Stratified Sampling

Stratified sampling involves dividing the population into homogeneous subgroups called strata (e.g., gender, age, regions, etc.). Then, a random sample is selected from each stratum, proportionally to its size in the population.

Formula for the stratified sample mean, \(\bar{X}_E\): $$\bar{X}_E = \sum_{h=1}^{L} \frac{N_h}{N} \bar{X}_h$$

Where:

\( L \): Number of strata.

\( N_h \): Population size in stratum h.

\( N \): Total population size.

\( \bar{X}_h \): Sample mean of stratum h.

Advantages:

Ensures that each subgroup is represented.

Improves the accuracy of estimates if the strata are homogeneous.

Systematic Sampling

In systematic sampling, every k-th element of the population is selected after choosing a random starting point. The selection interval (k) is calculated as:

$$k = \frac{N}{n}$$

Where:

\( N \): Population size.

\( n \): Sample size.

Formula for the systematic sample mean \( \bar{X}_S \) : $$\bar{X}_S = \frac{1}{n} \sum_{i=1}^{n} X_i$$

Where:

\( X_i \): Value of the i-th selected observation.

Advantages:

Easy to implement and quick to execute.

Works well in populations where data do not have cyclical patterns.