3.8.1 Variance and standard deviation (1): Commonly used indicators of deviation from the center 方差与标准差（一）：离中趋势之常用指标在线视频

下面我们讲 离中趋势指标
In this lecture, we are going to learn dispersion tendency index

离中趋势指标的话 它说明的情况
The dispersion tendency index describes a situation

跟集中趋势指标刚好反的
opposite to central tendency

它在反映总体里面
It represents the difference among

各个变量值之间的差异程度
variable values in the population

因为这些差异程度将导致
The difference helps us

我们对算术平均
to understand arithmetic mean

就是对集中趋势的理解
which is central tendency

也就是可能能够说明
That is to say, it shows

集中趋势指标的代表性
whether the representation of central tendency

高和低的有关问题
is high or low

离中趋势指标
Dispersion tendency index

它的计算的作用非常大
plays an important role in calculation

一 说明平均指标它的代表性高低
First of all, it indicates the representative quality of the mean

第二 还能说明我们的事物发展
Secondly, it predicts whether the development

是否均衡或者是否稳定
is stable or balanced

当然这个均衡 稳定
Stability and balance in statistics

都是时间和空间上
is measured in

都要进行测算
time and space

比如说
For example

我们的新产品的研制
in the development of a new product

它就要进行测定
there must be determination

比如说某种新产品 农产品
when it comes to a new product, agricultural product

它的试值
of its false position

它的推广
its promotion

你就要测定它
It must be determined

在时间上 它会不会变异
in time to see if there is any variation

在空间上 会不会变异
and in space, too, to see if there is any variation

但农产品相对还好一些
It is easier to deal with

如果出现了问题
agricultural products

它的方差太大 出现了问题
When the variance is high

弥补起来
and causes social costs

社会成本也不多
the costs will be lower

但是 如果是药品
But, if it is a pharmaceutical product

它推广起来 那出现的就是人命
it is mortal, with the power to cure or kill

所以在这里面
So in promoting pharmaceutical products

我们大家都很重视这个指标
dispersion tendency is highly valued

这类指标 它的计算
The calculation of this indicator

它反映的情况就是
represents the variation

所有的总体单位的变量值
among all the variable values of population units

在总体里面它们的差异程度
in the population

到底有多大
How much is the difference

那差异程度到底有多大
How much is the variation

我们大家首先来想
Think about it

首先就会想到第一个
The first thing comes to our mind is

它最大的差异有多大
what the biggest variation is

那就第一个指标 全距 也叫极差
The first indicator, range, also called extreme variation

它由R表示 它是用最大变量值
is represented by letter R, equal to the maximum variable value

减掉最小变量值
minus the minimal variable value

用这个数字来表示
We use this figure to describe

比如说 我们工人生产的产品
for example, the products of our company

比如说这个产品
Take the product size

用尺寸表示它的大小
for example, its size

它最大的生产这个产品的尺寸
use its largest size

最大尺寸用最小尺寸相减
minus its minimal size

它就它的极差是多大
to get the range

那说明它的产品质量
When it describes the product quality

极差越大 产品质量越差
the bigger the range, the worse the quality is

这是我们讲的第一个
This is the first indicator we have learned

反映离散程度的指标
to represent dispersion tendency

下面的反映离散程度指标
The following indicators representing dispersion tendency

肯定就是从第一个指标开始
originate from this first one

你发现第一个指标它的优点很明确
The first indicator, we can see, has a clear advantage

就是直观 告诉你
It tells us in an obvious way

这个变量值的离散程度有多大
the dispersion degree of the variable values

但是它的缺点
But it also

也出来了
has disadvantages

它只是讲了极端
It can only reflect the extreme variation

中间所有变量值它怎么变
It fails to show what the other variable values

差别有多大反映不出来
are like, or how much their variations are

那这样的话我们就把x{\fs10}i{\r}
In this case, what can we do

它的所有变动都反映出来
to represent all the variations

那怎么反应 大家想
of x{\fs10}i{\r}. We assume

那就要给一个对比
if we want to describe a variation

要找一个标准
we need to establish a standard

那就是x{\fs10}i{\r}-a
That is to say, x{\fs10}i{\r}-a

都跟a对比
is compared with a

a就是标准
a is the standard

那这个标准怎么确定呢
How can we establish the standard

那我们就可以取最大值
We take the maximum value

也可以取最小值
or the minimal value

也可以取中间值
or the median

也可以取它本身
or the figure itself

我们通过不断地试验
We try all these out

不断地证明
proving and calculating

最后确定了
and at last, we find

我们每一个变量值x{\fs11}i{\r}
we can compare the variable values x{\fs11}i{\r}

与算术平均数相比
with arithmetic mean

比完了以后我们把它
After that, we calculate the

比出来的差额来进行平均
average of the differences

来说明它们
to describe

总体的变量值的离散程度
the dispersion degree of the variable values in the population

但这里就有一个问题
But there is a problem.

我们知道
As we know

算术平均数它有一个特点
arithmetic mean has a characteristic

（公式如上）它永远等于0
(see from the formula above), it is always equal to 0

也就是说 小的数字减平均数
which means, for figures less than the average

它是为负 大的数字减平均数
their difference is negative, and for figures greater than the average

它为正 它正负抵消等于0
it is positive, and the offset is 0

为了让它正负不抵消 怎么办呢
If we want to avoid the offset, what can we do

我们就是说它的离差
We want to talk about the deviations

我们是第一步
The first step we take

第一种方法去处理的时候
by this mean, to deal with this issue

就取绝对值
is to use its absolute value

正的保持正
The absolute value of positive number is the number

负的变为正
The absolute value of the negative number is the number without the negative sign

那就把它的离差加起来
Now let us add these deviations together

就离差和加起来
Divide the sum of these deviations

除以它的总体单位数
by the population size

这样算出来叫平均差
to calculate the average deviation

也就是平均离差
or the mean deviation

我们这个上面写的 就是
As we have written here

（公式如上）
(see the formula above)

当然这是简单的 加权的话
This is the simple mean deviation. If there is weight

加上f除以底下的∑f
add f to divide ∑f

下面我们试试用加权平均差的方法
Now let us try to calculate the weighted mean deviation

对二妞班上
by analyzing

这次统计学考试成绩进行分析
the Statistics test scores in Er Niu’s class

在第四讲中
In our fourth lecture

我们已经通过
we have already learned

加权算术平均数的方法
how to calculate weighted arithmetic mean

得到了二妞班
We know

这次统计学考试的平均成绩
the mean of Statistics test scores in Er Niu’s class

也就是72.6分
is 72.6.

那么有了这个平均成绩
With this average grade

下面我们可以进一步计算
we can further calculate

其加权平均差
the weighted mean deviation

前面我们已经知道
In previous lectures

加权平均差的计算公式
we learned the formula to calculate WMD

（公式如上）
(see the formula above)

由此我们可以知道
From this formula we know

要计算出二妞班上
to calculate the WMD

这次考试的加权平均差
of test scores in Er Niu’s class

我们就需要得到
we need

（公式如上）的数据
data (see the formula above)

以及∑f的数据
and ∑f

大家下面看到的这张频数分布表上
Let us look at this frequency distribution table

已经将（公式如上）以及
it shows the values of (see the formula above)

（公式如上）的值
and (see the formula above)

计算出来了
the output here

这里要注意的是（公式如上）中的x
But we should keep in mind the x (see the formula above)

它实际上表示的是每一组的组中值
is actually the class mid-point of individual sets

而x拔是我们在第四讲中
and x bar is what we have, in the fourth lecture

已经计算出来的 72.6
calculated, 72.6.

如 50到60这一组
Take the group from 50 to 60 as an example

所计算出来的X-X拔
the X, X bar calculated

应该等于55-72.6=�17.6
should be 55-72.6=�17.6

通过这张频数分布表我们已经
From the frequency distribution table

可以看到
we can see

（公式如上）的加和是等于389.6
(see the formula above), the sum is equal to 389.6

而∑f等于50
and ∑f is equal to 50

将这两个数据分别代入得到
Substitute the two data, and we can get

A.D=7.792
A.D=7.792

也就是我们要计算得到的
which is the weighted mean deviation

加权平均差
we have calculated

这里需要特别说明的是（公式如上）
We need to clarify that (see the formula above)

它表示的是单个离差
this figure represents a single deviation

而（公式如上）它表示的是组离差
that figure (see the formula above) represents a class deviation

而（公式如上）
and (see the formula above)

则表示的是总离差
that figure is total deviation.

还有一种处理方法
Another way to deal with this

让它的负数变为正数
is to remove the negative sign of the negative numbers

正数保持正数
while maintaining positive numbers.

就是什么呢
In other words

它的离差 x-x拔的离差呢
the deviation between x and x bar

括号进行平方
square the bracket

平方了以后呢
After this operation

它就都保持正的
they all become positive.

把它平方和加起来
Add the quadratic sum together

除以它的总体单位数
and divide the sum by the population size

那得出的这个指标 得出的结论
the output, the indicator

得出的结果
the result from this computation

就是我们平时讲的方差
is what we call the variance

这个指标用的特别多
It is a widely used indicator

它就（公式如上）
This figure (see the formula above)

这个指标开根号以后就是标准差
the square root of this indicator is standard deviation

这个指标 在我们后面会使用非常多
We will come across this indicator later in the following lectures

这也是统计里面用得比较多的
It is also a widely used indicator

一个指标
in statistics

下面我们试试用加权方差的方法
Now let us try to calculate the weighted variance

对二妞班上这次统计学考试成绩
by analyzing the Statistics test scores

进行分析
in Er Niu’s class

首先根据方差的基本公式
First, according to the formula of variance

（公式如上）
(see the formula above)

（公式如上）
(see the formula above)

可以知道
we can see

要求其加权方差
in order to calculate weighted variance

就必须计算出
we have to first calculate

（公式如上）
(see the formula above)

∑f的值
the value of ∑f

下面我已经将所需计算的数值
Here all the values we need to calculate

显示在这张频数分布表上
are displayed on this frequency distribution table

我们只需要将它们
All we need is to substitute them

代入方差公式即可
into the formula of variance

但是 在这里要注意的是
But we must keep in mind

其中的x实际上是每一组的组中值
x here is actually the class mid-point of individual groups

它并不是每一组数据的真实值
rather than the true value of data in individual groups

也就是说
which means

在组距式数列中
in class interval distribution series

其每一组的组中值表示的是
the class mid-point of each group

这一组数据的平均值
is the mean of data in this group

而不是这一组数据的真实值
rather than the true value of the data

由此我们进而可以知道
From this, we can know that

通过加权方差所得到的σ平方
σ square calculated by weighted variance

它并不是总体的真实方差
is not the true variance of the population

而真实方差我们就需要通过计算
To find its true variance, we need to calculate

它的简单方差
the simple variance

（公式如上）
(see the formula above)

（公式如上）
(see the formula above)

我们在这里就不给大家进行计算了
I will skip the detailed computation here

有兴趣的同学
If you are interested

可以自己计算一下
you are welcome to do the computation by yourself

它的简单方差
The simple variance

那么肯定是与我们
is sure different from

刚刚所计算得到的加权方差的结果
the weighted variance, as the previous calculation

是不一样的
shows

那方差在计算的时候
So, in calculating variance

在实际过程中计算的时候
in the specific computation procedure

要注意了
we should pay attention

实际过程中
in the computation procedure

比如我们举的例子里面
like in the case we just mentioned

用的是组距数列计算的时候
When it is class interval series

这个方差它的组距数列用的是
its variance is the class mid-point

组中值
of the class interval series

也就是我们刚上面讲的
This is what we have talked about

上限加下限除以2的组中值
The class mid-point by dividing the sum of upper and lower limit by 2

来代表这一组的组平均数
can represent variance calculated from

计算出来的方差
the subgroup average of this class

它不是这个总体的真实方差
It is not the true variance of this population

总体的真实方差 其实
The true variance of this population, actually

就用刚才原始的
is the original

（∑x-x拔）平方除以n
（∑x-x bar）square divided by n

这才是它真实方差
This is the true variance

当然这是未分组资料
Of course only the ungrouped data

才有真实的方差
has true variance

那分了组的时候
After the data is grouped

那怎么办
what can we do

那我们就会下面有一个
Then we will have the

方差的一个数学性质
mathematical quality of the variance

大家注意一下
Please pay attention to this

那下面我们来讲方差的数学性质
Now let us talk about the mathematical quality of the variance

方差的数学性质就是
The mathematical quality of the variance

方差加法定理
is variance addition theorem

它就把未分组的资料
It separates ungrouped data

分成几个组
into groups

比如我们PPT里面讲的例子里面
as in the examples in our PPT

它就分成三个组
it has been divided into 3 groups

这三个组里面
In the three groups

我们马上就能看出来
we can see immediately

首先它没有分之前
first, there is a total deviation

它们自己有一个总的离差
before the data is grouped

也就是总方差
Namely, the total variance

已经算出来了
is known

第二 首先三个组之间就有差别
Second, the three groups are different

那这样的话
In that case

我们可以计算三个组之间的差别
we can calculate the difference among them

算出来的方差 叫组间方差
The variance from that calculation, is called variance between laboratories

以什么来代表来算
How to calculate this

以各个组的组平均数作为代表
Use the subgroup average as representatives

进行加权计算
to conduct weighted computation

大家看一下这个公式
Let us look at this formula

第二 组里面它的这些单位
Second, the units within the group

你看一下 它们之间也有差别
if you look closer, are different as well

那这个差别我们也可以算出方差
We can also use this difference to calculate variance

那这个方差是组内方差
That variance is variance within laboratory

那有三个组 就是三个组内方差
Since there are three groups, there are three variances within laboratory

那我们现在算出三个组内方差
Now let us calculate the three variances within laboratory

大家用这个公式算出来以后
After calculating them by this formula

再算一个总的组内平均方差
we can further calculate a total average variance within a group

就用这个公式
by this formula

用各组的组内方差进行加权平均
Use variances within laboratory for weighted mean

算出组内平均方差
and calculate the average variance within a group

大家发现通过这个计算
We can see from this calculation

总方差会等于组内平均方差
that the total variance is equal to the sum of

加上组间方差
variance within laboratory and variance between laboratories

这个就是我们讲的
This is what we call

方差的加法定理
variance addition theorem

刚才我们讲这个例子
In the case we just mentioned

用组距数列来计算方差 有一点像
to calculate variance by class interval series is similar

我们用组平均数
to calculating variance between laboratories

计算的组间方差
by subgroup average

但是这两者
But the two are

本质是不相同的
different in nature

但是形式上相同
despite their similarity in form

这是方差的加法定理
This is variance addition theorem

方差的加法定理的用途特别多
The variance addition theorem can be applied widely

我们可以用来进行一系列的
Let us perform a series of

分析和检验
analysis and tests

这个等到大家去看
You can find many

方差分析里面就有
easy-to-understand examples

举个比较简单的例子
in variance analysis

比如说化妆品
Take cosmetics for example

它的广告效果到底用
which advertising media

哪种广告媒体做 它的效果好
is best for its promotion

你就可以用这种方法来进行测定
You can test this by using this method

这是方差的有关问题
This is how variance is used in real life problems

方差开平方以后就是标准差
The square root of variance is standard variance

标准差它的单位
The unit of standard variance

跟平均差的单位
the unit of the mean deviation

和全距的单位
the unit of range

与总体单位的变量值的单位
and the unit of the variable values in the population

是一样的
are the same

哦 原来方差和标准差这么有用呀
Well, variance and standard deviation are truly useful tools

以后我就用它们来帮助老师
In the future, I will use them to help me

选择最稳定的学生
to choose the most stable student

来担任辅导差生的任务啦
to offer academic help to those in need

方差和标准差可不是万能的
But variance and standard deviation are not panacea

如果遇到均值不同
In a data set whose mean value varies

而标准差也不同的数据组
or whose stand deviation is different

它们是很难比较的
it is difficult to make the comparison

必须对它们进行一定的转换
unless these data are converted