Mathematics and Statistics 9(5): 697-710, 2021 http://www.hrpub.org DOI: 10.13189/ms.2021.090509 Choice of Strata Boundaries for Allocation Proportional to Stratum Cluster Totals in Stratified Cluster Sampling Bhuwaneshwar Kumar Gupt 1,* , F. Lalthlamuanpuii 1 , Md. Irphan Ahamed 2 1 Department of Statistics, North-Eastern Hill University, Shillong, 793022, India 2 Department of Mathematics, Umshyrpi College, Shillong, 793004, India Received June 11, 2021; Revised July 30, 2021; Accepted August 22, 2021 Cite This Paper in the following Citation Styles (a): [1] Bhuwaneshwar Kumar Gupt, F. Lalthlamuanpuii, Md. Irphan Ahamed , "Choice of Strata Boundaries for Allocation Proportional to Stratum Cluster Totals in Stratified Cluster Sampling," Mathematics and Statistics, Vol. 9, No. 5, pp. 697 - 710, 2021. DOI: 10.13189/ms.2021.090509. (b): Bhuwaneshwar Kumar Gupt, F. Lalthlamuanpuii, Md. Irphan Ahamed (2021). Choice of Strata Boundaries for Allocation Proportional to Stratum Cluster Totals in Stratified Cluster Sampling. Mathematics and Statistics, 9(5), 697 - 710. DOI: 10.13189/ms.2021.090509. Copyright©2021 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract In survey planning, sometimes, there arises situation to use cluster sampling because of nature of spatial relationship between elements of population or physical feature of land over which elements are dispersed or unavailability of reliable list of elements. At the same time, there requires technique and strategy for ensuring precision of the sample in representing the parent population. Although several theoretical cum practical works have been done in cluster sampling, stratified sampling and stratified cluster sampling, so far, the problem of stratified cluster sampling for a study variable based on an auxiliary variable, which is required in practice, has never been approached. For the first time, this paper deals with the problem of optimum stratification of population of clusters in cluster sampling with clusters of equal size of a characteristic  under study based on highly correlated concomitant variable  for allocation proportional to stratum cluster totals under a super population model. Equations giving optimum strata boundaries (OSB) for dividing population, in which sampling unit of the population is a cluster, are obtained by minimising sampling variance of the estimator of population mean. As the equations are implicit in nature, a few methods of finding approximately optimum strata boundaries (AOSB) are deduced from the equations giving OSB. In deriving the equations, mathematical tools of calculus and algebra are used in addition to statistical methods of finding conditional expectation of variance. All the proposed methods of stratification are empirically examined by illustrating in live data, population of villages in Lunglei and Serchhip districts of Mizoram State, India, and found to perform efficiently in stratifying the population. The proposed methods may provide practically feasible solution in planning socio-economic survey. Keywords Allocation, Gamma Probability Density Function, Cluster Size, Optimum Strata Boundaries, Stratified Cluster Sampling, Stratification Variable 1. Introduction In stratified sampling, a heterogeneous population is divided into a number of groups called strata which are within strata homogeneous and sample is selected from strata using suitable sample selection method; the method is used for administrative convenience and enhancing the precision of representation of the sample for the parent population. On the other hand, when the availability of reliable list of elements (units) of population is difficult or the elements are spatially dispersed in such a way that there requires lots of energy, time and cost while surveying the elements selected by simple random sampling, cluster sampling or area sampling is employed by grouping the contiguous elements or elements, which can be conveniently surveyed together without much extra effort, into clusters; then, the clusters are taken as