3.4.1 Calculating the average (1): Full expression of central tendency 计算平均数（一）：集中趋势之充分表达在线视频

我们开始讲数值计算
We will learn numerical computation today

讲数值计算之前我们要明确一个事
At the beginning of this lecture, I would like to remind you

我们的目的是要说明总体
that our purpose here is to describe the population

也就是要反映总体
and reflect the population

要反映总体的话
How to accomplish this mission

从哪些方面来反映
and what should it represent

初学者可能觉得无从下手
are not easy for beginners to grasp

告诉大家
To begin with

只从两个方面反映总体
there are two aspects

就能够把总体比较完整的反映出来
that can reflect the overall situation of the population

哪两个方面
What are they

一个是集中趋势
They are central tendency

一个是离中趋势
and dispersion tendency

我举一个例子
Here is an example

比如说 一个房地产公司
suppose a real-estate company

他想在某个地方进行房地产开发
is looking for property development projects

他就要了解他的潜在客户
He needs to find potential clients

对房产的需求 要求情况
their demand and investment requirements

他就委托一个调查公司进行调查
So the company commissioned a research company to investigate

调查公司给出两个资料
The research company submitted two sets of data

一个就是家庭平均人数2.7个人
One, averagely there are 2.7 members per family

第二个资料就是这个家庭的方差
Two, the variance of family

为0.7平方/人
is 0.7 square per person

这什么意思告诉他们呢
What does the data show

他就告诉你这里的平均数
It shows the average of number

这个地区想买你房子的
of its potential clients in this area

潜在顾客的平均家庭人数算2.7
is 2.7 per household

也就3个人左右
approximate to 3

但是这3个人左右
But this is the average number

每个家庭之间的人数的差别比较大
different from the actual family size

要注意
Keep this in mind

但是这个房地产公司开发的时候
But the real-estate company designs its project

只注意到平均人数
based on the average number only

他就觉得一家三口人
And imaging a two-bedroom apartment

他住两室一厅就比较好
is sufficient for a family of three

所以他大量开发两室一厅的房子
Therefore, two-bedroom apartments

但是最终结果
outnumber

一室一厅和三室一厅开发的比较少
one-bedroom or three-bedroom apartments

最后卖的时候就出了问题
At the time of sale

一室一厅和三室一厅供不应求
the demand for one-bedroom or three-bedroom apartment

两室一厅的人数
exceeds the supply,

要的人数比较少
while the supply of two-bedroom apartment is excessive.

这是为什么呢
How does this happen

他就去问那个调查公司
The company blames the research company

你好像给出来的资料有问题
for giving him the wrong information

他说不是
The research company denies

但你还有一个指标
by saying there is another indicator

你那个0.7平方人
of 0.7 square per person

就是家庭人数 他们的差别很大
representing the difference in household size

它的方差大
There is a big variance

要注意到这里的家庭
This shows that there are

他这里有两个可能
possibilities of

出现两个极端
two extremes

一个是单身家庭的人多
large number of one-person households

二是三代同堂的比较多
and large number of three-generation households

从这个例子上说明一个什么问题
What does this case tell us

说明我们要认识一个总体的话
It tells us that we need two kinds of data

从两个方面来认识
to describe the population

一个是统计的集中趋势
One is central tendency in statistics

总体的集中趋势
the central tendency of the population

第二个就认识总体的离中趋势
The other is dispersion tendency of population

好 现在我们开始来讲
Ok, now let us look at

总体的集中趋势
the central tendency of the population

总体的集中趋势 就是我们说
It is what we call

反映总体的一般水平的指标
the tendency represented by the indicator

所代表的趋势
that reflects the average value of the population

总体的一般水平
The average value of the population

我们讲的一般水平
as we call it

指的就是平均数
is the mean value

当然平均数 它是抽象到它个体差异
The mean value is the middle calculated

得出来的一般水平
by abstracting individual differences

在统计里面平均数又比较多
There are various mean values in statistics

它首先分为动态平均
First, there are consecutive mean

和静态平均数两大类
and static mean

动态平均数是在时间上进行平均
Consecutive mean is abstracted over periods of time

静态平均数是在空间上进行平均
while static mean, in a certain space

那么静态平均数
So, static mean

我们又把它分为两大类
is further divided into

一类是计算平均数
arithmetic mean

一类是位置平均数
and location average

好 现在我们开始讲计算平均数
Now let us learn how to calculate mean value

计算平均数先讲算术平均数
We will start with arithmetic mean

算术平均数它是最简单的
Arithmetic mean is the simplest

使用最多的平均数
and most widely used average

大家平时讲的
In real life

比如说 这些收入
when we say the five of us

大家五个人一起分
split the earnings equally

这个就是属于算术平均
it is a case of arithmetic mean

其它的我们讲的那个平均重量
And the terms average weight

平均年龄等等都是算术平均数
And average age are all arithmetic means

这个平均数它是用它的总体单位数
Arithmetic mean divides the sum of total amount of character value

去除每个单位的标志值总和
by population size

也就是我们那个公式里面讲的
as shown in this formula

（公式如上）
(see the formula above)

用这个公式来进行计算
This formula can calculate arithmetic mean

当然这是简单的算术平均数
This is the simple arithmetic mean

大家可以看例子
Let us look at this example

我们可以运用
We can use

简单算术平均数的方法
simple arithmetic mean formula

对二妞班上的总评成绩计算平均
to calculate the average test scores of Er Niu’s class

大家可以看下面的计算过程
Let me show you the computation

根据我们刚刚学过的
According to what we have learned

计算简单算术平均数的公式
the formula to calculate simple arithmetic mean

我们可以知道x拔
we know the X bar

是等于总体标志总量
is the population mark total amount

除以总体单位总量
divided by population size total amount

用数学表达式表示为（公式如上）
expressed in mathematical expression (the formula above)

这里总体标志总量表示的
the population mark total amount

就是二妞班上所有同学
Is the sum of Statistics test scores

在这次统计学考试的总评分
of all the students in Er Niu’s class

它分别为50+69+73
In other words, it is the sum of 50, 69, 73

一直加到69
and all the way to last on the list, 69

而总体单位总量
The population size total amount

表示的则是二妞班上
is the overall number of test takers

参加这次考试的总人数
in Er Niu’s class

根据这份成绩单我们可以看到
From this school report we can see

在这里n是等于50
n is 50

最后计算结果得到71.54
and the output is 71.54

我们还有加权的算术平均数
And there is also weighted arithmetic mean

加权的算术平均数是什么意思呢
What is weighted arithmetic mean

就是因为出现了
Weighted arithmetic mean

同一个变量值出现了几次
takes into account the frequency of a variable

出现几次 它的权数就是几
The frequency is the weight

进行加权平均
in calculating the weighted mean

所以这个公式就是我们看到的
That is the formula we can see

（公式如上）
(see the formula above)

在这里大家注意了
Please pay attention to this

这里有个重要概念就是权数
It is important to learn the concept of weight

权数的出现将影响到统计
The concept of weight will affect

后面的所有知识
our learning in the following lectures

权数指的是什么
What is weight

宽泛一点讲
Generally speaking

权数是权衡轻重作用的那个数
weight is used to express the relative importance

如果讲具体一点的话
Specifically speaking

它就是讲在计算平均指标的时候
it means in calculating the mean

衡量每一个变量值
the variables are weighted

对平均指标的贡献大小的那个数
to find the value that contributes to the average indicator

这就是权数
the weight

权数在现实生活里面应用的比较广
Weight is also widely used in real life

人大代表投票的时候
When NPC members vote

每个人的权数都是N分之一
they each have a weight of one Nth

每个人都有一票
Everyone has one vote

这时候的权数就相等
and the weight is equal

那平均起来的时候
So, to calculate the mean

其实用简单算术就能算出来
we only need simple arithmetic

但是如果说股票里面的股权的构成
But when it comes to ownership structure in stocks

那时候的每个股东他的权数不相同
the shareholders have different weight

那时候就加权进行平均
That is why we need weighted mean

那时候就体现了权重大小的作用
to show the function of weight here

下面我们试试用加权算术
Now let me show you how to

平均数的方法
calculate weighted arithmetic mean

来对二妞班上这次考试的
to measure the GPA of the test scores

平均成绩进行计算
in Er Niu’s class

根据我们之前所学过的
According to what we have learned

统计分组的方法
about statistical grouping

我们可以得到 按总评分成绩分组
we can form a frequency distribution table

所形成的频数分布表
based on the grouping of test scores

根据这张频数分布表
The frequency distribution table

我们可以很清楚的看到
shows clearly

这里使用的是组距分组的方法
it uses class interval grouping

而组距分组的方法
One of the important characteristics

一个重要的特征是
of class interval grouping

其中的每一个组
is that units here

不是用一个具体的变量值表示
are not expressed in a specific variable value

而是使用变量值的一个区间
but in a variable interval

表示一个组的分组
to indicate the division of a unit

而要显示每一个组的平均水平
But in order to indicate the average level of each unit

就要进一步引入组中值的概念
the concept of class mid-point is used

我们之前已经学过
We have learned in previous lectures

如何对开口组或者是闭口组的
about different ways to calculate

不同的形式来计算组中值
class mid-point in open-end and close-end classes

我们这里很明显显示的是闭口组
the one displayed obviously in a close-end class

也就是运用每一组的下限
So we can use the lower limit

加上每一组的上限除以2
plus the upper limit and divide by 2

得到这一组的组中值
to get the class mid-point

如我们这里第一组50到60
Suppose the first class is from 50 to 60

它的组中值是等于50+60/2=55
its class mid-point is (50+60)/2=55

其它的组中值的计算方法依此类推
And so on and so forth

分别得到65 75 85和95
The class mid-points of other units

这里每一个组的组中值的具体数值
are 65, 75, 85 and 95

实际上就是
This is in fact

加权算术平均数里的小x
the x in weighted arithmetic average

而根据加权算术平均数的公式
According to the formula to calculate weighted arithmetic average

（公式如上）
(See the formula above)

而要得到每一组的xf
the xf in each group

实际上就是用每一组的组中值
is in fact its class mid-point

乘以每一组的人数
times the number of members in each group

通过进一步计算
In this way

我们可以得到每一组的xf
we have calculated the xf of each group

分别等于110 1365 1200
is 110, 1365, 1200

765和190
765 and 190

Σf则等于所有的总人数
Σf is the total number of test takers

等于50
which is 50

最后通过计算得到x拔
And the x bar we have calculated

等于72.6
is 72.6

大家可以看到在这里
We can check the output

我们使用加权算术平均数
of weighted arithmetic average

所得到的结果
by calculation

和我们前面运用简单算术平均数
It is not the same number

所得到的结果71.54
as the output we got from simple arithmetic average

是不一样的
71.54

那么同学有没有想过
Do any of you wonder

为什么两者前后是不一样的呢
why there are two outputs

其实这里的关键就在于这个组中值
The key is the class mid-point

它表示的只是这一组数据的平均值
It shows the average value of the data in this set

它不能准确的表示这一组的
It cannot show the accurate

具体的数值
specific value of this set

下面我们再来简要介绍一下
At last, let us look at

算术平均数的数学性质
the mathematical quality of arithmetic mean

第一 算术平均数
Firstly, arithmetic mean

与总体单位数的乘积
multiplies population size

等于各标志值的总和
is equal to the total amount of its character value

用数学表达式表示为
In mathematical expression

（公式如上）或者说是
it is (see the formula above) or

（公式如上）
(see the formula above)

第二个数学性质为
The second mathematical quality

每个标志值加或减一个任意常数a
is for every character value to plus or minus an arbitrary constant a

则算术平均数也增加或减少
Consequently, the arithmetic mean is increased or decreased

一个任意常数a
by an arbitrary constant a

三 每一个标志值乘以或除以
Thirdly, every character value multiplies and is divided by

一个任意常数a
an arbitrary constant a

则算术平均数也乘以或除以a
Consequently, the arithmetic mean multiplies or is divided by a

第四个数学性质非常重要
The fourth mathematical quality is very important

它表示的是各标志值
It reveals that the sum of character values

与其算术平均数的离差之和等于零
and arithmetic mean dispersion is zero

如果我们对刚刚的
If we represent the mathematical quality

算术平均数的数学性质
of arithmetic mean

用数学表达式来表示
in mathematical expression

也就是（公式如上）或者是
it is (see the formula above) or

（公式如上）
(see the formula above)

下面我们来看一下
Now let us look at the

具体的论证过程
detailed reasoning

这里可以看到
We can see here

通过将（公式如上）展开
by expanding the (formula above)

可以得到（公式如上）
we can get (formula above)

又由于(公式如上)
and because of (formula above)

再进一步可以得到
we can again get

（公式如上）
(formula above)

最后得到零
and at last, the output, zero

同理我们也可以证明得到
Similarly, we can also prove

（公式如上）
(formula above)

进一步得到（公式如上）
can lead to (formula above)

最后结果也是零
and the output is also zero