Sampling Techniques

Core Principle: Selection occurs at regular intervals from an ordered list. It is often easier to implement than simple random sampling for large populations.

Methodology: A starting point is chosen randomly from the first $k$ elements, and then every $k$-th element is selected thereafter. The interval $k$ is calculated as $k = \frac{\text{Population Size}}{\text{Sample Size}}$.

Risk of Periodicity: If the population list contains a recurring pattern that matches the sampling interval, the sample will be highly biased (e.g., selecting every 7th day of a production cycle might only capture Mondays).

Core Principle: The population is divided into mutually exclusive subgroups called strata based on shared characteristics (e.g., age, gender, income).

Proportionality: To ensure the sample reflects the population structure, the number of individuals taken from each stratum must be proportional to that stratum's size in the total population.

Calculation: The number of individuals to select from a specific group is calculated using: $\text{Sample Size from Group} = \frac{\text{Size of Group}}{\text{Total Population}} \times \text{Total Sample Size}$

Check the Total: In stratified sampling problems, always sum the calculated sample sizes for each group to ensure they equal the required total sample size. Rounding errors can sometimes cause a discrepancy of $\pm 1$.

Identify the 'k': For systematic sampling, clearly define the interval $k$. If the question asks for a sample of $50$ from $1000$, your interval is $20$. Ensure the starting point is a random number between $1$ and $20$.

Bias Analysis: When asked to evaluate a sampling method, look for missing groups (under-coverage) or non-random selection processes (convenience) that might skew results.

Context Matters: If a population has distinct, diverse groups, stratified sampling is almost always the superior choice to ensure every group is heard.