Study on the Normal and Skewed Distribution of Isometric Grouping

Because of thinking only the number of numbers but not fitting function, it would be adequate to take further a field when calculating group numbers with empirical formula. We have proved the three theorems based on studying the normal distribution, and then reach the conclusion that there is a better method to do the same work. The method is simpler and more practical than empirical method and also works well with any skewed distribution.


Introduction
In various books on statistics, when discussing the isometric group, the empirical formula is treated as only a reference or put aside simply.It is attributed that the class interval is only relevant to the number of numbers not the shape of fit function in the empirical formula.
This paper analyzes the isometric group while conforming to the normal distribution of series, and derives simple and practical method to find class interval.Furthermore, the same formula also works well with any skewed distribution.

Histogram and the Upper Bound of Class Interval
2.1.Find Out Theorem 1 , and find out the minimum value 1 x  and maximum value N x  .And select proper that slightly less than c R is range, normally , and then divide into five or more groups.
The series is bell-shaped distributed in a symmetrical way.Set the axis of symmetry x   , its fit density function is , according to the sample numbers, divide it into groups, and the scope of each group is This diagram is called frequency distribution of classes of sample numbers.Obviously,  l, and then the histogram is made out (Figure 1).And , (Table 1)

Major Theorem
Theorem 1.If a group of numbers shows a normal distribution, its fit density is When we discuss the isometric group,  is the upper bound of a (a is the class interval).

1) Definitions Definition 1
We divide Their points are , Proof.By Lemma 2, there is then the intervals in i are similar intervals.By Lemma 2, the similar intervals are at least 2.

Work out the Way to Find Class Interval by Theorem 3
1) Arrange the data in ascending order, and calculate the average  and variance  of the numbers.
2) Find the closed range , and calculate , to the two points).
3) Find out the minimum similar intervals 1 , , n s s  ; and their numbers are 1 , , n s s  respectively.Let it be the minimum value, thus let i s s  .

4) Fix on the class interval : a
5) Grouping a group of numbers, link the groups up carefully.If it is done well, it can reflect the overall trend.The chain of numbers is , where 1 2 are included in one group.How to insert this class into the chain of data?We set a rule that this group should be included in , where If it is a skewed distribution, the above method is also available, but need to do twice, referring to the flowing example.

Example 1
By a sample survey of living conditions of urban households, we get the following numbers of per capita monthly household income (already arranged).
The minimum number is 610, and the maximum number is 2380.It is a marmal distribution progression.
It is suggested that the similar interval is closer to It is easy to see that there are six numbers between 1800 and 2000, which are 1840, 1860, 1870, 1880, 1940, and 1970, and be the class interval, then 6 × 33.3 = 200 is the valuation of .We could get the following distribution series after further arrangement.

Example 2
By a sample survey of living conditions of urban households, we get the following numbers of per capita monthly household income (already arranged).1) Table 3: The minimum number is 810, and the maximum number is 2380.And the average is 810 840 2300 2380 1497.2 54 where , , the closed range is (Table 4).
2) Because it is not completely symmetrical, we divide into two steps to finish it.From 800 to 1497.2, there are 27 numbers, and the average distance between them is    1050, 1070, 1080, 1100, 1120, 1120, 1160, 1200, 1240, 1280, 1300, and  We could further ascertain that there are two points on the right side of 1148.6 at most, which are 1160 and 1200.It is easy to find that there are 8 numbers in interval (1020, 1220) (see the fifth points in the third part), which are 1050, 1070, 1080, 1100, 1120, 1120, 1160, and 1200.Mathematics diagram Frequency distribution of the per capita monthly income available for living expenses of urban households in a certain city. (1020,1220) is a similar interval.And S = 8, [2] C. S. Wu, "Probability and Statistics," Higher Education Press, Beijing, 2004, pp. 128-144. 206.4 Vol. 40, No. 20, 2011, pp. 238-244.

Figure 1 .
Figure 1.The formation process of histogram.
There are at least two similar intervals in a normally distributed series [3].

Table 1 . Normal distribution of data (Example 1).
is not completely symmetrical, we just consider the data from 1500 to 2400.From 600 to 2400 there are 27 numbers, and the average distance between