Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

STATISTICS OF EDUCATION
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

INTRODUCTION

?统计学的发展史简介
?教育统计学的主要内容
?统计学中的基本概念
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1.1 统计学的发展史简介
1.1.1 统计学的起源
?STATISTICS(统计学 )一词源于法语 STATUS(状态 )

?H.Conring 创立国势学体系

?十七世纪，政治算术统计学在英国兴起。
?概率论的起源与发展。概率论的发展最早源于赌博
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
C.Huygens 著, 骰子赌博的理论,

?描述统计学的发展：生物统计学的影响
F.galton 主要研究平均值的偏差和回归问题
K.Pearson 在前者的基础上发展出许多描述统计方法：频数分

?推断统计学的诞生
R.A.Fisher 统计推断学的创立
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1.1.2 统计学的应用
1.1.3 统计学的发展
?理论和方法不断地被完善和深化

?计算机及相关的软件成为统计工作中不可少的工具
SPSS,SAS,DATA-TEST,STATA等软件
?发展成为独立的交叉性学科

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1.2 教育统计学的主要内容
1.2.1 统计学与教育统计学
1,统计学 (Statistics)

2,教育统计学 (Statistics of education)

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1.2.2 教育统计学的基本内容
1,描述统计 (Descriptive Statistics)

2,推断统计 (Inferential Statistics)

3.实验设计 (Experimental designs)

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

○ 如何计算样本容量
○ 确定怎样的实验对照形式
○ 如何实现实验组和对照组的等组化
○ 如何安排实验因素
○ 如何控制无关变量
○ 用什么统计方法处理及分析实验结果。
4, 描述统计、推断统计、实验设计三者之间的关系
?描述统计是推断统计的基础
?推断统计是描述统计的升华
?良好的实验设计是获得真实的，有价值的数据，并推测未知

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1.2.3 教育统计学的结构

?经常性资料
?调查数据
?实验数据
?历史资料
?测验数据
?统计图表
?集中量
?差异量
?相关量
?预测统计
predictive
?Z 检验
?T 检验
?非参数检验
?方差分析
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1.3 教育统计学中的基本概念
1,变量与常量 (Variable vs Constant)
2,随机变量 (Random Variable)

3,总体与样本 (Population vs Sample)
4,统计量与参数 (Statistic vs Parameter)

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
5,点计数据与度量数据
6,间断变量与连续变量
(Discrete variable vs Continuous variable)
7,名称变量、等级变量、等距变量及比率变量
(Nominal variable, Ordinal variable, Interval variable,
Ratio variable)
Nominal variable,Has values which differ in quality only
Ordinal variable,Has values ordered by quality
Interval variable,Has ordered by quality with equal-sized
interval between each
Ratio variable,Like interval variables but with a true zero
point
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

ORGANIZING AND PRESENTING DATA
?如何做表
PRESENTING DATA IN A TABLE
?如何做图
PRESENTING DATA IN A GRAPH
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2.1 数据的基本来源
1,经常性资料
2,专题性资料

2.2 数据初步整理的基本过程
1,分组 (如表 2.1)

１８．５－－１９．５ １５
１９．５－－２０．５ １０
２０．５－－２１．５ ７
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2,汇总

3,编统计图、统计表

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2.3 统计表
2.3.1 表的基本结构
?标题
?表号
?标目 (横标目、纵标目 )
?线条 (三栏一竖 )
?数字 (表的主要内容 )
?表注
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2.3.2 统计表的种类
1,简单表

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2,分组表

3,复合表

n S
４岁 市区 １６７ ６３,３７ １９,１７

６岁 市区 １６７ ９１,４７ １９,５３

X
X
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2.3.3 连续变量频数分布表列法
2.3.3.1 概念
1,频数

2,频数分布

3,频数分布表

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2.3.3.2 连续变量频数分布表的编制

135 134 129 133 131 131 131 134 125 128
135 127 127 133 130 132 132 129 124 132
122 124 127 131 137 132 133 134 124 128
135 133 131 123 115 132 134 138 124 132
128 136 127 120 125 131 136 127 124 129
129 132 138 125 131 120 121 144 128 133
128 127 130 120 121 122 127 121 125 130
140 121 126 130 122 128 127 125 127 131
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1,求全距 (Range,R)

2,决定组数与组距 (Intervals and Width)

3,决定组限

39.21029 ???组数 R
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
4,登记频数并计算

5,计算频数 (Frequency)

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

115 122 124 127 128 129 131 132 133 135
120 122 125 127 128 129 131 132 133 136
120 122 125 127 128 130 131 132 134 136
120 123 125 127 128 130 131 132 134 137
121 124 125 127 128 130 131 132 134 138
121 124 125 127 128 130 131 133 134 138
121 124 126 127 129 131 132 133 135 140
121 124 127 127 129 131 132 133 135 144
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

115-
118-
121-
124-
127-
130-
133-
136-
139-
142-
116.5
119.5
122.5
125.5
128.5
131.5
134.5
137.5
140.5
143.5
1
3
8
10
20
19
12
4
2
1

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2.3.3.2 累积频数和累积百分比分布表
1.区分几个概念

Mid-points

(Absolute) Frequency

Relative Frequency

(Absolute) Cumulative Frequency

Relative Cumulative Frequency
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2,累积频数和累积百分比分布表

115-
118-
121-
124-
127-
130-
133-
136-
139-
142-
116.5
119.5
122.5
125.5
128.5
131.5
134.5
137.5
140.5
143.5
1
3
8
10
20
19
12
4
2
1
.0125
.3750
.1000
.1250
.2500
.2375
.1500
.0500
.0250
.0125
1
4
12
22
42
61
73
77
79
80
1.25
5.00
15.00
27.50
52.50
76.25
91.25
96.25
98.75
100

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2.4 统计图
2.4.1 表示间断变量的统计图
1,直条图
?是利用条形的长短比较各种统计指标的大小 。

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
?绘制直条图注意点,
?图形的尺度必须以零点为起点，同时尺度上的任何单位必须

?条形的长短表示数量的多少。
?各条形的宽度必须相等，各条形之间的间隔应一致，一般为

?各条形的排列应有一定的顺序。
?直条的顶端和下端不要注写数字。
?在复合条形图和条形结构图中应采用不同的线纹或颜色加以

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2．圆形图
?圆形图的定义

?绘制步骤
?求各组成部分所占百分比
?求组成部分的中心角度数
?以圆的下半径 (或上半径 )为基线，按被比事物特定顺序，

?用不同线条或不同颜色将各扇形加以区别，并在各扇形内

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

40.2%
40.8%
15.9%
3.1%
144.72°
146.88 °
57.24 °
11.16 °
40.2%
40.8%
15.9%
3.1%

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2.4.2 表示连续变量的统计图
1．线形图
?定义

?绘制方法
?先画一条直角坐标系，横轴表示时间或自变量，纵轴表示

?描点：用直线连接相邻两点。 (按时间顺序连成线条即成 )
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

49
51
53
55
2.0
3.5
4.0
4.5
6.0
0
1
2
3
4
5
6
7

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2,频数分布图

?注意点,
?绘折线，不画光滑曲线
?图中相比较的线一般不超过五条，图中不用文字或数字表

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
?直方图
?用面积表示频数分布，用各组上下限的矩形面积表示各组

115-
118-
121-
124-
127-
130-
133-
136-
139-
142-
116.5
119.5
122.5
125.5
128.5
131.5
134.5
137.5
140.5
143.5
1
3
8
10
20
19
12
4
2
1
1
4
12
22
42
61
73
77
79
80
1.25
5.00
15.00
27.50
52.50
76.25
91.25
96.25
98.75
100

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
?作横轴：把上表第 (1)列的上、下限或第 (2)列的组中值分置

?作纵轴：在纵轴上表明尺度及其单位，以指示频数。
?在纵轴上定出各组频数高度，并在各组频数高度处划一横

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
0
2
4
6
8
10
12
14
16
18
20
115 118 121 124 127 130 133 136 139 142 145
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
0
2
4
6
8
10
12
14
16
18
20
115 118 121 124 127 130 133 136 139 142 145

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
?多边图
?特点：以纵轴上的高度表示频数的多少。
?绘制：以各组的中点为横坐标，以各组的频数为纵坐标

?图形的两端应该引至外侧一组的中点与基线相接。

0
5
10
15
20
25
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
0
2
4
6
8
10
12
14
16
18
20
115 118 121 124 127 130 133 136 139 142 145

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
?累积频数和累积百分比多边图
?累积频数多边图的绘制,
※ 作横轴

※ 作纵轴

※ 描点

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

115-
118-
121-
124-
127-
130-
133-
136-
139-
142-
116.5
119.5
122.5
125.5
128.5
131.5
134.5
137.5
140.5
143.5
1
3
8
10
20
19
12
4
2
1
1
4
12
22
42
61
73
77
79
80
1.25
5.00
15.00
27.50
52.50
76.25
91.25
96.25
98.75
100

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

S型曲线特殊应用是,

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
?将统计资料通俗化、形象化、使人一目了然、易于接收。
?有利于统计资料的比较对照，有助于科学分析研究。
?根据资料的性质和分析的目的，正确选择适合的图形。
?要写明图的标题和图号，文字明确扼要，写在图的正下方。
?图内应写出所依据的数字，如未标明数字则应附以统计表或

?图内资料配置一般应从左到右。
?图中如有必须解释的地方，可用图注加以说明。
2.4.3 统计图的功用
2.4.4 绘制统计图的规则
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1．将下例 125名中学生语文测验分数编成累积频数，累积百分

29.5

34.5

39.5

44.5

49.5

59.5

64.5

69.5

74.5

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

20－ 22.5 3 1
25－ 27.5 1 14
30－ 32.5 6 20
35－ 37.5 10 19
40－ 42.5 18 21
45－ 47.5 21 21
50－ 52.5 29 14
55－ 57.5 28 13
60－ 62.5 40 5
65－ 67.5 31 4
70－ 72.5 32 2
75－ 77.5 19 0
80－ 82.5 14 0
85－ 87.5 10 0
90－ 92.5 4 0

2、把下列甲乙两组学生化学成绩的分布制在同一直角坐标上，以资比较。
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

CENTRAL TENDENCY
?三种主要集中量
THREE MEASURES OF CENTRAL TENDENCY
?集中量的比较
COMPARING MEASURES OF CENTRAL TENDENCY
?对称分布和偏态分布
SYMMETRICAL AND SKEWED DISTRIBUTION
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3.1 概述
3.1.1 描述性统计 (DESCRIPTIVE STATISTICS)
?概念

?种类

?
?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3.1.2 集中量的概念及作用

?集中量的作用,

?集中量 (CENTRAL TENDENCY)的种类,

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1、概念

?
2．公式,
N
XX ??
( 算术平均数 = )

X=各观察值
N=观察值的个数
?
3.2 算术平均数 (MEAN)
3.2.1 算术平均数概念
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1,原始数据计算法

9,6,8,9,7,6,8,9,7,7。试计算这些幼儿的计算练

N
XX ??
767798679869 ???????????? X
)(6.71076 分??X
3.2.2 算术平均数计算

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

N
fxX ??

?fx
?f ?f
2,频数分布表计算法 (组中值计算法 )
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3.2.2 算术平均数的应用及其特点

(1)优点,
○ 反应灵敏，一组数据中任何一个数值发生或大或小的变化，

○严密确定。由同一组数据计算出来的平均数是同一个值。
○计算简便，只需四则运算。
○受抽样变动的影响较小。
○是计算方差、标准差、相关系数以及推断统计的基础。
(2) 缺点,
○ 易受两极端数值的影响 (只要一个极低值,就会下降，

○一组数据中某个数值模糊或不确切，就无法计算其 。
X
X
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

3.3 中位数 (MEDIAN,Md)
3.3.1 中位数概念
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1,原始数据计算方法

3,5,7,8,9,11,14

3.3.2 中位数 计算方法
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2,频数分布表计算法

6,9,10,11,12,14,13,17
Md= 5.11
2
1211 ??
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
○ 计算公式,Md=Lmd+( n1) (由小向大计算 ) ?
2
N
mdf
i

N表示总额数
n1表示小于中位数所在组下限的频数总和
i表示额数分布表上的组距
fmd表示中位数所在组的频数
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
○ 计算步骤,
a,求
b,确定中位数所在组

c,确定由中位数所在组取多少个频数，就能使由上往下
(或下往上的积累频数等于,即求 n1,n1为小于

n1= -23=1
242482 ??N
2
N
2
N
?2N
?2N
248
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
d,计算中位数所在组所取频数的距离

fmd是中位数所在组的频数
i=组距

?2N fmdi
23248 ? 75
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Md=L+( n1)× (由上往下数的频数 )
=80+( -23) × =80.71

U表示中位数所在组的上限
n2表示大于中位数所在组上限的频数总和

=85- =80.71
?2N fmdi
2
48 75
2
N fmdi
18248 ? 75
7
30
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

○由上往下计算 Md时，当小于某一组下限的累积频数正好等于

○由下往上计算 Md时，大于某一组上限的累积频数正好等

○中位数是百分数中的特例。
2
N
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

3.4 百分位数 (PP)
3.4.1百分位数概念
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3.4.2百分位数 的计算方法

Pp = Lp + (p*N - n)
pf
i

Pp表示百分位数
p表示与百分位数相对应的比数
N表示总频数
Lp表示百分位数所在组的下限
n表示小于百分位数所在组下限的频数总和
fp表示百分位数所在组的频数。
i表示组距。
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

3.5 众数 (Mo)
3.5.1 概念
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3.5.2 众数的计算方法
1、用观察法直接寻找粗略众数

○在一组原始数据 2,4,3,6,4,5,4其中频数出现最多的

○在频数分布表中，频数最多一组的组中值就是粗略众数
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2、用公式求理论众数的近似值 (p38)

Mo 3Md-2

Mo=3× 77.89-2× 78.20=77.27

? X

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3、练习一
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

115-
118-
121-
124-
127-
130-
133-
136-
139-
142-
1
3
8
10
20
20
11
4
2
1

3、练习二
?求众数
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3.6 平均数、中位数,众数 三者间的关系
3.6.1 集中量的适用范围
3.6.2 分布的不同形态影响三者间的关系
1、分布为正态或对称分布时,
Normal Distribution or Symmetrical Distribution
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2、分布为正偏态分布时,
Negative Skew Distribution
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3、分布为负偏态分布时,
Positive Skew Distribution
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

VARIABILITY
?DESCRIBE IN WORDS THE TERM VARIABILITY
?DEFINE THE RANGE AND SEMI-INTERQUARTILE
RANGE
?DEFINE THE VARIANCE
?DEFINE THE STANDARD VARIANCE
?COMPUTE ALL OF DIFFERENT VARIANCE
?COMPARE DISTRIBUTIONS WITH VARIANCE
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
4.1 差异量的概念
1,概念

A组,8,8,9,10,11,12,12 ( ＝ 10 )
B组,5,6,8,10,12,14,15 ( ＝ 10 )
C组,1,2,5,10,15,18,19 ( ＝ 10 ) X
X
X
RANGE AND SEMI-INTERQUARTILE RANGE
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Range,semi-interquartile range,variance,standard variance
Coefficient of variance
2,特点
3,种类
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1．概念,

2．计算,
(1) 原始 数据求全距 (R)＝最大值－最小值

4.2 全距
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3,频数分布表求全距,

(或 )最大一组与最小一组下限之差。

(1)

(3)

(4)

115-
118-
121-
124-
127-
130-
133-
136-
139-
142-
116.5
119.5
122.5
125.5
128.5
131.5
134.5
137.5
140.5
143.5
1
3
8
10
20
19
12
4
2
1
1
4
12
22
42
61
73
77
79
80
R＝ 143.5
－ 116.5=27

R=142－
115=27

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
4．全距的优缺点,

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
4.3.1 方差,

4.3,2 计算公式
＝ 或 S2
N
XX 2)－（?
N
XS 22 ?＝
X在这里,X－ 表示离差 〔 即每个数据与平均数的差数 〕

N表示总频数
2)(? ? XX
4.3 方差和标准差
Variance and standard variance
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
4.3.3 标准差

4.3.4 计算公式
1,原始数据计算法,
S= 或
1
2

〕－（
N
XX?
1
2
－＝ N
XS ?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

3.9、和 7.5
17
)1113(119119 222
?
?? ??〕－＋（）－（

＝ 11; ＝ 11; ＝ 11
1X 3
X
2X
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
X 离散程度 (S) 说明

(居中 )

X
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2,频数分布表计算法 (用于数据较多的分组资料 )

S=
2
2
2 )(
N
fx
N
f xS ???＝
22 )( NfxNfx ???

ｆ表示各组频数
N表示总人数
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

57.5
2

(2)

f
(3)
fx
(4)=(2 )
× (3)
(5)=(2)
× (4)

50-
52.5 2 52.5× 2 × 2
=12.25
55- 57.5 0 57.5× 0 × 0
60- 62.5 2 62.5× 2 × 2
65- 67.5 3 67.5× 3 × 3
70- 72.5 8 72.5× 8 × 8
75- 77.5 7 77.5× 7 77.5 × 7
80- 82.5 7 82.5× 7 × 7
85- 87.5 7 87.5× 7 × 7
90- 92.5 5 92.5× 5 × 5
95- 97.5 6 97.5× 6 × 6

f x2
52.5
2
2
22 )(
NfxN
f xS ???＝
=150
S=
22 )( NfxNfx ???
2
62.5
2
67.5
2
72.5
82.5
2
87.5 2
92.5
2
97.52
2
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(1)优点,

(2)缺点,

4.4 方差和标准差的应用及其优缺点
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

16岁儿童身高平均为 16.8公分，标准差为 6.52。试比较他

4.5 差异系数 (CV Coefficient of variance )
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1.差异系数 (VC)的概念

2.计算公式
X
SCV= × 100%

CV越大，表明离散程度越大 (数据的分布愈分散 )
CV越小，表明离散程度越小 (数据的分布愈集中 )

S表示标准差

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3,用途
(1) 比较不同单位 (现象 )的 (变异 )差异程度
CV = × 100%=0.0403× 100%=4.03%

CV × 100%=12.81%
79.48
25.6?

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(2)比较不同水平的同类现象 (单位 )的差异程度
CV
%16.4%1 0 027.1 2 058 ???岁
CV %03.4%1 0 0
88.1 6 1
52.6
16 ???岁

8岁组比 16岁组大些。
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(1)甲组小学生参加算术测验，平均分数为 20.50，标准差为
5.24；乙组小学生参加同样的测验，其平均分数为 34.80，

(2)小学一年级学生的平均身高是 118.8公分，标准差是 5.6公

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(3)可判断特殊差异情况

4．注意事项,
(1) 测试尺度上的单位必须是等距的。若不等距，则 CV的运

(2) 测试尺度的起点必须是绝对零点 。
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1.单用平均数尚有不足的，须用差异量数辅助说明。
2.平均数是一种代表性指标，它的代表性随差异量数的大小而

3.差异量有两种：绝对差异量与相对差异量。本章主要讨论的绝

S2

4.6 差异量与集中量的关系
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
4.7 峰态量
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

PROBABILITY AND PROBABILITY DISTRIBUTION
5.1 概率的一般概论
5.2 二项分布
BINOMIAL DISTRIBUTION
5.3 正态分布
NORMAL DISTRIBUTION
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
5.3.1 概念,

1,正态分布 (NORMAL DISTRIBUTION)
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Z=
?
XX ?
2,标准正态分布 (STANDARD NORMAL DISTRIBUTION)

2
2
2
1 Ze ???
?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

μ =0,σ=1 μ =0
？ 为什么标准正态分布的平均数为 0，标准差为 1 !
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3,正态分布的特点

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
4,正态分布曲线的面积与纵线

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
5,正态分布的应用

99%的学生的成绩的分布范围

4000人，平均分为 1700，标准差为 200。拟录取人数为
1000人，问录取的最低分数应为多少？
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

∴ 甲幼儿,Z = Z= = =0.64

X
S XX ?
S XX ?
S
XX ?
5.127078?
5.127083?
5.127065?

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

1.9
0.8
118
22.9
119
23

σ

(Z)

X
1.52
0.12
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

59
75
63
51
79
72
50
74
67
4
10
9

X
2.25
0.10
-0.44
0.75
0.50
0.56
1.91 1.31

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(2) 现为了使得年投资回报总额达到 14万元, 该选用哪种投资

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6、使用标准分数 (Z)应注意的问题,

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

SAMPLING DISTRIBUTION &
INFERENTIAL STATISTICS OF POPULATION MEAN
?随机抽样分布
THE RANDOM SAMPLING DISTRIBUTION
?平均数的随机抽样分布
THE RANDOM SAMPLING DISTRIBUTION OF THE MEAN
?标准误与总体标准差的关系
DEFINE THE STANDARD ERROR OF THE PROPORTION
AND STATE ITS RELATIONSHIP TO THE POPULATION
STANDARD DEVIATION
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
?总体平均数的参数估计
PARAMETER ESTIMATION OF THE POPULATION MEAN
?检验功效
POWER
?一类和二类错误
TYPEⅠ AND TYPEⅡ ERRORS
?假设检验的逻辑与过程
THE LOGIC AND PROCEDURE OF THE HYPOTHESIS
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.1 抽样分布
SAMPLING DISTRIBUTION
6.1.1 研究实例

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

___X
1、分析上述实例
?区分总体和样本
?区分参数与统计量及不同的表达方式
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

___X

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2、抽样分析

___
2X
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

1 500 σx1
2 500 σx2
3 500 σx3

i 500 σxi

k 500 σxk

∝ 500 σx ∝
___
2X
___
1X
___
3X
___
iX
___
kX
___
?X
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

THE SAMPLING DISTRIBUTION)图,

__x?

__x?

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.1.2 平均数抽样分布的几个定理
3、正态总体中，平均数的抽样

??? )( _ _ _X1,
nx
?? ?
__
2,
4、偏态总体中，当抽样容量较

__x? __x?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.1.3 样本平均数与总体平均数的离差统计量
__x?
__x?

___
iX

__
__
_ _ _
x
x
i
i
X
?
??
??
n
X i
i /
_ _ _
?
????
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
?当总体标准差已知时，平均数的离差统计量的计算,
?当总体标准差未知时，平均数的离差统计量的计算,
n
X i
i /
_ _ _
?
????

xx n
nS ?
1??

__xS
nSS x
x
/__ ?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1/__ ?? nS x
x
?

__
___
X
S
X
t
??
?
?t分布及特点
?自由度

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.2 总体平均数的参数估计
PARAMETER ESTIMATION OF POPULATION MEAN
6.2.1 点估计
POINT ESTIMATION
1、点估计的定义
2、点估计的评价标准
? 无偏性 (UNBIASED)
? 有效性
? 一致性
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.2.2 区间估计
INTERVAL ESTIMATION
1、区间估计的定义
2、区间估计的计算

?以平均数为中心,95%学生的成绩的分布范围；
?其成绩在 128到 132间的人数的比例；
?最高成 5%学生成绩的分布范围。
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
?从总体中抽取 25人，计算其平均成绩，该平均成绩在 128到 132

?从总体中抽取 25人，计算其平均成绩，该平均成绩以总体平均

?从总体中抽取 25人，计算其平均成绩，该平均成绩由高到低
95%概率下的分布范围；
?从总体中抽取 25人，计算其平均成绩，最高 5%的平均成绩的

?从总体中抽取 25人，计算其平均成绩，该平均成绩大于 135的

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

27名 10岁儿童，其平均身高为 134.2厘米，试估计该校 10岁儿童

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
4,总体标准差 ( σ )未知条件下的区间估计

?区间估计的基本原理
t值
μ
__xS
__
___
X
S
Xt ???
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

29.917，标准差为 3.926。试估计该校三年级学生总体平均成绩
95%和 99%的置信区间。
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.3 假设检验的基本原理和过程
THE LOGIC AND PROCEDURE OF HYPOTHESIS TESTING
1、假设检验的定义
2、假设检验的原理
6.3.1 假设检验的原理
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

REJECTION)

6.3.2 假设检验中的基本概念
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.3.3 假设检验中的基本过程
(1) 假定：首先假定两个样本所代表的总体之间没有差异，目前样

(2) 计算：选择适当的统计公式，正确计算统计值。
(3) 比较：将计算出的统计值与查表得到的临界值相比较，得出上

(4) 判断：按照差异显著性的判断规则，作出一定可靠程度的判断。
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.4 总体平均数的显著性检验
6.4.1 总体标准差 ( σ )已知条件下的总体平均数的显著性检验

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

103.3。能否认为受过良好早期教育的儿童智力高于一般水平？
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.4.2 总体标准差 ( σ )未知条件下的总体平均数的显著性检验

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.4.3 差异显著性的判断规则

5

0H
0H
H1
0H
H1
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.5 两类错误及检验功效
TYPEⅠ ERRORS, TYPEⅡ ERRORS AND POWER
6.5.1 两类错误
1,实例分析

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
=100
1.96 1.60
=103 ___?
?例 A假设检验的示意图
α /2=.025 α /2=.025
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
=106
-1.96 -1.06
=103 ___?
?例 B假设检验的示意图
α /2=.025 α /2=.025
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
μ1 =105 μ0 =100
1.60
=103 ___?
?例 A假设检验中所犯错误
1.96
α/2=.025 α/2=.025 β
β=.24
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
μ0 =100 μ1 =105
-1.06
=103 ___?
?例 B假设检验中所犯错误
α/2=.025 α/2=.025 β
-1.96
β=.24
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2,两类错误的定义
?α错误：假设是真而被拒绝，其大小与假设检验的显著性水

?β错误：假设是伪而被接受。
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3,两类错误的相互关系
?在我们做决策时两类错误客观存在；
?当一种错误在减小时，另一类错误在增加。
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
4,控制两类错误的方法
?合理安排拒绝区域的位置；
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
?扩大抽样的容量。
5、抽样容量要多大？

nx
?? ?
__
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.5.2 检验功效 ( POWER )
1、什么是检验功效 Power=1-β

Power:The probability of rejecting a false null hypothesis
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2、影响功效的因素
Power=1- β
?检验的形式
?样本的容量
?鉴别力 (EFFECT SIZE, d值 )
d
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3、依据功效的要求，确定样本的大小

μ1 =105 μ0 =100
1.96
α/2=.025 α/2=.025 β
N=71.91
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
6.6 练习
1,某人做 100个 5选 1的选题, 假如规定做对 95%的题目才算了

2,一位人类学家测量了取自某一海岛上的 100名男子的随机样

1) 求总体平均身高 μ 的 95%的置信区间；
2) 如果想把 95%的置信区间变得窄一些, 如想误差区间变为 0.5

3,假定飞机乘客体重的总平均值为 60公斤 。 标准差为 11公斤, 某

1) 55位乘客的飞行将会超重的机会是多少？
2) 要将超重的机率减小到 0.005,该飞机的标准载重量应为多少？
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(一 )什么叫推断统计
1.推断统计的 定义

2.推断统计与描述统计的关系,

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3.推断统计的 分类,

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(二 )推断统计的功能与目的
1,功能

2,目的

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(三 ) 推断统计在教育科学研究中的意义
1 推断统计是教育科研的基础
2 推断统计是一种科学的思维方法
3 推断统计是一种重要的学习工具。
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(一 )差异显著性检验 (假设检验 )的涵义
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
1.总体和样本,(p6)

2.参数和统计量,

ρ)称参数。

?
?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3,计量资料和计数资料 (p11～ 12)

4,随机误差与条件误差

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(二 ) 差异显著性检验的逻辑

1,差异显著性检验的推理逻辑是一种, 反证法, 。其基本思路是：
2.差异显著性检验的一般步骤,
(1) 假定：首先假定两个样本所代表的总体之间没有差异，目前样

(2) 计算：选择适当的统计公式，正确计算统计值。
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(3) 比较：将计算出的统计值与查表得到的临界值相比较，得出上述

(4) 判断：按照差异显著性的判断规则，作出一定可靠程度的判断。

○ 将 P>0.05作为, 差异不显著,,意即, 两个总体没有差异的可能性大于 5%”,
○ 将 P≤ 0.05作为, 差异显著,,意即, 两个总体没有差异的可能性小于

○ 将 P≤ 0.01作为, 差异非常显著,,意即, 两个总体没有差异的可能性

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(保留 0H

(即有把握 )说两

( 拒绝
0H

(即有很大把握 )说两

(拒绝
0H

P值 统计意义 判断
P>0.05 差异不显著
)
0.01<P≤0.05 差异显著
)
P≤0.01 差异非常显著
)
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

H0:μ1=μ2 H1:μ1≠μ2
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(三 ) 进行差异显著性检验应注意的事项
1,必须强调样本的代表性和样本资料的可比性。
2.差异显著性检验只适用于对随机误差的检验。
3.不同类型的数量资料，要用不同的检验方法。一般

4.正确理解差异显著性检验的结果 。
5,差异显著性检验只能通过样本的差异是否显著来推断

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(一 )什么是 Z检验

1.相关样本,
a.同一测验，对同一被试，前后两次测验。
b.同一测验，测基本情况相等的不同被试。

A A
B B
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2.独立样本,

(二 )Z检验的显著性水平表
*表示显著，指有把握 (即有大于或等于９５％的把握 )判断总体平

**表示非常显著，指有把握 (即有大于或等于９９％的把握 )判断总

ns表示两个样本的均数差异不显著，指没有把握判断总体平均数存

Z值 P值 显著性
Z<1.96 P>0.05 差异不显著 (ns)
2.58>Z≥1.96 0.01<P ≤ 0.05 差异显著 (*)
Z≥2.58 P≤0.01 差异非常显著
(**)
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(一 )假设：提出假设,

210, ?? ?H 211, ?? ?H
0H
H1
(二 )计算
(三 )比较
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(四 )判断
Z检验的差异显著性水平水平的判断原则表

(一 ) 两个独立大样本平均数差异显著性检验
Z≥
Z值 P值 差异的意义 判断
Z<1.96 P>0.05 差异不显著 (ns) 保留
2.58>Z≤1.
96
0.01<P≤0.
05

2.58 P≤0.01 差异非常显著 (**) 拒绝

0H
H1
0H
0H
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

nS

22,18.5,17,24,20.5,21.5、
19.5,19,19.5,19.5,23,20、
19.5,20,19,19.5,21,20、
20.5,22.5,19,22,22.5,20、
16.5,20.5,20,21.5,21,20
20,20.5,16.5,23,18,19,18.5,21、
17.5,22.5,18.5,22,19.5,17.5,20,22、
19,19,18,20.5,19,18.5,21,19,21、
19.5,20,22,20.5,21,18,20.5

) 20.3 19.77
1.61 1.576
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

1.提出假设,
2.选择检验统计量并计算其值,

210, ?? ?H 211, ?? ?H
3 0 8 9.1
32
5 7 6.1
30
61.1
77.193.20
22
2
2
2
1
2
1
21 ?
?
?
?
?
?
?
n
S
n
S
xx
Z
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
３,确定检验形式：两个独立大样本平均数差异显著性检验
4,统计决断,∵ 1.3089<1.96 P>0.05 ∴ 接受 H0

5.统计检验表,

x

nS

s)
P>0.05 女 32 19.8 1.576
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

1,提出假设,
2.选择检验统计量并计算其值
3,统计决断,∵ 7.62>2.58 ∴P<0.01 拒绝 接受
210, ?? ?H 211, ?? ?H

4-5岁 60 2078
5-6岁 50 38.72
**62.76 1 6.7
60
6 3 2.7
50
7 9 2.7
78.2072.38 ??
?
??Z
0H H1
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
4.结论：两样本平均数差异非常显著，两总体平均数差异显著。区内 4-5

5.统计检验表,

nS

4-5岁 60 2078 7.63
Z=7.62**
P<0.01 5-6岁 50 33.72 9.79
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(二 )两个独立大样本的百分率差异显著性检验

— Z

Z =
? ? ??
?
?
???
? ??
?
21
21
111
NNPP
PP
ee
—— 总的不良率
—— 总的良好率 eP? ?
eP?1
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

1,提出假设,
2.选择检验统计量并计算其值,

n

NnP?
116331 ?P
94452 ?P
21078?eP

210, ?? ?H 211, ?? ?H
Z=
? ? ??
?
?
??
?
?
??
?
21
21
111
NN
PP
PP
ee
= **8 9 7.2
94
1
1 1 6
1
2 1 0
1 3 2
2 1 0
78
94
45
1 1 6
33
94
1
1 1 6
1
2 1 0
781
2 1 0
78
94
45
1 1 6
33
?
?
?
??
?
? ??
?
?
?
?
??
?
? ??
?
??
?
? ??
?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3,统计决断,∵ =2.897>2.58 ∴ P<0.01拒绝 接受
4,结论：两样本百分率差异非常显著，两总体百分率差异显著。某

5.统计检验表,

NnP?
116331 ?P **897.2?Z
94452 ?P
21078?eP

P<0.01

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
§ 作业
1.某幼儿园大班男女幼儿身高情况表

N x
nS

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2：下表是两种教法教学后的效果统计，试问这两种教法的及格情况一

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(一 )什么是 t检验
—— 用两个样本平均数之间的差异来推断这两个样本所代表的

(二 )比较 Z检验,t检验,
Z检验 独立大样本 (n ≥ 30) 平均数、百 分率数

(N≤30)

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(一 ) 计算 t值
t检验公式,

)11(
2
)1()1(
2121
2
22
2
11
21
NNNN
SNSN
XXt
?
??
???
?
?
1X 2X
21NN
21SS
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(二 ) 根据样本容量查找临界值
(1)决定自由度

(2) 查, 简便 t检验临界值表, (简称, t值表, )，根据自由

(t值表见书 385页 )
(三 ) 判断样本平均数的差异是否显著
t检验的差异显著性水平水平的判断原则表
221 ??? nndf
df
05.0)(dft 01.0)(dft
t≥
t值 p值 显著性 判断
t<0.05 p>0.05 差异不显著
(ns)

0.05t > 0.01 0.01<P≤0.05 差异显著 (*) 拒绝零假设
0.01 P≤0.01 差异非常显著
(**)

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(四 ) 对总体平均数有无差异做出判断

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
[检验 ]
(1)计算 t值,
(2)根据样本容量找出临界值,

19.211 ?X
31.192 ?X
74.11 ?S 131 ?N
70.12 ?S 132 ?N
)11(
2
)1()1(
2121
2
22
2
11
21
NNNN
SNSN
XXt
?
??
???
?
?
? ? ? ? *79.2
13
1
13
1
21313
70.111374.1113
31.1919.21
22
?
?????? ???? ?????
??t
2421313 ????df
? ? 064.205.024 ?t ? ? 797.201.024 ?t查表得,
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(3)判断样本平均数是否有显著性意义,
(4) 对总体平均数有无差异做出判断,

(单位：千克 )。试根据测量结果做出判断。

∴ P<0.05，表示两个样本平均数的差异有显著性意义。
? ? ? ? 01.02405.024 ttt ??
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
[检验 ]
(1) 计算 t值,
(2)根据样本容量找出临界值,

(ns)
)11(
2
)1()1(
2121
2
22
2
11
21
NNNN
SNSN
XXt
?
??
???
?
?
? ? ? ? 48.1
34
1
28
1
23428
98.113433.2128
81.1662.17
22
?
?????? ???? ?????
??t
6023428 ????df
? ? 05.060t
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(3) 判断样本平均数是否有显著性意义,
(4)对总体平均数有无差异做出判断,
∴ P>0.05，表示两个样本平均数的差异不显著。因此，没有把握

(一 ) 两个样本的容量不要太小
(二 ) 两个样本的方差不能相差太大。否则，得出的结论难免会发生

?t
? ? 05.060t
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

1.从某幼儿园中随机抽取 4-5岁的幼儿 35人,5-6岁的幼儿 32

2.对某幼儿园大班幼儿进行思维能力训练, 为了了解这项思

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

1.两相关样本的原始分数差异显著性检验
= (皮尔逊 )
yxSNS
xy?? ?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

1 31 32
2 15 41
3 60 66
4 46 57
5 32 57
6 58 68
7 28 27
8 22 41
9 23 20
10 23 30
N=10
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(1)先计算这两相关样本的相关程度 ( )

?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(2) 根据样本容量找出临界值

=0.804**

(3) 判断两总体是否有显著性意义,
(4)对总体是否具有显著差异做出判断,
∴P<0.01,表示这两样本相关非常显著，两总体相关显著，

? ?? ?
? ? ?
??
???
2222 )()(
))((
YYNXXN
YXXYN
82 ??? Ndf
6319.005.0 ?? 7649.001.0 ??
7649.0804.0 ???
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2,两相关样本的平均数差异显著性检验

? ?
? ?1
22
1
?
?
??
NN
DD
xxt
e
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

1 20.5 20
2 18.5 17.5
3 22 19
4 23 19
5 20.5 20
6 21 21.5
7 21.5 22
8 23 20
9 23.5 19
10 22 19.5
11 22.5 20.5
12 19.5 17
13 18 16

1x

13?N 19.211 ?x 308.192 ?x
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(1)解,.这是两相关样本的平均数差异显著性检验。 (列表 )

1x

2x 21 xxD ??
13?N 19.211 ?x 308.192 ?x
?
?
?
?
75.76
5.24
2D
D

1 20.5 20 0.5
2 18.5 17.5 1
3 22 19 3
4 23 19 4
5 20.5 20 0.5
6 21 21.5 -0.5
7 21.5 22 -0.5
8 23 20 3
9 23.5 19 4.5
10 22 19.5 2.5
11 22.5 20.5 2
12 19.5 17 2.5
13 18 16 2
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(2) 根据样本容量找出临界值

(3)对总体是否具有显著差异做出判断,
∴ P>0.01,两样本平均数差异非常显著，两总体平均数差异

? ?
? ?1
22
1
?
?
?
?
? ?
NN
DD
xx
t e
2 5 7.4
1213
13
52.2475.76
3 0 8.191 9 2.21
?
?
?
?
?t
12113 ???df
055.3257.4
055.3
179.2
01.0
05.0
??
?
?
t
t
t
df=N-1
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
§ 作业
1：试问下列 9名幼儿语言表达能力和算术能力是否相关 (或有无关系 )

1 7.2 52
2 3.5 66
3 4.5 44
4 6.6 68
5 2.0 35
6 2.9 35
7 3.4 50
8 2.8 62
9 6.2 55
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2：试问此区内幼儿智力与体育训练有无关系

1x

2x编号
1 52 72
2 66 83
3 62 68
4 55 49
5 44 47
6 68 65
7 35 52
8 35 34
N=8
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

— — 判断实际观测到的频数与有关总体的理论频

2x
2x
2x
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(二 ) 检验的适用范围

1,检验可以用来检验各种实际频数与理论频数是否吻合

2x
2x
2x
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(1)实际频数 即：好 32；中 46；差 30
(2)理论频数 本例，假若该地区幼儿这三类健康状况的人数相同

? 3631?
2x
2x
2x
2x

2x
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2,检验可以用来判断两组或多组计数资料是相互关联还是彼此独

(1) 按年级分类：大班；小班
(2)按态度分类：喜欢；不喜欢
2x
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

2x
2x
2x
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(1)计算 值
1,检验实际频数与理论频数是否吻合的问题。 (如例 1之类的情况。 )

K表示实际频数在理论上应占总频数的比值。
2x
2x
2x ?
??
f
ff
e
ox
2
12 )(
?
0f
ef
KNfe ??
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

∴ ×
2x
ef
108?N
3
1?K
108?f e 3631 ?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(2) 在根据实际频数和理论频数编制一张统计表
(3) 将实际频数和理论频数带入公式，计算出 值
(ns)

? ?? f ff
e
ox
2
12 )(
? ? ? ? ? ? 11.2
36
3630
36
3646
36
3632 2222 ???????x
2x
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(二 ) 根据分类组数找出临界值
(三 ) 判断样本数的差异是否有显著意义 (下表 )
df=K-1(K表示组数 )
x2 检验的显著特性水平表
213 ???df
x2 的值 P值 显著性
x2 <0.05 P>0.05 ns(不显著 )
0.05 ≤ x 2<0.01 0.05≥P>0.01 *(显著 )
x2≥0.01 P≤0.01 **(非常显著 )
(X2值表 P412)
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

∵ x2< x2(df)0.05
∴ P>0.05,表示差异不显著
(四 )对总体频数有无差异作出判断

2x ? ? 9 9 1.52 05.02 ?x ? ? 210.92 01.02 ?x
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2 由实际频数直接计算 值的公式,

N表示样本容量的总频数；

2x
)1(
2
2 ?? ?
NN
f
cR
oNx
?
0f
RN 0f
0fCN
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

(1)先将实际频数列成一张双向表
(2)将各个数值带入公式,计算出 值
2x

Nc 89 26 115(N)
2x
)1(
2
2 ?? ?
NN
f
cR
oNx
399.1112655 202660 68955 358960 54115 22222 ????????? ??????????x **
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
(二 ) 根据分类组数找出临界值。
df=(R-1)(C-1) (R 表示横行的分组数 C表示 纵列的分组数 )
df=(2-1)× (2-1)=1
(三 ) 判断样本数的差异是否有显著意义 (下表 )
(四 )对总体频数有无差异做出判断
(1)判断样本频数的差异是否有显著性意义
∵ 查 值表得,
ｘ ２ (１ )0.05=3.841,ｘ ２ (１ )0.01=6.635
∴ P<0.01,表示差异非常显著
(2)结论：有很大把握说幼儿对这种治理游戏的喜爱程度与

∵,
∴ P<0.01，表示样本实际频数和理论频数的差异非常显著。
2x ? ?2
01.02x?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

RN
CN 116( N

58 42 16 )

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
[检验 ]

)1(
2
2 ?? ?
NN
f
cR
oNx 116?
*16.6
1
1656
10
1660
6
4256
14
4260
28
5856
32
5860
26 222222
?
???
?
???
? ?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ? ? 21312 ?????df

∵, ∴P<0.05,表示差异显著。 ? ?2 05.02x?2x
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

1,检验的基本条件：遵守分组归类原则 (分类完

2.样本容量的总频数应有足够多，如果大小就难以

2x
2x
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

2
2
2
1
2
1
2 1
?
? ?
n
S
n
S
x x Z
)
11
(
2
)1()1(
2121
2
22
2
11
21
NNNN
SNSN
XXt
?
??
???
?
?
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

? ?? ?
? ? ?
??
???
2222 )()(
))((
YYNXXN
YXXYN

? ?
? ?1
22
1
?
?
?
?
NN
DD
xx
t e
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

? ?? f ff
e
ox
2
12 )(

2x
2x
)1(
2
2 ?? ?
NN
f
cR
oNx
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
2,某幼儿园教师为了了解, 家园教育一致促进幼儿智力发展,

1,从某班中随机抽取 9名男孩,8名女孩，其智力成绩见下

)

Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
3,某幼儿园随机抽取 12对幼儿，用配对等组法进行某种营养措施

1 20 19.5
2 18 17
3 21 18
4 22 19
5 20 20
6 21 21.5
7 21.5 22
8 23 19
9 23 19
10 22 20
11 21 19
12 20 18
1x 2x
Department of Special Education ECNU
Department of Special Education ECNU
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
De
par
tm
en
t of
Sp
eci
al E
du
cat
ion
ECN
U
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
Statistics of Education
4.为了了解 4-5岁幼儿看图讲述的层次结构情况，某幼儿园老师随机抽

5.某幼儿园教师调查大、中、小班幼儿对唱歌、跳舞和画图最喜爱哪

0f

RN
CN