3.2.1 Frequency distribution: the initial appearance of the overall distribution characteristics 频数分布：初显总体分布特征在线视频

统计数列
Statistical Series

按品质标志分组形成的数列
There are series based on qualitative characters

为品质数列
called quality series

按数量标志分组
and on numerical indications

生成的数列为变量数列
called variable series

当然我们也可以统一为变量数列
Of course, we can regard both kinds as variable series

因为品质数列它最终结果
because the result of quality series

也可以量化为变量
can also be transformed into a variable

那变量数列它的构成形式
A variable series consists of

有两个基本要素
two fundamental elements

一 变量 二 次数
variable and number of times

或者 就变量或频数
Or called, variable and frequency

如果用相对数表示
When expressed as a relative number

那就变量和比率 或者变量和频率
it is variable and ratio, or variable and frequency

那在这里变量的表现形式不同的话
And if the variable takes different forms of expression

又有两类变量
there will be two subdivisions

或者我们统计分组
Or，in statistical grouping

一类是单项式数列
One of them is single-valued distribution series

单项式数列就是用一个数值
which uses one numerical value

来代表一组
to represent a set

还有一种叫组距式数列
and the other is class interval distribution series

它是用一个区间
which uses an interval

一个组来代表
to represent a set

比如说 我们年龄可以这样分
For instance, when we list ages

16岁 17岁 18岁 19岁
we can use individual numerical value of 16, 17, 18 and 19

分别代表一个组
to represent individual groups

这叫单项数列
that is single-valued distribution series

你也可以这样分
Or we can use the age box

20岁以下 21到30岁
under 20, 21 to 30

31到40岁
and 31 to 40

那这个20岁以下
These intervals, under 20

21到30 31到40
21 to 30, 31 to 40

这个是组距式数列
form class interval distribution series

这两组数列的优缺点
Both series have pros and cons

要看各自的分析方法
and should be used according to analysis methods

如果编表的来看
In preparing reports

如果那个总体的差异比较大
when there are considerable variations

单位的差异比较大
in population and units

你如果用
and if you use

单项式数列
single-valued distribution series

那么单项式数列编出的表
the table following single-valued distribution series

将变成哈达表
And it will become a Hadar table

太难看了
But that will be too long to read

用组距数列就编的比较紧凑
The table following class interval distribution series is more precious

但是组距数列编出来的变量数列
But the variable series from class interval distribution series

它计算出来的结果
is not as accurate as

没有单项式数列计算的结果准确
the result from single-valued distribution series

比如说用组距式数列
The average number

计算后面的平均数
calculated by class interval distribution series

那么平均数将可能是一个近似值
is more of approximate value

这个就是我们讲的
These are the two forms of

变量数列的两种形式
variable series that we have discussed

那变量数列里面最关键的数列
The most important variable series

用的比较多的数列
And the most used one

就是组距式数列
is class interval distribution series

组距式数列的构成要素里面
And there are a few things we need to pay attention to

有几个要注意
in the components of class interval distribution series

一个是组限 组距 组数
They are class limit, class interval, number of sets

组限与组距之间的关系
the relation between class limit and class interval

组距等于上限减下限
The class interval is upper limit minus lower limit

上限是这一组里面的最大值
Upper limit is the maximum value in this set

下限是这一组里面的最小值
and lower limit, the minimal value

当然这个变量的最大值
That the maximum value of the variable

减这个变量的最小值
minus the minimal value of the variable

属于这个变量的全距
forms the range of the variable

在这里我们每一组 它有两个数值
There are two values in each set

一个是上限 一个是下限
the upper limit and the lower limit

那哪一个代表这一组呢
So, which one is used to represent the set

我们就用组里面的组中值
We use the class mid-point

代表这一组的平均数
to represent the average number of the set

那组中值怎么算呢
How do we calculate class mid-point

就是上限加下限除以2
Divide the sum of the upper limit and the lower limit by two

大家看例子
Here is an example

成绩单上的考试成绩
The grades in our school report

是属于连续型变量
are continuous variables

而且变量值较多
with many variable values

因此要采用组距分组
So, it is better to use interval grouping

既然是组距分组
In interval grouping

那么就需要确定全距 组数 和组距
we need to set range, number of sets and class interval

下面我们就对这份成绩单上的
Now let us conduct statistical grouping

总评分数据进行统计分组
among the grades in the school report

当然有一些组它属于开口向上组
Some groups are upward open-end class

比如说 我们90分以上
like the group above 90

它就开口向上
is upward open-end

还有60分以下
And the group, below 60

就开口向下
is downward open-end

它只有上限或者只有下限
An open-end class only has upper limit or lower limit

那怎么办呢
What can we do about this

就用上限
We subtract

减去1/2相邻组的组距
half the class interval of its neighbor set from its upper limit

那下限
And in upward open-end class

开口向上的就下限
we add the lower limit

加上1/2相邻组的组距
to half the class interval of its neighbor set

大家看例子
Let us look at this example

在这里我们还要特别注意的是
I would like to remind you

如果全部数据中的最大值和最小值
if the maximum value and the minimal value

与其它数据相差悬殊
differ greatly from the other data in the set

那么在组距式数列中
there might be blank group

为避免出现空白组
in class interval distribution series

或个别极端值被漏掉
or danger of missing extreme values

第一组和最后一组可以采取
It is better to set the first and last group

某个值以下或某个值以上
to be open-end class

这样的开口组
above or below a certain value

如上述例子
As in the case above

采用开口组的形式可以表示如下
the open-end class can be set like this

在这里我们可以看到
As we can see

60分以下和90分以上
the groups of below 60 and above 90

所采用的都是开口组的形式
are set in the form of open-end

既然是开口组
Since they are open-end classes

那么我们就应当采用前面所讲过的
we should use the formula mentioned earlier

计算开口组组中值的公式
to calculate its class mid-point

也即组中值等于下限
which is its lower limit

加1/2邻组组距
plus half the class interval of its neighbor set

或者是组中值等于上限减去
or its upper limit minus

1/2邻组组距
half the class interval of its neighbor set

在这里60以下的组中值
The class mid-point of group below 60

实际上就是用
is

60-(70-60)/2=55
60-(70-60)/2=55

90以上这一组所对应的组中值
and of group above 90

应当等于这一组的下限
is the sum of its lower limit

加1/2邻组组距
and half the class interval of its neighbor set

也就是90减80等于95
that is, 90+(90-80)/2=95

至于闭口组组中值的计算方法
As for the class mid-point of close-end group

它应当等于1/2上限加下限
it is half of the sum of its upper limit and lower limit

例如 60到70这一组
Take the group of 60 to 70 for instance

它的组中值应当等于60加70除以2
its class mid-point is (60+70)/2

等于65
that is, 65

实际上闭口组组中值的计算公式
In fact, the calculating formula of mid-point in close-end class

还可以做进一步变形
can be further conversed

也即组中值等于
Because the mid-point

1/2上限加下限
is half of the sum of its upper limit and lower limit

展开得到1/2上限
it is also half of the upper limit plus

加上1/2上限减去组距
half of upper limit and then minus the class interval

最后变形得到上限减去1/2组距
and finally derived into upper limit minus half of the class interval

现在我们再把闭口组组中值
Now let us compare the calculating formula

刚刚所拓展得到的这一计算公式
of mid-point in close-end class

和我们之前所学过的开口组
with the calculating formula

组中值的计算公式做一比较
of mid-point in open-end class

一个是上限减去1/2组距
One is upper limit minus half of the class interval

另外一个是上限
the other is upper limit

减去1/2邻组组距
minus half of the class interval of its neighbor set

由此可以看到
We can see

闭口组组距的确定原则
the class interval of close-end group

是等于本组的组距
is equal to its class interval

而开口组组距的确定原则
while the class interval of open-end group

则是等于相邻组的组距
is equal to the class interval of its neighbor set

无论是开口组还是闭口组
As for class interval series formed

所形成的组距式数列
in both open-end class and close-end class

其每个组中的变量
the variables of the series

都是连续性变量
are continuous variables

在对连续性变量编制
We should also be aware that

组距式数列的过程中
in writing class interval distribution series

我们还要注意的是
of continuous variables

它们的相邻组限是重叠的
there is an overlap between neighboring sets

也即前一组的上限和后一组的下限
That is, the former group’s upper limit and the later group’s lower limit

是相同的变量
can be the same

那么大家可能就会问
Some of you may wonder

如果某个同学的成绩正好为70分
if a student’s grade is 70

那他应该计入哪一组呢
which group is this grade

这里大家只需要记住一个原则
There is a principle we should keep in mind

就可以解决上面的问题
when dealing with this kind of question

也就是上组限不在内原则
that is, the principle to exclude upper limit

它的意思是每一组的
It means, for each group

上限变量值的总体单位
the maximum variable value of the unit

是不包括在本组内
is not included in this group

而应该放在下一组
but should be included in the next group

通过上述对二妞班上统计学考试
If we use this method to sort out the scores of Statistics Tests

成绩进行分组
in Er Niu’s class

我们可以看到
we can see

它所得到的组距式数列
the class intervals shown between

每一组的组距是相等的
class interval distribution series are equal

那么我们在统计学里
In Statistics

称之为等距数列
we call it series of equal interval

而当各组组距不相等
When the class interval is not equal

所形成的组距数列
among the class interval distribution series

我们称之为异距数列
we call it series of unequal interval

在异距数列中
In series of unequal interval

反映次数在各组分布
the indicator that reflect the density of

密集程度的指标
frequency in each group

是次数密度
is called frequency density

所谓次数密度
Frequency density

它表示的是本组的次数
shows the proportion of frequency

与本组组距之比
to the class interval of this group

大家可以看下面的表
Let us look the table below

直观上我们可以看到
It is intuitively obvious that

100至120
the group from 100 to 120

这个组的单位数最多
has the most units

总体单位在这个组内最为集中
The density of the population unit seems the biggest in this group

但实际情况是否真的如此呢
Is it true

我们注意到这个组距数列
We have noticed the class interval series here

并不是一个等距数列
is not a series of equal interval

每一组的组距不尽相等
The class intervals of each unit vary

一般来说组距越长
Generally, the longer the class interval is

容纳的总体单位相应的也会更多
the more units it includes

就好比一间面积大的房子
in the same way that

要比一间面积小的房子
a classroom of larger area

能够容纳多一些学生
can hold more students than

是一样的道理
a classroom of smaller area

但我们并不能就此认为
But it is wrong for us to assume

大房子的学生更为密集
the density of students in the large classroom is higher

我们要看的应该是单位面积上
We should calculate the number of students

容纳的学生的多少
per unit area

同样的道理
Similarly

现在我们看到的数据
the data we are seeing now

有的组距是20
with class interval of 20

有的组距是10
and of 10

此时要看总体在各组中的分布特点
should be calculated by the distribution of units in each class

这个时候就要用次数密度
We should use the indicator of

这一指标来帮助我们了
frequency density to calculate

比如第一组为30除以20
For example, the first group is 30/20

等于0.15
it is 0.15

依次把每一组的次数密度计算好
Calculate the frequency density of these groups one by one

再观察他们的结果
and observe the result

我们可以看到90至100
We can tell that the group of from 90 to 100

这一组的频数密度为0.7最大
has the largest frequency density, 0.7

这说明这一组内的单位最为密集
This means the units in this group is the densest

而不是100至120这个组
rather than the group of from 100 to 120

通常情况下等距分组
Normally equal class interval

适用于变量值变动比较均匀的现象
is used when the variable value change is even

而异距分组
And unequal class interval

适用于变量值变动不均匀
is used when variable change is uneven

出现急剧增长或下降
with rapid growth or decline

波动较大的现象
in fluctuation