Stratified sampling examples

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===Example 1===
 
===Example 1===
  
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[[File:5.2.6-fig74.png|right|thumb|300px|Figure 1:  Illustration why  stratification is most efficient when the ''strata  means'' are as  different as possible]]
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This example shows stratified sampling of the example [[population]] in figure 1.
  
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Imagine the example population of <math>N=30</math> elements be subdivided into three strata as in figure 1. 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 variances are given in Table 1.               
 
  
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'''Table 1:''' Stratum parameters for the stratified example population.
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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.                
  
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<blockquote>
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'''Table 1''' Stratum parameters for the stratified example population.
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<div style="float:left; margin-right:2em">
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{| class="wikitable"
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:{| class="wikitable"
 
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|-
 
!''Stratum''
 
!''Stratum''
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!''<math>\sigma_h^2\,</math>''
 
!''<math>\sigma_h^2\,</math>''
 
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|1
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|align="center"|'''1'''
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|14
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|align="right"|14
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|4
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|align="right"|4
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|6.29
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|align="right"|6.29
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|3.49
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|align="right"|3.49
 
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|-
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|2
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|align="center"|'''2'''
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|8
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|align="right"|8
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|3
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|align="right"|3
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|10.13
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|align="right"|10.13
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|4.86
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|align="right"|4.86
 
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|-
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|3
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|align="center"|'''3'''
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|8
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|align="right"|8
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|3
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|align="right"|3
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|5.38
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|align="right"|5.38
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|2.48
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|align="right"|2.48
 
|}
 
|}
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</div>
 
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[[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>).]]
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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.
  
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'''Table 2''' Calculation of parametric population mean from the parametric strata means.
  
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:{| class="wikitable"
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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.
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'''Table 2:''' Calculation of parametric population mean from the parametric strata means.
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<blockquote>
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<div style="float:left; margin-right:2em">
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{| class="wikitable"
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|-
 
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!''Stratum''
 
!''Stratum''
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!''mean*weight''
 
!''mean*weight''
 
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|-
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|1
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|align="center"|'''1'''
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|6.29
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|align="right"|6.29
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|0.466667
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|align="right"|0.466667
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|2.9333
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|align="right"|2.9333
 
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|-
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|2
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|align="center"|'''2'''
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|10.13
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|align="right"|10.13
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|0.266667
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|align="right"|0.266667
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|2.7000
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|align="right"|2.7000
 
|-
 
|-
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|3
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|align="center"|'''3'''
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|5.38
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|align="right"|5.38
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|0.266667
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|align="right"|0.266667
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|1.4333
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|align="right"|1.4333
 
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|'''7.0667'''
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|align="right"|'''7.0667'''
 
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|}
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</div>
 
  
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[[File:5.2.6-fig75.png|right|thumb|300px|Figure 2:  Subdividing the   example population (arbitrarily) in three strata, for  illustration    purposes]]
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'''Table 3''' Calculation of parametric error variance of the estimated mean of the population for <math>n=10</math>.
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:{| class="wikitable"
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|-
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!''Stratum''
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!''fpc''
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!''<math>\sigma_h^2/n</math>''
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!''var per stratum<math>fpc*\sigma_h^2/n</math>''
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!''<math>var*W_h^2</math>''
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|align="center"|'''1'''
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|align="right"|0.769230769
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|align="right"|0.87244898
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|align="right"|0.67111461
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|align="right"|0.146154
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|-
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|align="center"|'''2'''
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|align="right"|0.714285714
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|align="right"|1.61979167
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|align="right"|1.15699405
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|align="right"|0.082275
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|-
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|align="center"|'''3'''
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|align="right"|0.714285714
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|align="right"|0.82812500
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|align="right"|0.59151786
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|align="right"|0.042063
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|-
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|align="right"|<math>var(\bar y)=</math>
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|align="right"|'''0.270492'''
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|}
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The error variance of the estimated mean is
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:<math>var(\bar y)=0.27049</math>
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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.
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==References==
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<references/>
  
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{{Construction}}
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{{SEO
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|keywords=stratified random sampling,strata,population,sampling technique,sub-population
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|descrip=Stratified  sampling is a method to subdivide a population into separate and more  homogeneous sub-populations called strata.
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}}
  
 
[[Category:Forest Inventory Examples]]
 
[[Category:Forest Inventory Examples]]

Latest revision as of 13:06, 26 October 2013

Forest inventory
Introduction to forest inventory
Forest mensuration
Introduction to sampling
Plot design
Sampling design
Estimation issues
Planning issues
UNFCCC Implications for Forest Monitoring
Forest Inventory Examples
  Forest Inventory Glossary

[edit] Example 1

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


Imagine the example population of \(N=30\) elements be subdivided into three strata [1]. 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 \(n=10\), taking \(n_1=4\) from the first stratum and \(n_2=n_3=3\) from the other two strata. The stratum parametric means and variances are given in table 1.

Table 1 Stratum parameters for the stratified example population.

Stratum \(N_h\,\) \(n_h\,\) \(\mu_h\,\) \(\sigma_h^2\,\)
1 14 4 6.29 3.49
2 8 3 10.13 4.86
3 8 3 5.38 2.48
Figure 2 Subdividing the example population (arbitrarily) in three strata, for illustration purposes (Kleinn 2007[2]).

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 \(n=10\) and the defined allocation of samples to the three strata.

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

Stratum Stratum mean Weight \((W_h)\) mean*weight
1 6.29 0.466667 2.9333
2 10.13 0.266667 2.7000
3 5.38 0.266667 1.4333
7.0667

Table 3 Calculation of parametric error variance of the estimated mean of the population for \(n=10\).

Stratum fpc \(\sigma_h^2/n\) var per stratum\(fpc*\sigma_h^2/n\) \(var*W_h^2\)
1 0.769230769 0.87244898 0.67111461 0.146154
2 0.714285714 1.61979167 1.15699405 0.082275
3 0.714285714 0.82812500 0.59151786 0.042063
\(var(\bar y)=\) 0.270492


The error variance of the estimated mean is

\[var(\bar y)=0.27049\]

which is considerably smaller than for simple random sampling with \(n=10\). That is: in this case, stratification makes sense and increases precision without increasing much the sample size. Stratification criteria must be known or decided on a priori.

[edit] References

  1. ↑ de Vries, P.G., 1986. Sampling Theory for Forest Inventory. A Teach-Yourself Course. Springer. 399 p.
  2. ↑ 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.

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