View source for Stratified sampling examples

===Example 1===

This example shows stratified sampling of the example [[population]] in figure 1.


Imagine the example population of <math>N=30</math> elements be subdivided into three [[strata]] <ref>de Vries, P.G., 1986. Sampling Theory for  Forest Inventory. A Teach-Yourself Course. Springer. 399 p.</ref>. Here, stratification has been done arbitrarily into three strata of size 14, 8 and 8. From this stratified population, we wish to take a sample of <math>n=10</math>, taking <math>n_1=4</math> from the first stratum and <math>n_2=n_3=3</math> from the other two strata. The [[stratum parametric]] means and [[variance|variances]] are given in table 1.                 

'''Table 1''' Stratum parameters for the stratified example population.

:{| class="wikitable"
|-
!''Stratum''
!''<math>N_h\,</math>''
!''<math>n_h\,</math>''
!''<math>\mu_h\,</math>''
!''<math>\sigma_h^2\,</math>''
|-
|align="center"|'''1'''
|align="right"|14
|align="right"|4
|align="right"|6.29
|align="right"|3.49
|-
|align="center"|'''2'''
|align="right"|8
|align="right"|3
|align="right"|10.13
|align="right"|4.86
|-
|align="center"|'''3'''
|align="right"|8
|align="right"|3
|align="right"|5.38
|align="right"|2.48
|}

[[File:5.2.6-fig75.png|right|thumb|300px|'''Figure 2''' Subdividing the example population (arbitrarily) in three strata, for illustration purposes (Kleinn 2007<ref name="kleinn2007">Kleinn, C. 2007. Lecture Notes for the Teaching Module Forest Inventory. Department of Forest Inventory  and Remote Sensing. Faculty of Forest Science and Forest Ecology, Georg-August-Universität Göttingen. 164 S.</ref>).]]
 
Calculation in stratified sampling is best done in tabular format, first per stratum and then combining the per-stratum results to the values / estimations for the entire population. The estimation of the [[mean]] is illustrated in Table 2 and results – as expected – in the parametric mean without stratification. Table 3 presents the calculation of the parametric [[error variance]] for <math>n=10</math> and the defined allocation of samples to the three strata.

'''Table 2''' Calculation of parametric population mean from the parametric strata means.

:{| class="wikitable"
|-
!''Stratum''
!''Stratum mean''
!''Weight <math>(W_h)</math>''
!''mean*weight''
|-
|align="center"|'''1'''
|align="right"|6.29
|align="right"|0.466667
|align="right"|2.9333
|-
|align="center"|'''2'''
|align="right"|10.13
|align="right"|0.266667
|align="right"|2.7000
|-
|align="center"|'''3'''
|align="right"|5.38
|align="right"|0.266667
|align="right"|1.4333
|-
|
|
|
|align="right"|'''7.0667'''
|}

'''Table 3''' Calculation of parametric error variance of the estimated mean of the population for <math>n=10</math>.

:{| class="wikitable"
|-
!''Stratum''
!''fpc''
!''<math>\sigma_h^2/n</math>''
!''var per stratum<math>fpc*\sigma_h^2/n</math>''
!''<math>var*W_h^2</math>''
|-
|align="center"|'''1'''
|align="right"|0.769230769
|align="right"|0.87244898
|align="right"|0.67111461
|align="right"|0.146154
|-
|align="center"|'''2'''
|align="right"|0.714285714
|align="right"|1.61979167
|align="right"|1.15699405
|align="right"|0.082275
|-
|align="center"|'''3'''
|align="right"|0.714285714
|align="right"|0.82812500
|align="right"|0.59151786
|align="right"|0.042063
|-
|
|
|
|align="right"|<math>var(\bar y)=</math>
|align="right"|'''0.270492'''
|}

  
The error variance of the estimated mean is

:<math>var(\bar y)=0.27049</math> 

which is considerably smaller than for [[simple random sampling]] with <math>n=10</math>. That is: in this case, stratification makes sense and increases [[Accuracy and precision|precision]] without increasing much the [[sample size]]. Stratification criteria must be known or decided on a priori.