Stratified sampling examples
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!''<math>\sigma_h^2/n</math>'' | !''<math>\sigma_h^2/n</math>'' | ||
| − | !''var per stratum<math>fpc\sigma_h^2/n'' | + | !''var per stratum<math>fpc\sigma_h^2/n</math>'' |
!''var*<math>W_h^2</math>'' | !''var*<math>W_h^2</math>'' | ||
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Revision as of 19:20, 16 December 2010
Example 1
This example shows stratified sampling by the example population from figure 1.
Imagine the example population of \(N=30\) elements be subdivided into three strata as in figure 2. 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 |
| \(var(\hat y)=\) | 0.270492 |
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