Arun

Stratified Sampling

The population is divided into homogeneous subgroups (strata), and samples are drawn from each stratum.

Mathematical Description

Let:

• Population divided into \( k \) strata
• Stratum \( h \) has size \( N_h \)
• Sample size from stratum \( h \) is \( n_h \)

Then:
$$ \sum_{h=1}^{k} N_h = N, \quad \sum_{h=1}^{k} n_h = n $$
Two common allocation methods:

1. Proportional Allocation:
   $$ n_h = \frac{N_h}{N} \times n $$
   
2. Optimal (Neyman) Allocation:
   $$n_h = \frac{N_h \sigma_h}{\sum N_h \sigma_h} \times n$$
   where \( \sigma_h \) is the standard deviation within stratum \( h \).

Example: Sampling students from different academic years to ensure representation.