是以纵横交叉的线条所绘制,用来表现据
资料的
形式。
统计图是指表现统计数据的各种图形。它是用点、线、面的位置、升降或大小来表达统计数据数量关系的一种形式。是显示统计数据的重要方式之一。
四、简答题 5~10道题
1、简述统计表的构成
2、统计表可以分成几种?
3、条形图、线图、圆形图、直方图、茎叶图、箱线图、雷达图主要适用什么场合?
4、简述众数、中位数、平均数的三者之间关系、特点和应用。
5、简述变异指标的作用
6、变异系数在什么情况下使用?
7、 经验法则与切比雪夫定理在什么情况下使用?切比雪夫定理内容是什么?
统计表的形式可以有多种,但作为一个完整的统计表,从结构上看,一般要由下面几个部分组成:表头、行标题、列标题、数据资料。除基本结构之外,为使用方便和规范化,在统计表的下方,对表中的一些指标或数据作必要的说明和解释,为附加部分。从统计表的内容上看,分为主词和宾词。主词通常为统计表所要说明的总体及分组,一般位于统计表的左边;宾词通常是用来说明总体的各项指标,一般位于统计表的右边,但主词和宾词的位置并不是固定的。
从不同角度,统计表可以分为不同种类。
按对主词是否分组及分组的程度分为: 简单表、简单分组表、复合分组表。
按统计数列的性质不同分为: 时间数列表、空间数列表、时空数列表。
两种图表都可以用来表示与一组或几组分类相关的数据,可以用于不同现象的比较,也可用于时间数列描述现象的发展趋势。线图主要用于显示时间数列的数据,以反映事物发展变化的规律和趋势;圆形图主要用于总体内部的结构,各组成部分所占比例等;直方图是用来表示频数分布的图形;箱线图用于显示未分组的原始数据或分组数据的分布;雷达图适合多维数据即多个变量的图示,在显示或对比各变量的数值总和时十分有用。
众数、中位数和平均数的关系
对称分布的资料
分配为钟形,轻微不对称的资料,三者关系式可以表示为:
其中,又分为左偏分布和右偏分布
众数、中位数、平均数的特点和应用
众数、中位数、平均数的特点和应用
众数是位置平均数,不受极端值影响,具有不惟一性。数据分布偏斜程度较大时应用,主要用于分类数据
中位数是位置平均数,不受极端值影响。数据分布偏斜程度较大时应用;主要用于顺序数据平均数是最主要测度值,使用了全部数据,易受极端值影响。数学性质优良.主要用于数值型数据。数据对称分布或接近对称分布时应用效果最好。
变异指标或标志变动度,主要描述以平均数为中心的总体各单位间变量值差异程度。即从另一个侧面说明了集中趋势测度值的代表程度。衡量平均数的代表性。如果标志变动度大,各变量值与平均数离差的总和就大,平均数的代表性,均衡性,稳定性就低;反之,标志变动度越小,平均数的代表性,均衡性,稳定性就高。
如果是数据的性质不同,比较不同标志的变异程度,不同组别数据离散程度,或者对比的双方具有不同的平均水平,必须要用变异系数进行比较。变异系数消除了数据水平高低和计量单位的影响,使得对比结果真实反映总体的离散程度。
经验法则适用数据对称分布时,切比雪夫定理对任何分布形状的数据都适用。
切比雪夫定理:对于任意分布形态的数据,根据切比雪夫不等式,至少有
的数据落在k个标准差之内。约有
个数据落在平均数k倍标准差范围外。其中k是大于1的任意值,但不一定是整数。切比雪夫定理是我们能够说出在平均数的若干倍标准差范围之内的数据值所占的百分比。
五、计算题 10~15道题
1、某地区甲乙两个企业生产三种产品的单位成本和总成本资料如下表
产品名称
单位成本(元)
总成本(元)
甲企业
乙企业
A
B
C
10
15
20
10000
30000
40000
15000
15000
40000
试比较哪个企业的总平均成本高?并分析出现该结果的原因。
参考答案:
甲企业总平均成本:
元
乙企业总平均成本:
元
甲企业总平均成本高。主要是产品结构不同。
2、2005年某月甲乙两个农贸市场某农产品价格、成交量、成交额的资料如下:
品种
价格(元/千克)
甲市场成交额(万元)
乙市场成交量(万千克)
甲
乙
丙
1.2
1.4
1.5
1.2
2.8
1.5
2
1
1
合计
-——
5.5
4
试问该农产品哪一个市场的平均价格高?并说明其原因。
参考答案:
甲市场平均价格:
元/千克
乙市场平均价格:
元/千克
甲市场平均价格高。因为成交量结构不同。通过计算可以得到,两市场在成交总量上相同。但甲市场价格较高的乙产品成交量(2万千克)占总成交量的50%,而价格最低的甲产品在乙市场中成交量占总成交量的50%,所以,甲市场平均价格高。
3、(选做题)根据下列企业工人的技术等级的数据资料确定中位数和众数
技术等级(级)
1
2
3
4
5
6
7
8
工人数(人)
22
38
75
87
64
20
12
7
参考答案:
中位数:
位置=n+1/2=(325+1)/2=163
计算累计次数:从第一组至第三组的累计次数=22+38+75=135
从第一组至第四组的累计次数=22+38+75+87=222
因此,中位数应该在第四组。根据资料,中位数Me =4级
众数Mo=4级(出现次数最多的变量值)
5、早在1988年末美国人口统计的
指出有2500万美国人早晨起床后在他们的家里办公。个人计算机的不断普及是越来越多的人在家里工作的重要原因之一。文章公布了当时在家里上班的人士有关年龄方面的资料。下面是这些人年龄资料的一个样本:
22 58 24 50 29 52 57 31 30 41
44 40 46 29 31 37 32 44 49 29
(1)、计算样本的平均数和众数
(2)、所有成年人总体的年龄中位数是40.5岁。问:根据上面样本的中位数来判断家里工作的人比所有成年人总体年轻还是年长?
(3)、计算第一个和第三个四分位数
参考答案
样本的平均数
=775/20=38.75岁
众数:Mo=29岁(出现次数最多)
中位数:将20个数据由小到大排序,22,24,29,29,29,30,31,31,32,37,40,41,44,44,46,49,50,52,57,58
中位数位置=(20+1)/2=10.5
中位数=(37+40)/2=38.5岁
根据计算结果可以看出,在家工作的人年轻一些。
第一个四分位数
的位置:(n+1)/4=5.25
=29+0.25*(30-29)=29.25
第三个四分位数
的位置:3(n+1)/4=3*(20+1)/4=15.75
=46+0.75*(49-46)=48.25
6、某信用社连续10年的贷款利率分别为:地1——3年为8%,第4——6年为10%,第7——8年为12%,第9年为10%,第10年为9%。计算在单利和复利情况下的平均年利率。
参考答案
单利情况下的平均年利率
复利情况下的平均年利率:
平均年利率为9.69%
7、某省各地区八五期间的收入水平的有关数据如下,
人均GDP水平(千元)
地区数(个)
1——3
3——5
5——7
7——9
9——11
1
3
5
2
1
根据表中数据计算该地区收入水平的标准差系数。
参考答案
(组中值分别为2,4,6,8,10)
8、检查800只电子元件中有736只为合格品,求平均合格率和标准差。
参考答案
平均合格率=736/800=92%
标准差:
9、某装配车间共有装配工人200人,某日对其日装配工件数进行统计,分组资料如下:
日装配工件数(件)
工人数(人)
4——6
6——8
8——10
10——12
12——14
25
40
85
35
15
合计
200
平均数
标准差
标准差系数
偏态系数和峰度系数。
参考答案
有关数据计算表
日装配工件数
工人数
f
组中值x
X*f
4——6
25
5
125
-3.75
351.56
-1318.3594
4943.8476
6——8
40
7
280
-1.75
122.50
-214.375
375.1563
8—10
85
9
765
0.25
5.31
1.3281
0.3320
10—12
35
11
385
2.25
177.19
398.6719
897.0117
12—14
15
13
195
4.25
270.94
1151.4844
4893.8086
合计
200
——
1750
——
927.50
18.75
11110.1562
平均数
=1750/200=8.75(件/人)
标准差
标准差系数
偏态系数
偏态系数 SK =0 为对称分布, SK> 0 为右偏分布,本数据基本呈现正态分布,略微有一点右偏。
峰度系数。
偏度系数<3,数据呈现扁平分布
10美国商务部经济分析局某一年的报告曾指出,当年加利福尼亚的人均收入是21884美元。如果标准差是6000美元,年收入是12500美元的某居民的Z分数是多少?年收入是50000美元的某居民的Z分数又是多少?解释这两个Z分数,评价这两个居民的年收入是否为异常值?
参考答案
年收入是12500美元的某居民的Z分数
收入是50000美元的某居民的Z分数
年收入是12500美元的某居民的Z分数为-1.564,说明与当年加利福尼亚的人均收入相比,低了1.564个标准差;收入是50000美元的某居民的Z分数为4.686,
他的年收入与当年加利福尼亚的人均收入相比,高4.686个标准差,由于超过了3个标准差,这个数据成为离群点(异常值)
11、在国外曾经销售的一种新型节能汽车每加仑平均行驶里程是27.5英里,假定标准差是每加仑3.5英里。那么
①利用切比雪夫定理计算每加仑行驶里程在20.5——34.5英里、18.75——36.25英里、17——38英里的新型汽车所占百分比。
②如果假定新型汽车每加仑行驶里程服从钟形分布是合理的话,那么每加仑行驶里程在20.5——34.5英里新型汽车所占百分比是多少?17——38英里呢?
参考答案
每加仑行驶里程在20.5——34.5英里新型汽车所占百分比:
先计算Z分数:(20.5-27.5)/3.5= -2
(34.5-27.5)/3.5= 2
根据切比雪夫定理
计算百分比:
至少有75%的新型汽车每加仑行驶里程在20.5——34.5英里之间。
同理可以计算每加仑行驶里程在18.75——36.25英里新型汽车所占百分比
Z分数:(18.75-27.5)/3.5= -2.5
(36.25-27.5)/3.5=2.5
=84%
至少有84%的新型汽车每加仑行驶里程在18.75——36.25英里之间。
每加仑行驶里程在17——38英里的新型汽车所占百分比
Z分数:(17-27.5)/3.5= -3
(38-27.5)/3.5=3
=89%
至少有89%的新型汽车每加仑行驶里程在17——38英里之间。
六、综合应用题 5~10道题
1、某银行3月份为100家企业贷款情况资料,见下表
贷款额(万元)
贷款企业数(个)
0——10
10——20
20——30
30——40
40——50
20
10
22
34
14
合计
100
计算平均每家企业的贷款额;(贷款额 众数;贷款额中位数。并判断数据分布的形态。)
参考答案
平均数为26.2
贷款额 众数为33.75
贷款额中位数29.09
左偏。
2、为获得某地区工人每小时工资方面的有关信息,调查了该地区25个工人,得到的数据如下:
11.5
8
9.9
13.15
9.15
8.4
13.65
6.85
13.1
12.05
11.75
7.05
15.35
6.65
8.45
10.05
9.05
11.1
13.1
5.85
10.25
11.9
14.7
9.2
9.8
计算工人每小时工资的平均数、中位数、众数、全距、标准差、偏度、峰度
参考答案
平均
10.4
标准误差
0.519495
中值
10.05
模式
13.1
标准偏差
2.597475
样本方差
6.746875
峰值
-0.8075
偏斜度
0.109199
区域
9.5
最小值
5.85
最大值
15.35
求和
260
计数
25
3、公路损失数据协会《伤害和碰撞数据记录》根据某一年存入
的保险索赔数,对汽车型号进行了等级排序,指数等级平均为100。低等级说明该型号的汽车更安全、更好。下面是20种中型卡车和20种小型卡车的等级数据:
中型卡车等级数据
81
81
100
76
91
60
103
68
93
51
119
81
127
58
82
91
68
75
128
82
小型卡车的等级数据
73
103
108
122
100
119
118
96
127
108
103
133
100
109
120
80
124
113
102
140
分别计算中型卡车、小型卡车的最小值、最大值、第一四分位数、第二四分位数
第三四分位数;
画出箱线图
根据你的汇总资料,比较说明中型卡车和小型卡车的安全性能。
参考答案
中型卡车
小型卡车
最小值
51
73
第1四分位数
73.25
101.5
第2四分位数
81.5
108.5
第3四分位数
94.75
120.5
最大值
128
140
箱线图略
根据四分位数的计算结果,50%的中型卡车等级在73.25——94.75之间,75%的中型卡车等级在94.75以下,25%的中型卡车等级(5个数据)超过了94.75;从箱线图来看,略偏但基本呈现正态。
小型卡车的结果表明,50%的小型卡车等级在101.5——120.5之间。从原始数据看,只有三个等级数据在100以下,有两个正好100。平均等级及等级以下只占25%,或者说75%的小型卡车等级都在平均等级以上。图形偏度大。指数等级高的次数比重大。
根据数据分析结果,中型卡车安全性能明显优于小型卡车。
4、以下数据给出的是道森供应公司和J.C克拉克批发公司两家公司交付定货天数的数据。
道森供应公司交货天数:
11 10 9 10 11 11 10 11 10 10
克拉克批发公司交货天数
8 10 13 7 10 11 10 7 15 12
哪一家供应公司交货时间更稳定、更可靠。用数据支持你的结论。
参考答案
克拉克批发公司
平均
10.3
标准误差
0.817177
中值
10
模式
10
标准偏差
2.58414
样本方差
6.677778
峰值
-0.35087
偏斜度
0.359289
区域
8
最小值
7
最大值
15
求和
103
计数
10
道森供应公司
平均
10.3
标准误差
0.213437
中值
10
模式
10
标准偏差
0.674949
样本方差
0.455556
峰值
-0.28299
偏斜度
-0.43364
区域
2
最小值
9
最大值
11
求和
103
计数
10
从数据计算结果看,平均交货天数都为10.3天,但标准差不同,道森供应公司标准差为0.674949,克拉克批发公司标准差为2.58414,由于平均数相同,因此可以认为道森供应公司交货时间更稳定、更可靠。从全距看,结论相同。
5、某职员每天可以有两种方法去上班。公共交通和汽车。每种方法所需要的样本时间记录如下(单位:分钟)
公共交通:28 29 32 37 33 25 29 32 41 34
汽车 :29 31 33 32 34 30 31 32 35 33
应该选用那一种方法更合适?请解释
参考答案:
公共交通上班平均时间=32
汽车上班平均时间=32
公共交通上班时间标准差=4.64
汽车上班时间标准差=1.93
应选汽车。变异程度小。
6、我国2002年某月某日的上海及深圳证券交易所67只绩优股的市盈率资料如下:
21.94
36.2
28.66
21.58
26.05
46.52
34.18
13.58
19.44
22.61
24.00
41.61
21.58
25.10
25.33
17.70
28.63
53.24
39.27
70.55
23.89
22.95
18.32
21.91
21.00
26.26
23.61
26.69
25.44
13.43
17.07
33.18
31.72
18.63
19.71
22.22
12.38
32.00
20.40
27.07
24.76
11.06
16.59
15.00
42.48
44.07
43.26
21.31
22.89
14.38
57.00
16.71
15.68
28.97
20.90
35.69
18.91
7.07
15.40
25.67
15.51
19.23
21.90
13.22
23.58
40.92
7.35
根据以上数据对67只股票的市盈率进行描述性分析,并对分析结果写成一段文字。(提示:可通过图形,相关指标进行分析)
参考答案:
文字描述:可以从集中趋势,离中趋势,分布形态及结论和看法。
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