Sampling from a Population

Population and Sample

Understanding the distinction between a population and a sample is fundamental in statistics. The population includes all cases of interest, while a sample is a subset of the population from which data are collected.

• Population: All individuals, objects, or cases relevant to a study.

• Sample: The subset of the population actually observed or measured.

• Sample Size (n): The number of cases in the sample, denoted by n.

• Census: When data are collected from every member of the population, the sample is the population itself.

Example: If a survey is conducted among 362 students, the sample is the 362 students who participated.

Additional info: Census data are rare due to cost and difficulty, but are more feasible for small populations (e.g., professional athletes).

Statistical Inference

Statistical inference is the process of using sample data to make generalizations about a population.

• Generalization: Applying sample results to the population.

• Key Point: The reliability of inference depends on the quality and representativeness of the sample.

Bias in Sampling

Bias occurs when the data collection process causes the sample to differ systematically from the population, undermining the validity of generalizations.

• Sampling Bias: The method of selecting the sample causes it to differ from the population.

• Response Bias: The responses obtained are inaccurate due to factors like wording, social desirability, or deliberate misreporting.

• Variability vs. Bias: Variability refers to random differences between samples; bias is systematic and cannot be corrected mathematically.

Types of Sampling Bias

Several specific types of sampling bias can affect the validity of a study:

• Undercoverage Bias: Certain members of the population are systematically excluded from the sample. Example: A poll that only includes people with telephones excludes those without, potentially skewing results.

• Self-Selection Bias: Individuals choose to participate, often those with strong opinions. Example: A survey where only passionate respondents reply.

• Nonresponse Bias: Selected individuals do not respond, possibly those with less interest or strong feelings. Example: A mailed survey with a low response rate.

Comparison Table: Types of Sampling Bias

Additional info: Undercoverage and nonresponse are distinct: undercoverage excludes cases from the start, while nonresponse includes them but they do not participate.

Response Bias

Response bias arises when the responses collected are inaccurate, often due to the way questions are asked or social pressures.

• Wording Bias: The phrasing of questions can influence responses.

• Social Desirability Bias: Respondents may answer in a way they believe is socially acceptable.

• Deliberate Lying: Respondents intentionally provide false information.

Example: Asking "heads or tails?" versus "tails or heads?" can change the likelihood of the response.

Random Sampling

Random sampling uses chance in a formal way to select the sample, ensuring every individual has an equal probability of selection.

• Random Sample: Every individual in the population has an equal chance of being selected.

• Simple Random Sample (SRS): Each group of size n has an equal chance of being chosen. Example: Assigning ID numbers and using a random number generator to select participants.

• Convenience Sample: Sample is chosen based on ease of access, not randomness; may not represent the population.

Additional info: Random sampling eliminates sampling bias but not response bias. SRS is the gold standard but often impractical for large populations.

Anecdotal Evidence

Anecdotal evidence is based on personal observation or unsystematic samples, often from convenience samples. It is not considered scientific evidence.

• Key Point: Anecdotal evidence cannot be generalized to the population.

• Example: A single person's experience with a treatment does not prove its effectiveness for all.

Summary Table: Sampling Methods

Key Formulas

• Sample Size:

• Population Size:

Additional info: The mathematics of random sampling allow for quantifying variability, but bias cannot be mathematically corrected.