Real Life Applications of the Skewed

Transcription

1 Chapter 8 Real Life Applications of the Skewed Distributions 8.1 Introduction In the last few chapters, i.e., from the chapter 2 to chapter 7, we have described various distributions and their properties. In this chapter we try to fit some real life data to the above mentioned distributions. 8.2 Fitting of the Skewed Distributions on the basis of Real Life Data For the fitting of skewed distributions, here we have consider the demographic structure of Scheduled Caste (SC) and Scheduled Tribe(ST) population among the various types of inhabitants in urban and rural area in Assam. Data relating to the study areas been 67

2 68 8. Real Life Applications of the Skewed Distributions taken from the journal Census of India 2001 Table A5, A6, A8, A9, and A10, published by the Directorate of Census Operations, Assam, Govt, of India. According to the census report, the various statistics relating to population census in Assam are given as Total number of districts: Total number of sub-districts: Total number of villages: Total number of inhabited villages: Total number of towns: Total number of statutory towns: Total population in Assam: Total Male population in Assam: Total female population in Assam: Overall literacy rate in Assam: 23 ( at present 28) ,312 25, ,655,528 13,777,037 12,878, % (Male-71.3%, Female-54.6) Table 8.1: Census 2001 Report of Assam Here, an attempt is made to study the distribution of SC and ST population which are scattered in the various districts, both in rural and urban areas in Assam. In the study we have consider only the urban area - mainly under municipality- corporation, Municipality- board, Town committee, township area under various corporation namely under oil refinery, paper mills etc. To study the such distribution, here we have consider a random sample of urban areas which covers all the districts by stratified random sampling using proportional allocation. Altogether a random sample of size 170(MC-1, MB-29, TC- 134 CT-6), of urban places is considered to study the distribution of population structure

3 8.2. Fitting of the Skewed Distributions on the basis of Real Life Data 69 of the said castes in Assam. Again, the statistics related to scheduled tribe and scheduled caste populations in Assam are as follows: Percentage of SC Population Percentage of ST population Literacy rate of SC population Literacy rate of ST population = 6.85 = = 66.8(Male=75.7, Female=51.1) (Male=72.3, Female=52.4) Table 8.2: Percent distribution of SC and ST Population In the following we set the tabular form of the distribution of SC population and ST populations in urban area in Assam and the distribution of SC population and ST populations in rural area in Assam. Finally, we have fitted these data with the above mentioned distributions Study of SC populations in urban area in Assam From table 8.7, we have m[ Mean of SC population/unit of urban area = m'2 = The estimated value of A for various distributions are as follows: 1. For skew Logistic distribution, estimated value of A = The estimated value of A = for the skew Rayleigh distribution. 3. In case of skew Weibull distribution for c = 2, estimated value of A =

4 70 8. Real Life Applications of the Skewed Distributions % of SC population in the area No. of urban places No. of area under MC, MB, TC, CT Upto 2 17 MB-4 TC-12 CT MC-1 MB-3 TC MB-9 TC-42 CT MB-8 TC-29 CT MB-4 TC-15 CT MB-1 TC-7 CT MB-0 TC-1 CT-0 40 & above 0 MB-0 TC-0 CT-0 Total 170 Table 8.3: Distribution of SC populations in urban area in Assam On the basis of estimated values of A for various distributions, we have fitted the theoretical frequencies of the respective distributions, which are mentioned in the following tables. The theoretical distributions are mentioned as follows: 1. Dist-I: Skew logistic distribution 2. Dist-II: Skew Rayleigh distribution 3. Dist-III: Skew Weibull distribution It is clear form the table (8.7) that skew Rayleigh distribution is fitted better that the standard Rayleigh distribution. In the figure (8.1) shows the pattern of the fitting of the data mentioned in the table (8.7). So in this regard, it may conclude that

5 8.2. Fitting of the Skewed Distributions on the basis of Real Life Data 71 % of ST population in the area No. of urban places No. of area under MC, MB, TC, CT Upto 2 24 MB-7 TC MB-16 TC-45 CT MC-1 MB-2 TC MB-1 TC-16 CT MB-1 TC MB-1 TC MB-1 TC-3 60& above 01 TC-1 Total 170 Table 8.4: distribution of ST populations in urban area in Assam the distribution of the ST population in urban area of Assam follows Rayleigh skew distribution Study of ST populations in urban area in Assam For table 8.9, we have m!x = Mean Proportion of ST population/ Unit of urban area = m'2 = The estimated value of A for various distributions are as follows: 1. For skew Logistic distribution, estimated value of A = For Skew Rayleigh distribution, estimated value of A = 0.02.

6 72 8. Real Life Applications of the Skewed Distributions % of SC population in the area No. of urban planes Upto Above 20 0 Total 170 Table 8.5: Distribution of SC populations in rural area in Assam 3. In case of skew Weibull distribution for c = 2, estimated value of A = On the basis of estimated values of A for various distributions, we have fitted the theoretical frequencies of the respective distributions, which are mentioned in the following tables. The theoretical distributions are mentioned as follows: 1. Dist-I: Skew logistic distribution 2. Dist-II: Skew Rayleigh distribution 3. Dist-III: Skew Weibull distribution It is seen from the table (8.9) that Rayleigh distribution is fitted better than the standard Rayleigh distribution. Also the figure 8.2 represents the pattern of the distributions of the data mentioned in the table (8.9). So in this regard, it may conclude that the distribution of the ST population in urban area of Assam follows Rayleigh skew distribution.

7 8.2. Fitting of the Skewed Distributions on the basis of Real Life Data 73 % of ST population in the area No. of rural places Upto Above Total 170 Table 8.6: Distribution of ST populations in rural area in Assam Study of the distribution of SC populations in rural area in Assam For table 8.11, we have m'j = Mean Proportion of ST population/ Unit of rural area = m'2 = The estimated value of A for various distributions are as follows: 1. For skew Logistic distribution, estimated value of A = For Rayleigh skewed distribution, estimated value of A = In case of skew-weibull distribution for c = 2, estimated value of A = On the basis of estimated values of A for various distributions, we have fitted the theoretical frequencies of the respective distributions, which are mentioned in the following

8 74 8. Real Life Applications of the Skewed Distributions tables. The theoretical distributions are mentioned as follows: 1. Dist-I: Skew-logistic distribution 2. Dist-II: Rayleigh skewed distribution 3. Dist-III: Skew-weibull distribution It is clear from the table (8.11) that Rayleigh distribution is fitted better than the standard Rayleigh distribution. Also the figure 8.3 represents the pattern of the distributions of the data mentioned in the table (8.11). So, in this regards it may conclude that the distribution of the SC population in rural area of Assam follows skew-logistic distribution Study of the distribution of ST populations in rural area in Assam For table 8.13, we have m'j = Mean Proportion of ST population/ Unit of rural area = m'2 = The estimated value of A for various distributions are as follows: 1. For skew Logistic distribution, estimated value of A = For Rayleigh skewed distribution, estimated value of A = In case of skew-weibull distribution for c = 2, estimated value of A = 0.64.

9 8.2. Fitting of the Skewed Distributions on the basis of Real Life Data 75 On the basis of estimated values of A for various distributions, we have fitted the theoretical frequencies of the respective distributions, which are mentioned in the following tables. The theoretical distributions are mentioned as follows: 1. Dist-I: Skew-logistic distribution 2. Dist-II: Rayleigh skewed distribution 3. Dist-III: Skew-weibull distribution The figure (8.4) shows the pattern of the fitting the data mentioned in the table (8.13), where it is observed that the Skew-logistic distribution is found to be better fitted distribution than the other two distributions. So, in this regards it may conclude that the distribution of the ST population in rural area of Assam follows skew-logistic distribution.

10 76 8. Real Life Applications of the Skewed Distributions % of SC population in the area No. of urban places (observed) Theoretical Frequencies Rayleigh Skew Rayleigh Upto & above Total Parameter cr A = Degrees of freedom 5 5 x Table 8.7: Distribution of SC populations (Theoretical) in urban area in Assam Distribution d.f. p-value Conclusion Dist-I 5 < highly significant Dist-II significant at 5 percent level Dist- III 5 < highly significant Table 8.8: p-values of the distributions obtained from the table(8.7) of the figure (8.1)

11 8.2. Fitting of the Skewed Distributions on the basis of Real Life Data 77 Presentation of fitted skewed distribution Observed )ist- I Dist-li *"'*.. Dist- 111 Figure 8.1: Fitting of distributions of the data in table (8.7)

12 78 8. Real Life Applications of the Skewed Distributions % of ST population in the area No. of urban places (observed) Theoretical Frequencies Rayleigh Skew Rayleigh Upto & above Total Parameter er = A = 0.02 degrees of freedom x Table 8.9: Distribution of ST people of urban areas in Assam Distribution d.f. p-value Conclusion Dist-I 5 < highly significant Dist-II highly significant Dist- III 5 < highly significant Table 8.10: p-values of the distributions obtained from the table(8.9) of the figure (8.2)

13 8.2, Fitting of the Skewed Distributions on the basis of Real Life Data 79 so Presentation of fitted skewed distribution for ST population(urban) Figure 8.2: Fitting of distributions of the data in table (8.4)

14 80 8. Real Life Applications of the Skewed Distributions % of SC population in the area No. of rural places (observed) Theoretical Frequencies Rayleigh Skew Rayleigh Upto Above Total Parameter a = A = 0.51 degrees of freedom 3 4 x Table 8.11: Distribution of SC people of rural areas in Assam Distribution d.f. p-value Conclusion Dist-I 4 < significant Dist-II highly significant Dist- III 4 < P < highly significant Table 8.12: p-values of the distributions obtained from the table(8.11) of the figure (8.3)

15 8.2. Fitting of the Skewed Distributions on the basis of Real Life Data 81 Presentation of fitted skewed distribution Figure 8.3: Fitting of distributions of the data in table (8.11)

16 82 8. Real Life Applications of the Skewed Distributions % of ST population in the area No. of rural places (observed) Theoretical Frequencies Dist-I Dist-II Dist-III Upto Total Parameter degrees of freedom x Table 8.13: Distribution of ST people of rural areas in Assam Distribution d.f. p-value Conclusion Dist-I 5 < highly significant Dist-II 3 < highly significant Dist- III 3 < highly significant Table 8.14: p-values of the distributions obtained from the table(8.13) of the figure (8.4)

17 8.2. Fitting of the Skewed Distributions on the basis of Real Life Data 83 Presentation of fitted skewed distribution» i for ST population(rural) **** Ob rvt d 'dmmhimmsk- 0fgt~j DM-II «* " * Dist-ii Figure 8.4: Fitting of distributions of the data in table (8.13)