Population & Sampling

Simple Random Sampling (SRS)

• Method: Every member of the population has an equal and independent chance of being selected. This is typically achieved by assigning a unique number to every member in the sampling frame and using a random number generator to pick the sample.
• Pros/Cons: It is the most unbiased method but requires a complete and accurate sampling frame, which may not always be available.

Systematic Sampling

• Method: Selecting members from a sampling frame at regular intervals, such as every $k$-th individual (where $k = \text{Population Size} / \text{Sample Size}$). The starting point must be chosen randomly.
• Risk: If the sampling frame contains a hidden periodic pattern that matches the sampling interval, the sample will be highly biased.

Stratified Sampling

• Method: The population is divided into mutually exclusive subgroups called strata (e.g., by age, gender, or income). A random sample is then taken from each stratum in proportion to its size in the population.
• Benefit: This ensures that all key subgroups are represented in the final sample, reducing sampling error for diverse populations.

Quota Sampling

• Method: The researcher ensures the sample contains a specific number of individuals from different categories (e.g., 50 men and 50 women). Unlike stratified sampling, the selection within these categories is not random.
• Usage: Often used in market research or street polling where a quick, diverse sample is needed without a formal sampling frame.

Convenience Sampling

• Method: Selecting individuals who are easiest to reach or most available to the researcher. For example, a student surveying their classmates to represent all students at a university.
• Risk: This method is highly prone to bias and rarely produces a representative sample, as it excludes anyone not immediately accessible to the researcher.

• Parameter vs. Statistic: A parameter is a numerical value describing a population (e.g., population mean $\mu$), while a statistic is a numerical value describing a sample (e.g., sample mean $\bar{x}$). Statistics are used to estimate parameters.

• Identify the Frame: Always check if a sampling frame exists. If there is no list of the population, probability methods like SRS or Systematic sampling cannot be used effectively.

• Check for Periodicity: In systematic sampling problems, look for cycles in the data. If a list of daily sales is sampled every 7th day, you might only be looking at Mondays, which would bias the results if sales vary by day of the week.

• Proportionality in Strata: When calculating stratified samples, use the formula: $n_h = (N_h / N) \times n$, where $n_h$ is the sample size for the stratum, $N_h$ is the stratum population, $N$ is the total population, and $n$ is the total sample size.

• Bias Detection: If a question describes a sample taken from a specific location (like a shopping mall) or a specific time, identify it as convenience sampling and note that it likely suffers from selection bias.