9.1.2 Correlation coefficient: quantify the degree of correlation 相关系数：量化相关程度在线视频

同学们 下面我们开始讲
Fellow students, now we start our lecture

相关分析里面的核心的内容
on the core content in correlation analysis

那就相关系数的测定
That is the measurement of the coefficient of correlation

那讲相关系数测定的时候
When it comes to the measurement of coefficient of correlation


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我们一定要搞清楚
we must make clear

相关分析的特点
the characteristics of correlation analysis

相关分析的特点和回归分析的特点
which are different

是不同的
from the characteristics of regression analysis

相关分析的特点它有三个
Correlation analysis has three characteristics

第一就是两个变量
First, both variables

都属于随机变量
are random variables

也就是X是随机变量 n
Namely, X is a random variable

Y也应该是随机变量
so should Y be

第二个特点相关分析里面的变量
The second characteristic is, the variables in correlation analysis

X与Y只能计算出一个相关系数
X and Y, can only figure out a coefficient of correlation

就是X对于Y来讲
which means the degree of correlation

它的相关程度是多少
between X and Y

如果是a
If it is a

那反过来Y对X来讲
then on the contrary the coefficient of correlation of Y with respect to X

它的相关系数也是a
is also a

那就相关系数是唯一
In short, the coefficient of correlation is unique

第三个特点就是说
The third characteristic is

在计算相关系数的时候
during the calculation of coefficient of correlation

它要求的资料
it requires equal access to data

X如果是抽样调查的资料
If X is the data of the sampling survey

那么Y也应该是抽样调查的资料
then Y should also be the data of the sampling survey

这就是相关分析三个特点
These are the three characteristics of correlation analysis

一、X与Y都属于随机变量
I Both X and Y are random variables

二、相关系数唯一
II The coefficient of correlation is unique

三、X与Y获取资料的方式要相同
III X and Y must access data by equal means

在讲相关系数计算的时候
When it comes to the calculation of coefficient of correlation

大家一定要注意一点
everyone must pay attention that

首先要进行理论分析
the first step is theoretical analysis

来确定变量与变量之间
to decide whether there exists

是否真正的存在我们讲的相关关系
any correlation between variables as we talk of

因为两组数据
Given two sets of data

你随便进行计算
however you calculate

它都能产生相关系数
they can always produce a coefficient of correlation

比如说 下面有一些例子
Below are some examples

大家看一下
for everyone to examine

一位印度的统计学家
An Indian statistician

他通过计算发现
finds through calculation

说印度的粮食产量跟石油价格有关
that the grain output in India is correlated with petroleum prices

每当世界的石油价格下降或上升
Whenever the world’s petroleum prices drop or rise

印度的粮食产量
India’s grain output

就是呈现着负相关的变化
varies, rendering a negative correlation

我们知道印度的粮食产量
As known, India’s grain output

与石油的价格
should have nothing to do with

应该是没有关系的
petroleum prices

这种计算出来的相关系数
Coefficient of correlation calculated therefrom

肯定是伪相关
must be indicative of a spurious correlation

还有俄罗斯曾经发现过
Still, Russia once discovered

有个地方
some place

有一种候鸟一来
where the babies born were all boys

它这个地方生的小孩都是男生
once a breed of migratory birds arrived

这候鸟一走以后
Once the birds flew away

这个地方生的都是女生
the babies born there were all girls

他说小孩的性别跟那个候鸟有关
He correlated the babies’gender with the migratory birds

这个好像分析也没有发现什么问题
It seems no problem was found through this analysis

它们之间没有存在着
There existing no

实际的相关关系
real correlation between them

它也属于伪相关
such is also a spurious correlation

在国内也有一些人
There are also some individuals in China

通过一些例子发现说
who conclude via some examples

中国啤酒的价格
that China’s beer price

跟我们统计教师的工资
is highly correlated with

有高度相关
the salary of our teachers of statistics

这个好像
This sounds

也不是我们相关分析里面
as if it were not the real correlation

所讲的我们真正的相关关系
we are talking about correlation analysis

因为中间还有一个
Because there is also

起着连接的变量
a connecting variable in between

也就是我们国家的经济增长
namely the economic growth of China

因为我们国家的经济增长
As China’s economic growth

一直朝上增
is always upward

经济总量越来越大
economic aggregate is greater and greater

劳动生产率越来越高
labor productivity is higher and higher

效率越来越高的时候
and as efficiency is higher and higher

所有的用工工资都增加
all employment salaries increase

所以统计老师的薪水也增加
with no exception to the salary of teachers of statistics

那相应的消费品
It follows that the prices of the corresponding consumer goods

啤酒的价格等等的价格都会增加
say beers, also tend to increase

所以我们讲
So we say

相关分析一定是要在
correlation analysis must

理论分析的前提下
be conducted

才能进行相关分析
under the premise of theoretical analysis

才能计算相关系数
to work out the coefficient of correlation

那相关分析
Regarding correlation analysis

我们可以通过下面几个方法
we can use several methods as below

来进行分析相关
to perform it

来体现X与Y之间是否有关系
and find out whether there is any relation between X and Y

是否存在相关关系
or whether there exists any correlation

第一我们可以画出相关图
First, we can plot a correlogram

比如X与Y之间的相关
For example, to find out whether there is any correlation

是否有关系
between X and Y

我们在直角坐标系上
we can plot a correlogram

画出相关图
in the rectangular coordinate system

看看X变化
and check whether

Y是不是随着它呈线性相关变动
Y varies

还是呈正相关还是呈负相关
in linear correlation to X

发生变化
and whether the correlation is positive or negative

这是相关图
This is about the correlogram

第二还有相关表
Second, we have a correlation table

我们通过表格进行分组
Via the table, we can group X and Y

X作为一列 Y作为一列
into two distinct columns

进行对比来看
By making a comparison, we see

X变化 Y怎么办
how Y varies as X varies

看看它们之间是否存在着同方向
and whether they vary in the same

还是反方向等等变化
or opposite directions

这是第二类分析 相关分析
This is the second type of correlation analysis

第三类相关分析
The third type of correlation analysis

我们就比较好直观的
is relatively intuitive

就是我们计算它的相关系数
By calculating the coefficient of correlation

我们算出来X与Y
we figure out the exact degree

到底有多密切的相关程度
to which X and Y are closely correlated

这个计算
In doing this calculation

我们先从我们的图里面来看
we first see in the graph

我们是从直角坐标系里面
We start by calculating the mean of X

先计算X的平均值
in the rectangular coordinate system

再计算Y的平均值
and then calculate the mean of Y

把所有的散点分成了四个象限
All scattered plots fall into the four quadrants

大家可以看一下这个图
Everyone can take a look at this graph

分成了四个象限以后
After they fall into the four quadrants

大家我们来看一下
Let’s have a look at

第一象限和第三象限的变量的点
the points of the variables in Quadrants I and III

我们就散点
We subtract

它们分别都跟它自己的
the mean of the corresponding variables

所对应的变量的平均数相减
from the value of the scattered points

把这两个离差相乘
and then multiply both deviations

就是说X减掉X的平均数
Namely, by multiplying X minus the mean of X

去乘上Y减掉Y的平均数
by Y minus the mean of Y

我们会发现
we will find

第一象限的离差积
the product of the deviations in Quadrant I

和第三象限的离差积都属于正号
and the product of the deviations in Quadrant III is both positive

我们就可以看出来什么呢
What can we conclude, then

就是X随着X的增加Y也在增加
As X increases Y increases

然后X与Y它属于正相关
and Y is positively correlated with X

刚好我们这个离差的积也是正号的
It just happens that the product of the deviations is positive

所以我们说
So we say

X与Y是同方向
X and Y vary in the same direction

这就是我们第一个
In summary

相关系数计算的时候
when calculating the coefficient of correlation

先确定方向
we first determine the direction

离差积的方向
The direction of the product of deviations

与我们X与Y的相关方向是相同的
is consistent with the direction of the correlation between X and Y

第二 看看它的程度
Second, we see the degree

（公式如上）
(by the formula above)

这个变量的大小
Which factor is the magnitude of this variable

首先跟哪个有关
correlated with in the first place

就跟散点的个数有关
It is correlated with the number of scattered points

因为它们都属于正的
since both are positive

个数越多 这个数值就越大
The greater the number the greater the value

我们要消除个数的影响的话
To eliminate the effect of number

统计里面很简单
we have very simple statistical techniques

通常就是在这个式子下除以一个N
A usual technique is to divide this expression by N

就消除了个数的影响
and thus the effect of the number is eliminated

算出平均 它的平均大小
As we figure out the mean

是不是消除了个数影响以后
does this expression fully reflect

我们这个式子就充分反映
the relation between X and Y

X与Y之间的关系呢 不是
after the effect of number is eliminated No

我们还发现
We also find

（公式如上）
(the formula above)

其实就是协方差
which is actually about the covariance

我们在前面讲方差的时候
While talking about variance previously

就发现方差的大小
we found the magnitude of variance

它除掉与变量的离散程度大小有关
is not only dependent on the discrete degree of a variable

还有变量的本身数值大小有关
but also on the value of the variable itself

也就是X本身数值大
In other words, a greater value of X

算出来的它的方差就大
coincides with a great variance of it

同理在协方差里面
By the same token

协方差的大小
the magnitude of covariance

除掉我们说X与Y的
is not only dependent on

关系密切程度有关
the degree to which X and Y are closely correlated

还有X本身的大小
but also on the magnitude of X

Y本身的大小有关
and Y themselves

我们参照标准差系数的计算方法
In reference to the method of calculating the coefficient of standard deviation

在标准差下面除以一个算术平均数
we divide the standard deviation by an arithmetic mean

就把它的X也就变量的
to eliminate the influential factor

本身影响因素去掉 剔除掉
of the variable X itself

在我们协方差里面
For covariance

我们也同样
we process by the same measure

我们去除以 分别除以
by dividing

X的标准差 和Y的标准差
the standard deviation of X and Y, respectively

除以X的标准差 和Y的标准差
We do so

就是为了剔除X和Y
just in order to eliminate

各自本身数值本身大小
the effect of the respective value of X and Y

对我们协方差的影响
on the covariance

我们刚才是从一三象限来看
Just now we saw from Quadrants I and III

二四象限大家来看 也一样
The same goes for Quadrants II and IV

X减掉X的平均数
The product of X minus the mean of X

乘上Y减Y的平均数
and Y minus the mean of Y

他们的方向
must be

他们的符号 一定是负的
negatively directed and signed

那刚好我们在坐标里面看发现
It just happens, as we find in the coordinates

X增加 二四象限里面表现Y是减少
that as X increases, Y decreases, as manifested in Quadrants II and IV

所以X与Y属于负相关
So X and Y are negatively correlated

所以符号相同
having the same sign

那再计算它的相关系数的
When calculating the coefficient of correlation

大小的时候
between them

我们把这个离差之积加起来
we add up the products of the deviations

它的大小也跟个数有关
The sum is dependent on the number

也跟它X与Y本身数值大小有关
as well as on the values of X and Y

那这样的话
This way

我们也要分别去除以N个数
we shall divide it by the number N

和标准差X的标准差
the standard deviation of X,

和Y的标准差
and the standard deviation of Y, respectively

那这样的话
In result

我们就得出了一个相关系数公式
we derive a formula for coefficient of correlation

大家看一下
Let’s have a look

（公式如上）
(at the formula above)

也就是协方差除以下面两个标准差
That is to divide the covariance by the two standard deviations below:

X的标准差 和Y的标准差
the standard deviation of X, and the standard deviation of Y

大家注意
Note that

这个公式它计算的
what this formula calculates

是X与Y之间的线性相关系数
is the coefficient of linear correlation between X and Y

如果X与Y之间不是线性相关
If X and Y are not linearly correlated

而是非线性相关的话
but nonlinearly correlated

那这个公式就会失效
then this formula becomes invalid

比如我们讲
For instance, if

通过相关系数计算出来的
it turns out r=0

结果是r=0
as the coefficient of correlation is calculated

是否说X与Y之间不存在关系呢
can we conclude there is no relation between X and Y

不 你只能说
No You can only conclude

说看通过r=0
judging by r=0

说X与Y之间不存在线性关系
there exists no linear relation between X and Y

可能存在非线性关系
and there may exist a nonlinear relation

比如X与Y之间的关系是圆
For instance, the relation between X and Y is a circle

我们讲的圆的关系
When it comes to the relation of circle

大家知道
everyone knows

在高中的时候学过圆的方程
the equation of circle as learned at senior high school

你如果拿着这个积差法
When referring to this formula for the coefficient of correlation

计算的相关系数公式里面来看
by the product-deviation technique

它刚好放了这个直角
you will find it just involves the four quadrants

坐标里面的四个象限里面
of the rectangular coordinate system

刚好每个象限都是1/4
each of which occupies a quarter

它算出来的结果
The result works out

X与Y的线性相关系数就等于0
is that the coefficient of linear correlation between X and Y equals zero

但是我们在高中的时候知道
But we knew at senior high school

X与Y之间是函数关系
that the relation between X and Y is a functional relation

它是完全相关
a complete correlation

它不是没有关系
It is not that they have no relation

它只是非线性关系里面的完全相关
but just that they bear a complete correlation, one of the nonlinear relations

这是相关系数计算的时候
This is the first thing to notice

使用的时候要注意的第一个
during the application and calculation of the coefficient of correlation

第二个我们计算的时候大家注意
A second thing we shall pay attention while calculating

相关系数计算出来的结果
is that the coefficient of correlation

是唯一的X对于Y
has a unique result of calculation for X versus Y

比如是A的相关系数
For instance, if A is the coefficient of correlation

Y对X的结果也是A
the result for Y versus X is also A

并且他们的数值在正负一之间
Moreover, its value ranges between -1 and 1

但是大家知道
However, everyone is aware

这里面搜集来的资料
that the data here

都是我们讲随机搜集来的资料
are randomly collected

也是样本资料
They are also sample data

样本资料要说明总体的情况
In order for the sample data to indicate the status of the population

我们要进行假设检验
we shall conduct a hypothesis test

因为你获取的资料
because whether the data you access

它的信息量是不是充足
have sufficient information content

也就是我们是否是大样本等等
namely whether we have a large sample

才能说明在总体里面
can explain whether there really exists

X与Y真正存在着
a correlation between X and Y in the population

我们讲的相关关系吗
as we are talking about

你随便找两个点来计算
Choose two arbitrary points

A X{\fs12}1{\r} Y{\fs12}1{\r}
A X{\fs12}1{\r} Y{\fs12}1{\r}

B X{\fs12}2{\r} Y{\fs12}2{\r}
and B X{\fs12}2{\r} Y{\fs12}2{\r}

你去计算相关系数
to calculate the coefficient of correlation

那相关系数一定是等于一
and you will find the coefficient of correlation always equals one

因为我们大家在数学里面知道
As known in mathematics

两点之间一定是画出一条直线
a straight line is bound to exist between two points

所以我们在计算出来的相关系数
So before using

进行使用之前
the coefficient of correlation we have figured out

你必须进行假设检验
we must conduct a hypothesis test

那相关系数的假设检验有两种
There are two types of hypothesis testing on the coefficient of correlation

大家可以看下面的
Everyone can read the following

我们给出来的方法和步骤
procedure and steps we have presented

以及相应的公式
with corresponding formulae

这是我们计算相关系数
This is the second issue

或者使用相关系数
we shall pay attention to

要注意的第二个问题
while calculating or using the coefficient of correlation

那第三个问题我们注意
The third issue we shall notice is

相关分析里面分析的关系
the relation analyzed in correlation analysis

X与Y的关系是直接关系
is a direct relation between X and Y

不能是间接关系
and cannot be an indirect relation

大家一定要注意
Everyone must pay attention

你算的时候
the coefficient of correlation

出来的相关系数
you figure out

一定是X与Y的直接关系
must reflect the direct relation between X and Y

第四个就相关系数的大小
Fourth, the magnitude of the coefficient of correlation

我们有一些叫高度相关
may reflect a high correlation at times

我们好多 大家去看
There are many cases for everyone to see

写的文章或者是搞的研究
In essays or studies

他们讲某某变量与变量之间
what the authors relate is something like high correlation

存在高度相关
or low correlation

还是低度相关等等
between variables

这个是相关程度的方面
This concerns the degree of correlation

我们也有一个大致的标准
There is also a general standard

大家可以参考
for everyone to refer to

其实在统计里面
Actually, in statistics

一般不太利用形容词
such adjectives

高度相关或者是弱相关等等
as highly correlated or weakly correlated are rarely used

我们就是把它的数值拿出来
We simply render the value

由用户 由我们的读者来进行判断
for our readers to make a judgment

这是我们相关分析的主要内容
So much for the main content of correlation

今天的课就讲到这里
and so much for today