Statistics 简明教程

Statistics - Stratified sampling

该检查策略用作环境的一部分，其中可以毫不费力地将总体划分为群体或阶层，这些群体或阶层彼此明显不同，但群体中的组成部分在某些属性方面是同类的，例如，学校的学生可以根据性别、开设课程、年龄等基础进行阶层划分。在这种情况下，总体最初被划分为阶层，然后从每个阶层中获取随机样本。分层抽样有两种类型：按比例分层抽样和按比例分层抽样。

This strategy for examining is utilized as a part of circumstance where the population can be effortlessly partitioned into gatherings or strata which are particularly not quite the same as one another, yet the components inside of a gathering are homogeneous regarding a few attributes e. g. understudies of school can be separated into strata on the premise of sexual orientation, courses offered, age and so forth. In this the population is initially partitioned into strata and afterward a basic irregular specimen is taken from every stratum. Stratified testing is of two sorts: proportionate stratified inspecting and disproportionate stratified examining.

  1. Proportionate Stratified Sampling - In this the number of units selected from each stratum is proportionate to the share of stratum in the population e.g. in a college there are total 2500 students out of which 1500 students are enrolled in graduate courses and 1000 are enrolled in post graduate courses. If a sample of 100 is to be chosen using proportionate stratified sampling then the number of undergraduate students in sample would be 60 and 40 would be post graduate students. Thus the two strata are represented in the same proportion in the sample as is their representation in the population. This method is most suitable when the purpose of sampling is to estimate the population value of some characteristic and there is no difference in within- stratum variances.

  2. Disproportionate Stratified Sampling - When the purpose of study is to compare the differences among strata then it become necessary to draw equal units from all strata irrespective of their share in population. Sometimes some strata are more variable with respect to some characteristic than other strata, in such a case a larger number of units may be drawn from the more variable strata. In both the situations the sample drawn is a disproportionate stratified sample. The difference in stratum size and stratum variability can be optimally allocated using the following formula for determining the sample size from different strata Formula${n_i = \frac{n.n_i\sigma_i}{n_1\sigma_1+n_2\sigma_2+…​n_k\sigma_k}\ for\ i = 1,2 …​k}$ Where − ${n_i}$ = the sample size of i strata. ${n}$ = the size of strata. ${\sigma_1}$ = the standard deviation of i strata. In addition to it, there might be a situation where cost of collecting a sample might be more in one strata than in other. The optimal disproportionate sampling should be done in a manner that ${\frac{n_1}{n_1\sigma_1\sqrt{c_1}} = \frac{n_2}{n_2\sigma_1\sqrt{c_2}} = …​ = \frac{n_k}{n_k\sigma_k\sqrt{c_k}}}$ Where ${c_1, c_2, …​ ,c_k}$ refer to the cost of sampling in k strata. The sample size from different strata can be determined using the following formula: ${n_i = \frac{\frac{n.n_i\sigma_i}{\sqrt{c_i}}}{\frac{n_1\sigma_1}{\sqrt{c_i}}\frac{n_2\sigma_2}{\sqrt{c_2}}…​\frac{n_k\sigma_k}{\sqrt{c_k}}}\ for\ i = 1,2 …​k}$

Example

Problem Statement:

Problem Statement:

一个组织有 5000 名员工，分为三个级别。

An organisation has 5000 employees who have been stratified into three levels.

  1. Stratum A: 50 executives with standard deviation = 9

  2. Stratum B: 1250 non-manual workers with standard deviation = 4

  3. Stratum C: 3700 manual workers with standard deviation = 1

如何以最佳分配方式以不成比例的基础抽取 300 名员工的样本？

How will a sample of 300 employees are drawn on a disproportionate basis having optimum allocation?

Solution:

Solution:

使用针对最佳分配的不成比例抽样公式。

Using the formula of disproportionate sampling for optimum allocation.