Stratified sampling examples

Example 1

Figure 1: Illustration why stratification is most efficient when the strata means are as different as possible

Imagine the example population of \(N=30\) 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 \(n=10\), 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

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

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