3.8.2 Variance and Standard Deviation (2): Commonly Used Indicators of Deviation Trend 方差与标准差（二）：离中趋势之常用指标在线视频

在这里面平均差和标准差
Mean deviation and standard deviation

它虽然是解决了第一个指标
helps to resolve the deficiency

全距的缺陷
of range, the first indicator

不能反映总体所有变量值的变动
by reflecting the variance of all the variable values

它现在解决了
in the population

能够反映总体所有变量值的变动
It can show the changes of variable values in the population

但是它自己的缺陷也出来了
but it also has its own problem

它自己的缺陷在哪里
What is its flaw

因为我们本来是用这个变量
Because we meant to use this variable

来反映变量的离散程度
to reflect the dispersion degree of variables

如果这个变量值
If the variable value

这个指标数值大
the indicator, is high

比如说标准差的数值大
say, the standard deviation is high

说明它的离散程度就大
which means the dispersion degree is large

平均数的代表性就低
the representativeness of this mean is low

也就是事物发展就不均衡不稳定
which means the object under observation is unstable and unbalanced

是不是我们可以反过来推,
Consequently, we can see

就是我们讲的离散程度大
if the dispersion degree is large

数值一定大
the value must be large

能不能反过来
But is it true vice versa

我们说不能够反过来
No, it is not true

为什么
Why

因为平均差和标准差
the mean deviation and standard deviation

它的大小受两个因素的影响
are affected by two factors

一个是说刚才讲的离散程度大
One is what we just mentioned, the dispersion degree

它的数值就大
When dispersion degree is large, its value is large

第二个它还受到一个影响
The other factor that influences

标志数本身大小
the value of standard deviation

标志数本身x{\fs10}i{\r}
is designation number x{\fs10}i{\r}

如果它本身大
If designation number is large

那么平均差也大
the mean deviation is large

算出来的标准差也大
and the calculated standard deviation is large

这个大家可以去看相应的例子
There are corresponding examples for you to see

那为了消除标志值本身大小
In order to eliminate the influence of designation number

对平均差和标准差的影响
on mean deviation and standard deviation

我们就要用第四个指标
we must employ a fourth indicator

第四个指标就是
The fourth indicator is

标准差变异系数
coefficient of variation of standard deviation

或者平均差变异系数
or coefficient of variation of mean deviation

我们用(字符如上)或者（字符如上）表示
We use (the symbol above) or (the symbol above) to represent

叫（字符如上）它就是用
(The symbol above) is expressed as

（公式如上）
(the formula above)

（字符如上）就是用
And (The symbol above) is expressed as

（公式如上）
(the formula above)

这个目的在于什么
What is purpose of doing this

它第一为了消除
Firstly, it is to eliminate the

标志值本身大小的影响
influence of designation number

当然它还可以用于对比
Of course, it can also be used to compare

两个不同总体的
the representativeness

平均数的代表性
of means from two populations

比如中国的工人平均收入
such as the representativeness

与日本工人的平均收入
between the average income of Chinese workers

这两个平均数代表性哪个高
and that of the Japanese workers

这时候你因为
In this case

它除掉离散程度大小以外
beside the dispersion degree

还受到单位不同
the two means are also influenced by

我们用人民币元 他们用日元
monetary units of CNY and JPY

这时候用标准差变异系数的话
The use of coefficient of variation of standard deviation

它就会消除单位
helps to eliminate the influence

也能消除变量值
of units, variable values

x本身大小的影响
and x itself

再就是第四个标志变异度指标
The fourth indicator to mark the variation degree

也就是离散程度指标
is measures of dispersion

离散程度指标使用的时候
When we use measures of dispersion

大家也要注意它的有关问题
we also need to pay attention to several issues

比如说我们刚才讲的
including, what we mentioned earlier

方差的使用
the variance

我们用组距数列用的时候
When we deal with class interval series

一定要注意
we must keep in mind

你这里不是组平均数
this is not the class mean

用的是各组的代表值
but the representative values of individual groups

离散系数就是从相对数角度
Coefficient of variation uses relative number

来反映数据的离散程度
to reflect the dispersion among data.

通过这个定义我们可以看到
From its definition we can see

离散系数的特点是
the characteristics of coefficient of variation

第一它可以用于比较
It is used to, first, compare

两个总体平均水平不同
the average level of two populations

第二它可以用于
and to, again, compare

两个总体性质不同
the nature of two populations

或计量单位不同
or the measuring units

通过我们前面学过的
According to what we learned before

反映数据离散程度的绝对指标
the absolute numbers used to reflect the dispersion among data

我们可以进一步得到
We can further calculate their relative numbers

它们的相对指标
Their relative numbers

也就是标准差系数
namely the coefficient of standard deviation

平均差系数
and the coefficient of mean deviation

当然在实际过程中
Of course, in practice

我们使用最多的
the most widely used

还是标准差系数
is still the coefficient of standard deviation

那所谓标准差系数
The coefficient of standard deviation

它是表示的是一组数据的标准差
refers to the ratio of standard deviation

与其对应的平均数之比
to its corresponding average of a set of data

用数学表达式来表示的是
Its mathematical expression is

（公式如上）
(see the formula above)

（公式如上）
(see the formula above)

如果是我们要计算平均差系数
If we need to calculate the coefficient of mean deviation

那么它就等于
it is equal to

（公式如上）
(the formula above)

无论是平均差系数
Both the coefficient of mean deviation

还是标准差系数
and the coefficient of standard deviation

它们表示的含义都是相同的
have the same implication

也就是 这些离散系数越大
which is, the larger the coefficient of variation is

那么就说明数据越分散
the more dispersed the data is

x拔 也就是平均数的代表性就越小
and the lower is the representativeness of the mean, x bar

如果离散系数越小
If the coefficient of variation is low

那么就说明数据越集中
it shows the data is centralized

x拔的代表性就越大
And the representativeness of x bar is high

大家可以看到
We can see here

我们这里显示的是
the two sets of data display

幼儿组和成人组的身高数据
the height of young children and adults

我们希望比较的是
We want to compare

这两组数据哪一组数据更集中
which set of data is more centralized

首先我们要分别计算
First we need to calculate the respective

成人组和幼儿组的算术平均数
arithmetic mean, and their standard deviation

和他们的标准差
of both sets

具体的计算过程 我们就不详述了
I will skip the detailed computation here

大家可以看到
We can see

我们直接给出的是计算的结果
the output from the calculation here

x拔等于168 σ等于2.83
x bar is 1168 σ equal to 2.83

那么幼儿组的x拔等于73
The x bar of the group of young children is 73

σ等于1.41
σ equal to 1.41

根据这个结果我们可以看到
The result from the calculation tells us

成人组和幼儿组的算术平均数
the arithmetic means of the two sets

是不相同的
are different

因此我们进一步
So we need to further

就要计算它们的标准差系数
calculate their coefficient of standard deviation

根据我们前面学过的
Using the formula of coefficient of standard deviation

标准差系数的公式
we learned in previous lectures

我们可以计算得到成人组的
we can figure out the coefficient of standard deviation

标准差系数等于
of the group of adults is

2.83/168 =1.68%
2.83/168 =1.68%

幼儿组的标准差系数
and the coefficient of standard deviation of young children’s group

等于1.41/73=1.93%
is 1.41/73=1.93%

根据这两个计算结果
Between the two outputs,

我们可以看到
as we can see,

1.68%小于1.93%
1.68% is less than 1.93%

也就说明
which means

成人组的这组数据更整齐
the data of the set of adults is more even

或者说更集中
or more centralized

方差在现实生活里面
Variance is widely used

用的特别多
in real life

比如高考的时候
Take college entrance examination for example

因为大家都知道我们现在高考
We know that the college

录取的时候都用的是
admission nowadays is based on

原始分录取
the raw scores

就是你四门分数
which is the sum of 4 subject scores

考试的四门分数加起来
Add the four subject scores

就作为你的高考分数
for college admission reference

这个是否公平
Is it fair

因为这里面有一个假定
There is a presumption

假定你每门课考试的难易程度
that suppose the difficulty of each subject test

相同的情况下
is the same

这是合理的
It is fair then

如果这四门考的时候
But if among tests of the four subjects

里面有一门难
there is one particularly difficult

或者有一门容易
or one particularly easy

这四门课是不能加起来的
then it is unfair to add the four test scores together

加起来就对偏科的人
It is unfair to people with partial sections

有些是不利 有些是有利
Sometimes, it is in their favor and sometimes is to their disadvantage

这时候最好是用标准分
It is better to use standard scores

标准分就在这里使用了标准差
The standard score functions as standard deviation here

就用你的分数去减掉平均分数
by minus your score with the average score

除以标准差
and then divide it by standard deviation

那你就算的出来
you will know

你跟平均数相比你怎么样
your position compared to the mean

所以标准差它的用途非常大
So standard deviation is really useful

大家可以看看相应的
After class, you can read about

比如说 正态分布底下的
say, in normal distribution

标准差的有关使用情况
the use of standard deviation

比如管理学里面用的三个σ原理
the three σ theory in Management

现在发展到六个σ原理的质量控制
which has evolved into 6 σ theories quality control

这些都是用标准差来进行计算的
These are all measured by standard deviation

好 标准差就讲到这里
Ok, this is what we need to learn about standard deviation today