Stratified sampling examples
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===Example 1=== | ===Example 1=== | ||
| − | This example shows stratified sampling of the example population in figure 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 | + | 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. | '''Table 1''' Stratum parameters for the stratified example population. | ||
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!''<math>\sigma_h^2\,</math>'' | !''<math>\sigma_h^2\,</math>'' | ||
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| − | |1 | + | |align="center"|'''1''' |
| − | |14 | + | |align="right"|14 |
| − | |4 | + | |align="right"|4 |
| − | |6.29 | + | |align="right"|6.29 |
| − | |3.49 | + | |align="right"|3.49 |
|- | |- | ||
| − | |2 | + | |align="center"|'''2''' |
| − | |8 | + | |align="right"|8 |
| − | |3 | + | |align="right"|3 |
| − | |10.13 | + | |align="right"|10.13 |
| − | |4.86 | + | |align="right"|4.86 |
|- | |- | ||
| − | |3 | + | |align="center"|'''3''' |
| − | |8 | + | |align="right"|8 |
| − | |3 | + | |align="right"|3 |
| − | |5.38 | + | |align="right"|5.38 |
| − | |2.48 | + | |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]] | + | [[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. | + | 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. | '''Table 2''' Calculation of parametric population mean from the parametric strata means. | ||
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!''mean*weight'' | !''mean*weight'' | ||
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| − | |1 | + | |align="center"|'''1''' |
| − | |6.29 | + | |align="right"|6.29 |
| − | |0.466667 | + | |align="right"|0.466667 |
| − | |2.9333 | + | |align="right"|2.9333 |
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| − | |2 | + | |align="center"|'''2''' |
| − | |10.13 | + | |align="right"|10.13 |
| − | |0.266667 | + | |align="right"|0.266667 |
| − | |2.7000 | + | |align="right"|2.7000 |
|- | |- | ||
| − | |3 | + | |align="center"|'''3''' |
| − | |5.38 | + | |align="right"|5.38 |
| − | |0.266667 | + | |align="right"|0.266667 |
| − | |1.4333 | + | |align="right"|1.4333 |
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| − | |'''7.0667''' | + | |align="right"|'''7.0667''' |
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!''<math>var*W_h^2</math>'' | !''<math>var*W_h^2</math>'' | ||
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| − | |1 | + | |align="center"|'''1''' |
| − | |0.769230769 | + | |align="right"|0.769230769 |
| − | |0.87244898 | + | |align="right"|0.87244898 |
| − | |0.67111461 | + | |align="right"|0.67111461 |
| − | |0.146154 | + | |align="right"|0.146154 |
|- | |- | ||
| − | |2 | + | |align="center"|'''2''' |
| − | |0.714285714 | + | |align="right"|0.714285714 |
| − | |1.61979167 | + | |align="right"|1.61979167 |
| − | |1.15699405 | + | |align="right"|1.15699405 |
| − | |0.082275 | + | |align="right"|0.082275 |
|- | |- | ||
| − | |3 | + | |align="center"|'''3''' |
| − | |0.714285714 | + | |align="right"|0.714285714 |
| − | |0.82812500 | + | |align="right"|0.82812500 |
| − | |0.59151786 | + | |align="right"|0.59151786 |
| − | |0.042063 | + | |align="right"|0.042063 |
|- | |- | ||
| | | | ||
| | | | ||
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| − | |<math>var(\bar y)=</math> | + | |align="right"|<math>var(\bar y)=</math> |
| − | |'''0.270492''' | + | |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 precision without increasing much the | + | 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. | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
| + | |||
| + | {{SEO | ||
| + | |keywords=stratified random sampling,strata,population,sampling technique,sub-population | ||
| + | |descrip=Stratified sampling is a method to subdivide a population into separate and more homogeneous sub-populations called strata. | ||
| + | }} | ||
| + | |||
[[Category:Forest Inventory Examples]] | [[Category:Forest Inventory Examples]] | ||
Latest revision as of 13:06, 26 October 2013
[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
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
- ↑ de Vries, P.G., 1986. Sampling Theory for Forest Inventory. A Teach-Yourself Course. Springer. 399 p.
- ↑ 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.
