3.7.1 Position average: a robust expression of central tendency 2 位置平均数：集中趋势之稳健表达2在线视频

这个就属于位置平均数
This is location average

大家注意了
We should keep in mind

使用这些平均数的时候
when we use these kinds of average

一定要遵循一些基本原则
there are certain principles to follow

第一个原则就是
The first principle is

一 同质性
homogeneity

平均数的对象那个总体
The population of this average

单位具有同质性
should be homogeneous in quality

前面我们讲过
As we have mentioned in earlier lectures

总体的第一个特点大量性
one of the characteristics of population is volume

第二就是同质性
The other is homogeneity

同质性指的就是
Homogeneity means

它们属于这个同一个总体的
they should belong to the same population

这样的话
Only in this way

我们计算出来的平均数
can the calculative average

才有代表性
be representative

比如说平均收入
Take the average income for example

有些地方公布出来的平均收入
The average income released in some places

就不具有同质性
violates homogeneity

有些 比如说有些人置疑
In some cases, there are doubts

农民人均收入
about the average income of farmers

农民人均可支配收入
Per capita disposable income of farmers

它好多不是农村的收入
includes data that is not rural income

它好多是什么
What is this data?

工业收入或者投资收入
It might be industry revenue and capital gains

这个能不能计算进去
It should not be included.

再一个因为它
Moreover, it is because

这些投资的人呢
the investors

他可能不是农民了
may not be farmers anymore

他已经是转换了
They might have become

他可能是金融家
financiers

也可能是企业家
and entrepreneurs

但只是因为在计算的时候
But he is registered as farmer

他的身份就是他的户口
in household registration

还属于农民
when the calculation is done

所以它把他的收入也计算进来
And his data is counted as well

但是我们如果从统计的
But judging by the homogeneity principle

总体的同质性来讲
in statistics

这一部分人不应该属于农民
their data should not be counted

所以这个收入里面
as the income of farmers anymore

在计算的时候特别要注意
We should always keep in mind

平均数使用计算的时候
when we need to calculate the average

一定要注意总体的同质性
we must ensure the homogeneity of the population

不然的话它会夸大
Or else, there will be exaggerations

或者是缩小平均数的那个代表性
or reduce the representation of the average

第二个问题呢
Another thing to remember is that

我们要用组平均数
we should use subgroup average

来补充说明总平均数
to supplement the population average

因为我们可能看
Sometimes, we might find

总平均数并不高
the population average is not high

但是你看
But we might also discover

各组的组平均数都很高
the subgroup averages are high

这是什么原因呢
What is the reason for this

这就是因为出现了权数
Because there is weight

因为权数在这里
Because weight has

会起着不同的作用
played its role in the calculation

这个在美国就出现过
This happened in the U.S.

美国劳务工组织就认为
The labor workers organizations in the U.S.

男女就业方面没有性别歧视
used to deny discrimination against sex in employment

并且可能还对女性更有优惠
They even claimed it was favorable to women

他们有一套数据
They showed a set of data

但是美国妇女组织
But National Organization for Women

拿出的数据就不是
had different data

她拿出的数据就是
Their data showed

美国妇女在各个部门
women were discriminated in every department

她觉得就业方面受到歧视
and in employment

她就业率要低于男性
Women’s employment rate was lower

这个时候就是组平均数
There was a contradiction

和总平均数发生了问题
between population average and subgroup average

所以我们在使用总平均数的时候
So when we use population average

用组平均数来进行补充说明
we should supplement it with subgroup average

这两类的分的情况
The two types of averages

是根据具体的要求来确定
should be defined by specific requirements

使用的要求对象来确定的
and by the requirements in operation

等一下我们会发现
Soon we will discover that

我们使用的平均数特别多
there are various averages, including

有算术平均数
arithmetic mean

调和平均数 几何平均数
harmonic mean, geometric mean

中位数 众数 四分位数
median, mode and quartile

这么多平均数
There are so many types of average

它们的使用的时候一定要注意
that we must pay attention to their application

它们使用的场合
on different occasions

它有些地方只能用一种
Sometimes, only one type of average is needed

比如说我们讲的人均收入
such as in calculating per-capita income

人均支出 消费支出
per capita expenditure and consumer expenditure

比如我举一个例子
Here is an example

它有一个农场
There is a farm

那个农场那个税务官员来问他
When the tax official asks the farmer

你们的收入是多少
what is your income

他说我这个农场的人均的
The farmer replies that the average annual income

年均收入是5000美元
per capita is $5000

但是他跟朋友介绍的时候
But when the farmer talks to his friend

我们这个农场的人
he says the workers on this farm

年均收入是10万美元
has an average annual income of $ 100,000

这是什么原因造成的
Why is there such a difference

这两个平均数都没错
Both of the averages are correct

他前面用的是中位数
But the one the farmer tells the official is median

后面用的是算术平均数
And the one the farmer talks to his friend is arithmetic mean

因为算术平均数受到
And arithmetic mean is affected

极端值的影响
by outliers

受到极端值的影响就是
Here, this is how it is

这个地方是一个农场
This is indeed a farm

农场就是养牛的
a cattle raising ranch

但是这个地方空气挺好
But it is located in a place with good air quality

住了两个亿万富翁
Two billionaires live in this area

就是这些穷人跟两个亿万富翁
And both the farmers and the billionaires

加起来算算平均每个人10万美元
are counted into the population, whose average is $100,000 per capita

但是如果由少
If we arrange the list

由收入最少的往最大的排队
from lowest to highest

一直排列
in this order

处于中间位置的那个人
the person in the middle position

加起来的收入就是5000美元
has an income of $5000

所以说我们讲
Therefore, we can say

每个平均数
each type of average

它是要有一定的使用特殊场合
is used to suit certain purposes

比如我们讲的衣服 帽子 鞋
In the case of clothing, hat and shoes

这个他使用的平均数
the average is given

在工厂里面生产的时候
for the reference of the manufacturer

他是用平均数生产
The average is needed in production arrangement

因为他也不知道
because the manufacture does not know

每一个顾客的要求尺寸
the size of every individual customer

他用平均数算的话
What kind of average

他用什么平均数呢
is the manufacture going to use

他不能用算术平均数
He cannot use arithmetic mean

因为算术平均数算出来的话呢
Because the size calculated by arithmetic mean

没有人能穿
is not realistic

他用的是众数
He uses mode

就是大多数人能够穿的尺码
which represents the size of most people

所以大家注意
So we should remember

等一下讲的这么多的平均数里面
to pay attention to the occasion

使用场合一定要注意
that requires the application of average

这是我们讲的平均数的有关情况
This is what we have learned about average

平均数的有关情况里面
There is one more thing

再要补充的就是中位数 众数
about the quantitative relations

和算术平均数它们之间
among median, mode

有一定的数量关系
and arithmetic mean

这个关系它是
This relation is summarized

我们通过现实的一些例子
and calculated from many examples

来测算出来的
in real life

但是这个数值不一定严格
But the numerical result is not rigorous

大家可以看看相应的公式
We can look at the relevant formula

大家就看那相应的公式
Look at the relevant formula

那公式里面有
which shows

比如说中位数它等于
say, the median is equal to

算术平均数什么 什么
arithmetic mean and the like

与它们之间的关系
Look at their relationship

众数跟算术平均数之间的什么关系
What is the relationship between mode and arithmetic mean

而算术平均数与众数和中位数
And what is the relation among arithmetic mean

之间的关系
mode and median

但是如果画出一个钟型图来的话
If there is a bell curve

你就能看出来
you can easily figure out

中位数总在中间
that median is always in the middle

而只是说分布的时候要看
And we look closer at the distribution

如果是正态分布
In normal distribution

中位数 众数 算术平均数
median, mode and arithmetic mean

是在一个点上
are at the same point

它们三者会相等
meaning they are the same

但是如果右偏的话呢
In right-skewed distribution

那中位数在中间
median is still in the middle

众数呢是在左边
with mode on its left

算术平均数在右边
arithmetic mean, right

如果是左偏的话呢
In left-skewed distribution

刚好反的
it is the opposite

中位数在中间
with median in the middle

算术平均数在左边
arithmetic mean, on its left

众数在右边
and mode, right

大家可以看看相应的图形
These are the corresponding charts

下面我们来试试
Now let us try to

用刚刚对二妞班上的
verify the theory

平均成绩所计算的结果
using the calculated GPA

来判断一下它们是属于
of Er Niu’s class to check if it is

对称分布 左偏分布
symmetrical distribution, left-skewed distribution

还是右偏分布
or right-skewed distribution

我们来看一看
Let us see

算术平均数（计算如上）
the arithmetic (see the calculation above)

中位数（计算如上）
median (see the calculation above)

众数（计算如上）
and mode (see the calculation above)

通过这三个指标大小的不同
From the difference in the three indicators

我们可以看到
we can see

（关系如上）
(the relationship above)

由此它显示的应该是右偏分布
thus, the chart should display right-skewed distribution

好
Ok

这就是我们讲的集中趋势指标
this is what we have learned about central tendency