3.5.1 Calculating the average (2): Full expression of central tendency 计算平均数（二）：集中趋势之充分表达在线视频

除掉算术平均数以外
Besides arithmetic mean

第二个平均数就是调和平均数
there is another average called harmonic mean

大家注意调和平均数
Pay attention to this

它是把所有的变量值变量
Harmonic mean is the reciprocal of

进行倒数
the variables

就是x先倒一下 倒成1/x
It turns x into 1/x

倒数以后 完了以后
and then

进行算术平均
conducts the arithmetic mean calculation

平均完了以后再把它倒回去
And again find the reciprocal of the output

所以这个平均数叫做
Therefore, harmonic mean is

倒数平均数的倒数
the reciprocal of the arithmetic mean of the reciprocals

这个平均数呢 大家注意了
We should keep in mind that this mean

这个平均数 它在现实生活里面
in real life

不管是自然现象 社会现象
in both natural and social phenomena

还没有找到它自己的运用对象
has few chances for application

但是出现在这里
But here

它主要是用于
it is mainly used

因为统计资料给出来的时候
when special occasions

出现了
arise

就是特殊的情况下
in statistical information

用于计算算术平均数的
It is used to measure a form variety

一种变形而使用
of arithmetic mean

大家可以看看例子
Here is an example

如果大家去过菜市场
If you have ever been to the food market

就应该知道菜市场的蔬菜
you should be aware that price of vegetables

在不同时间段的价格
at the market

一般是不一样的
varies by time

早上的蔬菜一般最贵
with price in the morning being the highest

而晚上的蔬菜一般最便宜
and in the evening, the lowest

我们这里假设二妞的妈妈
Suppose, Er Niu’s mother

分别在早上 中午和晚上
spends 1 yuan, respectively,

花了一块钱
in the morning, at noon and in the evening

买下这一类蔬菜
on the same kind of vegetable

那么让我们来求一求
let us calculate

这一类蔬菜的平均价格
the average price of this vegetable

根据公式我们可以得到
Following the formula, we can get

变量值倒数分别为
the reciprocal of variable

1/x{\fs10}1{\r}=2.5
1/x{\fs10}1{\r}=2.5

1/x{\fs10}2{\r}=4
1/x{\fs10}2{\r}=4

1/x{\fs10}3{\r}=5
1/x{\fs10}3{\r}=5

然后我们再对这一变量值的倒数
Then we use the reciprocal of the variable

计算算术平均数
to calculate the arithmetic mean

最后再求这一算术平数的倒数
At last, we find the reciprocal of the arithmetic mean

也就是我们这里计算得到的H
H, the output of the calculation

等于（计算如上）
is equal to (see the calculation above)

最后得到一斤两毛六的平均价格
The average price is 0.26 yuan per jin

下面我们再来看一看
Now let us look at

加权调和平均数是如何计算的
how to calculate weighted harmonic mean

我们还是用刚刚举的例子
We use the same example

来进行演示
to show

同样还是二妞的妈妈
Suppose, Er Niu’s mother

在菜市场买这一类蔬菜
buys a vegetable at the food market

假设这一类蔬菜的价格仍然不变
Assume the price of the vegetable stays the same

只是二妞的妈妈
But Er Niu’s mother

在早上和中午所购买蔬菜的金额
spends different amount of money buying the vegetable

发生了变化
in the morning and at noon

下面让我们来看一看
Now let us look at

在这个例子中
in this case

这一类蔬菜的平均价格为多少
how much the average price of this vegetable is

这里我们使用
Here we use

加权调和平均数的公式来进行计算
the formula of weighted harmonic mean to calculate

也就是H等于（公式如上）
H is equal to (see the formula above)

（公式如上）
(see the formula above)

（计算如上）
(see the calculation above)

最后得到一斤三毛三的平均价格
and the average price is 0.33 yuan per jin

通过加权调和平均数的公式
By using the formula of weighted harmonic mean

我们可以看到
we can see

在这里m实际上是等于xf
m here is equal to xf

也就是购买这一类蔬菜的价格
which means the price of the vegetable

和数量相乘
multiplies the amount purchased

得到的就是购买这一类蔬菜
is equal to the actual amount of money

所花费的实际金额
spent on buying the vegetable

由此我们可以进一步得到一个结论
Therefore, we can draw the conclusion

也就是当m等于xf时将其代入
when we substitute xf with m of the same value

最后H等于x拔
H is equal to x bar

从这里我们可以看到
Here we can see

调和平均数它实际上是
harmonic mean is in fact

算术平均数的变形
a form of variety of arithmetic average

这是调和平均数
This is harmonic mean

再一个就是几何平均数
Another average is geometric mean

几何平均数的话
Geometric mean

它是n个变量的连乘积开n次方
is the nth root of the continued product of n numbers

这是简单几何平均数
This is simple geometric mean

那就是大家看到的G等于
as we can see G is equal to

（公式如上）
(see the formula above)

（公式如上）
(see the formula above)

这是简单的
This is the simple geometric mean

如果要加权
When weight is involved

它也存在着加权几何平均数
there is also weighted geometric mean

它是（公式如上）
it is (see the formula above)

（公式如上）
(see the formula above)

开根号是Σf次方
the square root means the power of Σf

这个是几何平均数
This is geometric mean

几何平均数在后面
We will focus on geometric mean

我们的时间序列里面的
in the lecture on consecutive average

动态平均数的时候会讲
in time series

比如说平均发展速度
Take average development speed for instance

它就要用几何平均数来进行计算
it needs to be measured by geometric mean

几何平均数使用的对象大家注意
We should keep in mind that the geometric mean

它使用的是比例平均 速度平均
can only be used

才能用几何平均数
to work with ratio and speed

下面我们来看一看
Now let us look at

几何平均数在实际中是如何运用的
how geometric mean is used in real life

如某流水生产线
Suppose there is an assembly line

有前后衔接的五道工序
with five steps in a working procedure

某日各工序产品的合格率分别为
The other day, the respective qualification ratio is

95% 92% 90%
95% 92% 90%

85% 和88%
85% and 88%

试求整个流水生产线产品的
try to figure out the average qualification ratio

平均合格率
of the whole assembly line

要求平均合格率
To calculate the average qualification ratio

就只能使用几何平均的方法
the geometric mean should be used

也就是（公式如上）
as in (formula above)

（公式如上）
(formula above)

等于(计算如上)
is equal to (calculation above)

也就是这一流水生产线产品的
and this is the average qualification ratio

平均合格率
of this assembly line

由此我们可以看到
We can see

几何平均数的计算
the application of geometric average

它是适用于反映特定现象的
is used to represent the average level

平均水平
of a certain phenomenon

也就是说反映现象的总体标志值
That is to say, the population mark total amount

是各单位标志值的连乘积
is the continued product of unit character value

所以当用于计算比率或速度的
Therefore, when calculating the average

平均数的时候
of ratio or speed

我们一般使用的是几何平均数
we prefer to use geometric mean

而不是算术平均数
instead of arithmetic mean

最后再给大家留一个问题
At today’s assignment, think about this question

如果上题中
If, in the case we just discussed

不是由五道连续作业的工序
rather than an assembly line

组成的流水生产线
of five steps of working procedure

而是五个独立作业的车间
it was five independent workshops

并且各车间的合格率
and the qualification ratios

和前面是相同的
remain the same

同时我们又假定
Meanwhile we assume

各车间的产量相等 都为100件
the yield of the workshops is the same, 100

那么我们来试着求一求
Try to figure out

该企业产品的平均合格率
the average qualification ratio of the factory

这三种平均数它在使用的时候
These three means are used

各有自己的对象
in different applications

或者叫特殊的场合
or on different occasions

这三个平均数如果在
In these three means, when

自然数的计算下
calculated with natural number

它应该是
it seems

调和平均数最小
the harmonic mean is the smallest

几何平均数次之
geometric is smaller

算术平均数是最大的
than arithmetic mean

下面我们来看一下
Now let us look at

这一结果的推导过程
the reasoning

假设有两个不等的数值
Suppose there are two range of values

x{\fs10}1{\r} x{\fs10}2{\r}
x{\fs10}1{\r} x{\fs10}2{\r},

那么（计算如上）
then (see calculation above)

它肯定是大于等于0
it must be greater or equal to 0

进一步变换可以得到
Further transformation shows

（计算如上）
(see the calculation above)

（计算如上）
(see the calculation above)

那我们当然知道
and we know

（计算如上）
(see the calculation above)

（计算如上）
(see the calculation above)

也就是几何平均数
it is a geometric mean

将这一公式继续变形
If the formula is transformed again

我们就可以得到
as we can see

（计算如上）
(see calculation above)

进一步变形可以得到
and after transformation we get

（计算如上）
(see calculation above)

那我们当然可以看到
we sure know

（公式如上）
(see the formula above)

实际上就是调和平均数
it is a harmonic mean

通过刚刚的论证过程
The reasoning above

我们可以得到x拔
helps us get the kind of relation

大于等于G 大于等于H
that is, x bar is greater or equal to G

这样的一种关系
and greater or equal to H

推广到有限的几个变量值
It is also valid if extended to

也同样成立
a limited number of variables

这三个平均数
We refer to the three means

其实我们统称为计算平均数
as numerical average

计算平均数怎么讲呢
Why do we call it numerical average

它就是说所有的x
It means all the x

就是x{\fs10}i{\r}中 i等于1到n
in x{\fs10}i{\r}, i is equal to 1 to n

n个变量值都加入到计算过程中来的
and N variables are averages

平均数
included in the calculation

所以它叫计算平均数
So, it is called numerical average

它有一个非常大的特点
A very obvious feature of this average

它不稳健
is that it is unstable

就是像刚才我们举的例子
As in the precious example

平均收入 平均消费支出
the average income and average spending

它会受极端值的影响
are affected by extreme values

所以这个大家要注意
We should keep this in mind

它不稳健
It is not stable

如果要稳健的那些平均指标的话
If we want to calculate a stable mean

那就是我们下面要讲的位置平均数
we need location average, which we will cover in our next lecture