6.5.1 Sampling distribution: the cornerstone of sampling inference theory 抽样分布：抽样推断理论的基石在线视频

大家好
Hello, everyone

欢迎回到轻松学统计的课堂
Welcome back to the Easy Learning Statistics Class

这一讲呢
In this lecture

我们要给大家介绍
we’ll introduce you

抽样分布的概念
the concept of sampling distribution

那么什么是抽样分布呢
Then, what is sampling distribution

首先第一个问题
The first issue

需要大家能够理解
you need to understand is

什么是样本统计量
what sample statistics are

那么样本统计量呢
Sample statistics

我们是对总体参数而言的
are relative to the population parameters

大家都知道
We know

总体有很多特征
population has a lot of characteristics

比如说总体的均值
Let's say the average of the population

总体的方差
variance of the population

总体的比例等等
proportion of the population and so on

那么我们说这一系列
The series are used

说明总体特征的这些指标
to describe these indicators of the population characteristics

我们都把它称其为总体的参数
We call them the parameters of the population

那么样本统计量呢
Then, sample statistics are used to

就是说明样本特征的这些指标
describe these indicators that characterize the sample

比如说说明一个样本
For example, to describe a sample

我们有样本均值 样本方差
we have the sample average, sample variance

样本比例
sample proportion

那么这些都是样本统计量
These are all sample statistics

那么举例子
let's give an example

比如说我们要在一个班
Say, we want to select

40名同学当中
from 40 students of a class

抽取10名同学出来
10 students

那40名同学
Then, the 40 students

这个总体的
are the population

比如说平均成绩
For example, average scores

男生的比例
proportion of male students

成绩的方差
score variance

这些都是总体的参数
these are all parameters of the population

就总体是确定的
The population is certain

那么这些参数呢
what about these parameters

它也是确定的取值
The value range is also certain

那么我们要随机抽取一个
Then, we're going to pick randomly

10个人构成的这个样本
ten students to constitute the sample

那我抽取这10个同学
So, let me draw these ten students

那我就可以算他们的
I can calculate

这个样本的平均成绩
the average score of this sample

那大家想一想
Think about

我这个样本的平均成绩
the average score of my sample

它是一个固定的值吗
Is it a fixed value

或者它是一个随机变量
or it's a random variable

那么我们需要大家一定能够理解
So we need you to understand that

样本均值是随机变量
the sample average is a random variable

它为什么是随机变量呢
Why is it a random variable

因为我们从40个同学当中
Because from 40 students

随机去抽取这10名同学
we randomly select 10 students

这10名同学
These ten students

构成了一个随机样本
constitute a random sample

换句话说
In other words

我可能会抽这10个同学
we may select these ten students

也可能会抽那10个同学
or those ten students

那么这10个同学
In this way, these ten students

他是随机抽取的
are picked out randomly

就导致Xbar它是一个变化的量
This average Xbar is a variable

那么也就是
That is to say

是一个随机变量
it is a random variable

那么换句话说呢
In other words

我们说
Let’s say

样本比例 样本方差
sample proportion, sample variance

那么一样的原因
due to the same reason

他们都是随机变量
are random variables

也就是说
That is to say

我们刚才所讲
as we said just now

总体参数呢
the population parameters

都是固定的取值
are all fixed values

因为总体呢
Because population

我们说它是确定的一个总体
is a definite population

而样本统计量呢
Sample statistics

我们说因为它是样本的一个函数
are a function of the sample

就是它是通过样本来计算的
They are calculated based on the sample

而你的样本呢
Then as your sample

又是一个随机的样本
is a random sample

所以我们说样本统计量
we say sample statistics

就是随机变量
are a random variable

那么我们一提到随机变量
When we mention random variables

我们怎么样能够全面
how we comprehensively and

准确地刻画一个随机变量
accurately draw a random variable

所有的特征呢
by all its characteristics

大家通过学习概论与数理统计
Through the study of probability and mathematical statistics

大家应该有印象
you should remember that

那么我们想充分的了解一个
if we want to thoroughly understand

随机变量
a random variable

就要知道这个随机变量的概率分布
we need to know the probability distribution of this random variable

所以这个时候呢
So, at this point

就提出了我们抽样分布的概念
we put forward the concept of the sampling distribution

抽样分布呢
Sample distribution

就是指的样本统计量的概率分布
is the probability distribution of the sample statistics

也就是说如果你能够把这个
That is to say, if you can figure out

样本统计量它的概率分布搞清楚了
the probability distribution of the sample statistic

那这个样本统计量所有的特征
then, all characteristics of the sample statistic

我们都非常的容易搞清楚了
can be easily made clear

所以这里大家注意
So, we should pay attention to

我们这一章的名字
the name of this chapter

叫做抽样与抽样分布
is sampling and sampling distribution

那么这里的抽样分布
Sampling distribution here

就是指的样本统计量的概率分布
refers to the probability distribution of the sample statistic

当然那么抽样分布呢
Of course, sampling distribution

它往往是一种理论的分布
is often a theoretical distribution

它怎么得到的呢
How is it obtained

我们说理论上就是说
Theoretically

如我们重复选取容量为n的样本
If we repeatedly select the sample with size n

就依据这个容量为n的样本
then, all calculated statistics

所计算的统计量
and all possible values

它所有可能取值
based on this sample with size n

就会形成一个
we form a

相对频数分布1
relative frequency distribution

这就是它的概率分布
That's its probability distribution

换句话说
In other words

比如说我们刚才的那个Xbar
For example, Xbar we have just talked about

那么我们说Xbar有很多种取值
can have many values

如果我们能够把Xbar
If we can find out

所有可能的取值都找到
all possible values of Xbar

那么Xbar的概率分布
then, we obtain its

也就出来了
probability distribution

事实上
In fact

这也就是它的抽样分布
that is also its sampling distribution

所以这个就是我们所讲的
This is the concept of

抽样分布的概念的意思
sampling distribution we are talking about

那么抽样分布呢
Then, sampling distribution

我们说对于我们后续的这个
to the subsequent

统计推断
statistics inference

有非常重要的意义
is of very great significance

因为我们只有了解了样本统计量的
Only when we know about the probability distribution of

概率分布
sample statistics

我们才知道
can we know

样本统计量它相对比较稳定的
some relatively stable characteristics of

一些特征
sample statistics

所以样本统计量的概率分布呢
So the probability distribution of the sample statistics

是我们进行统计推断的理论基础
is the theoretical basis for statistical inference

同时也是抽样推断科学性的
and also an important base for the scientificity of

重要依据
sampling inference

下面呢
Next

我们给大家举一个例子
let's give you an example

让大家来理解
for you to understand

到底什么是抽样分布
what the sampling distribution is

它是如何形成的
How it is formed

那么我们来一起看一下这个例子
Let’s see the example

假设呢
Assume

我们有一个非常简单的总体
we have a very simple population

那么在这个总体当中呢
The population have

我们只有4个元素
only four elements

这4个元素呢
The four elements are

分别是1 2 3 4
1, 2, 3, 4

那么这个时候我们总体单位数
Then, the number of population units

当然就是4个了
are of course four

那么我们通过这样一个确定的总体
From such a certain population

我们是可以计算
we can calculate

总体的均值和方差的
the average and the variance of the population

怎么算均值呢
How to calculate the average value

我想大家都非常清楚
I think you know very well

那么在这里就是(1+2+3+4)÷4
It is (1+2+3+4)÷4

那么等于2.5
It is equal to 2.5

这就是总体均值
That's the population average

它是确定了一个取值
A value is determined

那么接下来我们大家也可以做方差
Next, we can do about the variance

比如说σ方等于什么呢
For example, what the σ squared is equal to

就用我们每一个变量值
We take each of the variable values

都跟均值做差
to calculate the difference from the average

然后呢把这个差做平方和
Then, use the differences to calculate the sum of squares

然后再除以我们数据的个数
Then, divide it by the number of data,

那么就得到了总体的方差
and we get the variance of the population

这μ和σ平方都是总体的参数
The μ and σ squared are all parameters of the population

接下来我们就要做抽样了
And next we're going to do the sampling

我们打算呢
We are going to

从这个总体当中
from the population

随机抽取n=2的一个简单随机样本
select randomly a simple n=2 random sample

那么我们要在重置抽样的条件下展开
We're going to conduct it under the condition of sampling with replacement

那大家想一下
Just think

从4个里面
from four

随机的抽两个
select two randomly

采取简单随机抽样的方式
by simple random sampling method

并且要求重复抽样
Repeat sampling is required

那么大家想一想
Then, think

我们有多少个可能的样本呢
how many possible samples we can have

当然就是首先我先抽第一个
First, I select the first

先抽第一个样本单位
Select the first sample unit

我有几个选择呢
How many choices do I have

我有4个选择
I have four choices

再接下来因为我是重复抽样
Next, because I’m doing repeat sampling

抽到了这一个呢
I select one

我又把它放回去
and I put it back

所以我第二次抽的时候
Then, when I do the second sampling

我仍然有4种可能性
I still have four choices

所以4×4=16
Therefore, 4×4=16

所以我们能够抽取的可能的样本数
So the number of possible samples that we can select

有16个
is 16

那么大家看一下
Let’s see

这就是我们可能的16个样本
that's the 16 samples that we could have

那么所有可能的样本
So all the possible samples

我们找到了
have been found out by us

那么在每一个样本下
For each sample,

我们都可以计算
we may calculate

这个样本的样本均值
its d

所以大家再来看这张表
Let’s see this table

这张表呢
The table

就展示了
shows

每一个样本下的样本的均值
the d of each sample

那我想问一下
I'd like to ask you

样本均值的概率分布
has the probability distribution of the sample average

现在是不是出来了
been obtained

当然是对吧
Of course, right

我们Xbar所有可能的取值
All possible values of Xbar

我们现在都找到了
have been found out

那么我们现在只要做一个
Then, we need only to do

相对频数分布就可以了
the relative frequency distribution

那么大家就看到样本均值的
Then, we will see the sampling distribution of

抽样分布
d

那么现在呢
Now

我们再来把总体的分布
let’s compare the population distribution

和抽样的分布
with the sampling distribution

我们来对比看一下
Compare them

在总体分布下
under population distribution

我们刚才计算了
We have just calculated that

μ=2.5 σ平方=1.25
μ=2.5 σ squared=1.25

这个大家刚才都做了
You did that just now

那么刚才呢
Then, just now

我们又看到了样本均值的概率分布
We also saw the probability distribution of the sample average

那么我们同样可以计算
We can also calculate

样本均值的均值
the average of the sample average

所以大家看
You see

我是用μ （下标Xbar）来表示
I use μ (subscript of Xbar) to represent

也就我要把Xbar所有可能的取值
so I'm going to add up all the possible values of Xbar

加起来
add them up

再除个数
then divide it by the number of values

刚才我们说了我们有16个取值
We just said we have 16 values

那我把这16个取值都加起来除16
So let me add up all 16 values and divide the sum by 16

这个就是Xbar的均值
This is the average of Xbar

那么看起来非常巧
So it seems very coincidental

Xbar的均值
The average value of X bar

就是总体的均值2.5
is the population average of 2.5

这个我们当然也是可以
Of course we can use

通过理论推导
theoretical derivation

出来的这样的一个结果
to get such result

再接下来呢
And then

我们看一下Xbra的方差
let's look at the variance of Xbra

Xbra的方差应该怎么算呢
How to calculate the variance of Xbar

我们Xbar有16个取值
Our Xbar has 16 values

我们把每一个取值
We take each value

都跟Xbar的均值来做差
to get its difference from the average of Xbar

然后把这些差呢
Then we do the sum of squares of these differences

我们做平方和之后再平均
and then average it out

那么这个过程呢
The process

就是一样的
is the same

其实就是做方差的过程
If fact, it is the process of doing variance

那么这个方差做出来呢
the resultant variance

是0.625
is 0.625

那它跟我们总体的方差1.25之间
Then, what’s its relationship with

有什么关系呢
the population variance 1.25

这就是我们下面会提到的内容
This is what we are going to talk about next

下面呢
Then

我们要给大家实质上呢
in fact, they are

是两个重要的这个定理
are two important theorems

那么这两个重要的定理呢
These two important theorems

会给我们Xbar抽样分布
will give the Xbar sampling distribution

非常重要的信息
very important information

因为不可能我们每一次抽样
Because we cannot find out in every sampling

都像我刚才
as I did just now

需要把所有可能的样本找到
all possible samples, and

然后再去计算Xbar的均值
then calculate the average of Xbar

Xbar的方差
and the variance of Xbar

当然也没有这个必要
Certainly not necessary

那么我们会有两个定理
So we have two theorems

来给大家一些结论
To give you some conclusions

来帮助大家
so as to help you

认识Xbar的这个抽样分布
know this sampling distribution of Xbar

那么第一个重要的结论呢
The first important conclusion

我们可以把它称其为正态分布的
can be called the regeneration theorem of

再生定理
normal distribution

这个定理呢
This theorem

我们是可以很容易的
can be easily

推导出来的
derived

就是当总体服从正态分布的时候
When the population obeys normal distribution

比如说我们看
for example

X服从均值为μ 方差为σ平方的
X obeys a normal distribution

这样一个正态分布
with a average of μ and a variance of σ squared

那么来自于这个
Then, for the sample of size n

总体的容量为n的样本
from the population

它的这个样本的均值
the average of the sample

Xbar是一定服从正态分布的
Xbar is sure to obey normal distribution

并且Xbar的数学期望
Besides, the mathematical expectation of Xbar

也就是它的均值为μ
or its average is μ

方差为n分之σ平方
The variance is σ squared / n

那么我们刚才的例子呢
From the example

大家也可以看到这一点
we can see

就是说
that is to say

如果总体服从正态
if the population obeys normal distribution

那你从中抽取的这个
then, the simple random samples

简单随机样本
selected from this population

那么这个样本的均值Xbar
must have an average Xbar

也一定服从正态
that obeys normal distribution

那么从理论推导上来看
From a theoretical derivation perspective

就是我从这个整体当中
if I select from a population

抽取容量为n的样本
samples of size n

那么我这个样本可以计为
then, the samples can be marked as

X1 X2 一直到Xn
X1, X2 up to Xn

那么我要用X1到Xn
Then I use X1 up to Xn

来做这个Xbar
to do the Xbar

也就是我把X1到Xn都加起来
So I'm adding up X1 to Xn

再除以n
and divide the sub by n

得到Xbar
to get Xbar

那当然Xbar是服从正态分布的
So, of course, Xbar obeys normal distribution

大家曾经看到正态分布的这个特性
You've seen this property of a normal distribution before

就是说如果一个随机变量
That is to say if a random variable

服从正态分布
obeys normal distribution

那么它的线性的函数
then it's a linear function

也是服从正态的
will also obey normal distribution

这个就是我们为什么
This is why

当总体服从正态的时候
when the population obeys normal distribution

Xbar也服从正态
the Xbar will also obeys normal distribution

另外呢
Besides

我们也很容易推导出
we can also derive easily

Xbar的数学期望为μ
the mathematical expectation of Xbar is μ

那么大家想一下
Think it over

我们X1+X2 一直加到Xn
we add up X1, X2 all the way till Xn

那么这个X1到Xn都服从正态
all X1 up to Xn obey normal distribution

那么你加起来除以n
Then, you add them up and divide the sum by n

大家做一下他的数学期望
You may do its mathematical expectation

当然会等于μ
Of course, it will be equal to μ

那么当然我们也可以计算
Of course, we can calculate this way

X1一直加 加到Xn÷n
add X1 to Xn and divide by n

它的方差了
to get its variance

那么在方差推导的时候
In variance derivation

大家要注意
we should note

那么我们有一个1/n
if we have a 1/n

然后上面这个分子上是
at the numerator

X1+X2 加到Xn
we add up X1+X2 up to Xn

那么我们的1/n
then, the 1/n

在拿出这个方差的计算公式的时候
when we take out the calculation formula of variance,

拿出这个D
take out this D

我们用D来表示方差了
We use D to represent the variance

那么拿出来的时候
when it is taken out

它是n平方分之一的
it is 1 / n squared

所以这样的话呢
Therefore

我们Xbar的方差
the variance of Xbar

就是n分之σ平方
is σ squared / n

这是第一条非常重要的定理
This is the first very important theorem

再接下来呢
Then

我们还有一个
we have a

还经常会用到的这样一个结论
conclusion that is frequently used

这就中心极限定理
That's the central limit theorem

那么中心极限定理
Then what the central limit theorem

告诉我们什么呢
tells us about

就是说如果总体的分布不知道
That is, if the population distribution is unknown

那就可以是一个任意的总体了
it can be an arbitrary population

你从一个任意总体当中
When you select from an arbitrary population

抽取容量为n的样本
samples of size n

只要n充分大
as long as n is sufficiently big

那么我们的样本均值的抽样分布呢
the sampling distribution of the sample average

就会近似服从正态分布
will approximate a normal distribution

并且呢它的均值就是总体均值
and its average is the population average

就这个样本均值的这个均值呢
Then, the average of the sample average

就是我们之前提到这个任意总体的
of such arbitrary population

这个总体均值μ
is the population average μ

它的方差呢
Its variance

就是总体方差的1/n
is 1/n of the population variance

这就是中心极限定理
That's what the central limit theorem

告诉我们的
tells us about

下面呢
Next

大家看一下这张图
let’s have a look at this picture

首先大家可以看到
First, you will see

我们紫色的这条线
the purple line

其实这个就是一个任意总体了
This is essentially an arbitrary population

很明显
Obviously

它肯定不是服从正态的
it definitely does no obey normal distribution

它有一些适度的右偏
It is a little properly right-skewed

那么我们从中抽取一个
Then, we select a

容量为n的样本
a sample of size n

只要样本容量足够大
As long as the sample size is sufficiently big

那么我们Xbar的抽样分布
the Xbar sampling distribution will

就趋于正态分布
intend to normal distribution

当然大家还可以计算Xbar的均值
And of course, you can also calculate the average of the Xbar

Xbar的方差
the e Xbar variance

所以这个是中心极限定理
Therefore, this is the conclusion told by

告诉我们的结论
the central limit theorem

接下来呢
Then

我们也可以看一下
we may have a look at

就是我们给大家举的这个例子
the example given here

这个是在不同的总体的
Under the populations

形态下
of different patterns

当n逐渐增大的时候
when n gradually increases

那么我们Xbar的这个概率分布
the probability distribution of Xbar

我们说也就是Xbar的抽样分布
or the sampling distribution of Xbar

那么我们说
is in the process of

越来越趋近于正态分布的这个过程
gradually approaching the normal distribution

这个大家可以看一下
Let’s have a look

那么这三列呢
at these three columns

我们说总体呢
We refer to the populations

分别是
Let’s take

比如说第一列
the first column

我们说总体是服从一个均匀分布
The population obeys a uniform distribution

然后呢第二列呢
For the second column

我们总体服从一个兔耳分布
the population obeys a rabbit ear distribution

第三列呢
For the third column

我们总体服从一个指数分布
the population obeys an exponential distribution

那么这样的话呢
In this case

我们说随着n的增大
with the increase of n

那么我们Xbar抽样分布呢
the Xbar sampling distribution

我们说越来越趋近于正态
tend to approximate normal distribution

总结一下
Make a summary

那么关于样本均值的抽样分布
what contents about the sampling distribution of sample average

大家需要了解的内容呢
that need to be understood

就是我们刚才给大家讲的
are the two conclusions

主要的这样两个结论
we just talked about

首先呢
First

当总体服从正态
when the population obeys normal distribution

那么Xbar一定服从正态
then, the Xbar will be sure to obey normal distribution

当总体是非正态
When the population is non-normal

但是如果是大样本的话
but the sample is large

那么样本均值呢
the sample average

我们说是近似服从正态的
will approximately obey normal distribution

那么在小样本的情况下呢
If the sample is small

我们要跟大家说
we should say

这个情况呢
this case

是比较特殊的
is special

那么我们样本均值呢
Then, the sample average

往往它是服从非正态分布的
often obeys non-normal distribution.

那么这种情况呢
This is the case

是我们没有办法进行讨论的情况
that we cannot discuss

那么我们在之后呢
Later

会接触到这个小样本分布
we may contact the small sample distribution

就是T分布
That is T distribution

那么那个前提条件呢
The preconditions for that

我们说是非常严格的
are very strict

就是当总体服从正态
That is, when the population obeys normal distribution

而且呢
and

我们说方差未知
the variance is unknown

并且呢
and

是小样本的情况下
the sample is small

我们的Xbar呢
then, the Xbar

我们说它是服从T分布的
obeys T distribution

这个是比较特殊的一个情况
This is a special situation

前提条件 正态总体
Preconditions are normal population

方差 总体的方差未知
unknown population variance

而且呢是小样本
and small sample size

那么Xbar呢
In this case, Xbar

我们说它是服从T分布的
obeys T distribution

就是这个时候
In this case

我们不论是做估计
no matter when we do estimation

还是做检验
or test

都要注意这种比较特殊的情况
we should pay attention to this special situation

再接下来呢
Next

我们要给大家介绍的是
we will introduce you

样本比例的抽样分布
the sampling distribution of sample proportions

其实就是样本比例概率的分布了
It's just the probability distribution of sample proportions

既然大家能够理解样本均值
Now that you understand the sample average

是一个随机变量
is a random variable

那么当然
then, of course

样本比例呢
sample proportion

它也是一个随机变量
is also a random variable

既然是随机变量
Since it's a random variable

那么它当然就会有它的概率分布
it of course has its probability distribution

那么关于样本比例的抽样分布
About the sampling distribution of sample proportion

大家需要了解的一个结论就是
one conclusion you need to know is that

中心极限定理告诉我们
the central limit theorem tells us that

当n充分大
when n is sufficiently big

那么在这个地方呢
at this point

我们一般要求的更加严格一点
we usually want to be a little bit stricter

一般要求nP≥5
It is required that nP≥5

并且n(1－P)≥5
and n(1－P)≥5

那么这种情况呢
Only in this case

我们才视为大样本
it is regarded as a large sample

在大样本的情况下呢
In the case of a large sample

我们的样本比例
the sample proportion

是近似服从正态的
approximately obeys normal distribution

服从什么样的正态呢
What kind of normal distribution does it obey

我们说他服从均值为大P
The normal distribution has an average greater than P

就是总体比例
that is the population proportion

然后它的方差呢
And what about its variance

（公式如上）
(The formula is as above)

这样的一个正态分布
It is this kind of normal distribution that it obeys

那么我们在之后的这个估计
In the future estimation

和检验当中呢
and test

都会经常用到
we will frequently use

这样的一个结论
this conclusion

这个是需要大家注意的
This need our attention

再接下来呢
Next

我们还要跟大家介绍的
we will introduce you

就是样本方差的抽样分布
the sampling distribution of sample variance

我们刚才提到了
We mentioned just now

样本均值
sample average

样本比例都是随机变量
The sample proportions are all random variables

那么样本方差
What about sample variances

它当然也是随机变量了
Of course, they are also random variables

这里呢
Here

我们给大家一个条件
we give you a condition

就是如果总体服从正态
If the population obeys normal distribution

然后呢
Then

你这个样本呢
The sample

是一个简单随机样本
is a simple random sample

那么我们说这个
Then

统计量
the sample statistics

就是（公式如上）
(The formula is as above)

S平方就是我们的样本方差
S squared is our sample variance

再除以σ平方
Divide it by σ squared

是总体方差
to get the population variance

那么这个随机变量
Then, this random variable

我们说它是服从自由度
obeys a chi-square distribution

为(n-1)的卡方分布的
with a (n-1) degree of freedom

那么也就是这样一个式子
This formula is about

那么关于总体方差的
the population variance

估计和检验当中
In estimation and test

我们会用到这个结论
we will use this conclusion

这个是需要大家非常熟悉的内容
This is the content that you should be very familiar with

好 这一讲我们就讲到这里
OK, so much for this lecture

谢谢大家
Thank you