Stratified sampling examples

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Example 1

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

Figure 1 Example population (de Vries 1986)

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

Figure 2 Subdividing the example population (arbitrarily) in three strata, for illustration purposes

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 effort. Stratification criteria must be known or decided on a priori.

References

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.