Sampling Methods and Bias in Statistics: Study Guide

Sampling in Statistics

What is a Random Sample?

A random sample is a fundamental concept in statistics, ensuring that every individual in a population has an equal chance of being selected. This is crucial for obtaining representative data and minimizing bias.

• Definition: A sample is random if every individual in the population has an equal chance of being selected.

• Importance: Random sampling prevents bias and makes the sample representative of the population.

• Example: To study student opinions about cafeteria food, pick names from the entire student list using a random number generator. NOT a random sample: Asking only people who eat at the cafeteria.

Sampling Methods

1. Simple Random Sample (SRS)

Simple Random Sampling is the most basic form of random sampling, where every possible group of individuals has the same chance of being chosen.

• Definition: Every group of n individuals has the same chance of being chosen.

• Example: A teacher puts all 25 student names in a bag, shakes it, and pulls out 5 names.

• Other methods: Use a random number generator or a random table to select individuals.

2. Stratified Sampling

Stratified sampling involves dividing the population into subgroups (strata) based on shared characteristics, then randomly sampling from each stratum. This ensures representation from all important subgroups.

• Definition: Population is divided into strata (e.g., by major, age, gender), and a random sample is taken from each stratum.

• When to use: When groups differ in important ways, but all should be represented.

• Example: College students grouped by major. Randomly pick 10 students from Science, 10 from Business, 10 from Arts.

• Benefit: Ensures all major groups are represented.

3. Cluster Sampling

Cluster sampling is used when the population is spread out. The population is divided into clusters, some clusters are randomly chosen, and all individuals in those clusters are included in the sample.

• Definition: Population is divided into clusters (groups), then some clusters are randomly chosen. Everyone in those clusters is included.

• When to use: When the population is too spread out to sample individuals easily.

• Example: A college has 50 classes. Randomly select 5 classes and survey all students in those classes.

• Benefit: Faster and cheaper than stratified sampling.

4. Systematic Sampling

Systematic sampling selects every kth individual from a list after a random starting point. It is easy to implement but requires that the list has no hidden pattern.

• Definition: Select every kth individual from a list after a random starting point.

• Example: A list of 500 customers is ordered randomly. Start at the 3rd person, then choose every 10th (3rd, 13th, 23rd, ...).

• Benefit: Easy to implement, but must ensure the list has no hidden pattern.

Comparison: Stratified vs. Cluster Sampling

Both methods divide the population into groups, but differ in how samples are selected:

• Stratified Sampling: Randomly sample individuals from every group (stratum).

• Cluster Sampling: Randomly select entire groups (clusters), then sample all individuals within those groups.

Non-Random Samples & Bias

Non-random sampling methods can introduce bias, making the sample unrepresentative of the population.

• Voluntary Response Bias: People choose themselves to respond, often those with strong opinions. Example: Online product reviews—usually only very happy or very angry customers respond.

• Convenience Sampling: Choose people who are easy to reach. Example: Surveying only the first row of students in a class.

• Response Bias: People give inaccurate answers, often due to embarrassment or social desirability. Example: Underreporting drinking habits.

• Nonresponse Bias: Some people do not respond, and those who do may differ from those who don’t. Example: Mail survey about income—those with very high or low income may not respond.

• Undercoverage: Some groups are missed in the sample. Example: Telephone survey using landlines misses younger people who only use cell phones.

Sampling Frame

The sampling frame is the list from which the sample is actually chosen. It is important that the sampling frame closely matches the population of interest.

• Definition: The list from which the sample is actually chosen.

• Example: If the population is all registered voters, the sampling frame might be the list of people who voted in the last election.

Retrospective vs. Prospective Studies

These terms describe the timing of data collection in observational studies.

• Retrospective Study: Looks at past data. Example: Checking high school GPA of today’s business owners.

• Prospective Study: Collects data into the future. Example: Following a group of freshmen through 4 years of college.

Observational vs. Experimental Studies

• Observational Study: Observe without interference. Example: Record how many hours students study per week.

• Experimental Study: (Not detailed in the notes, but for completeness) Researchers actively impose a treatment or intervention and observe the outcome.

Examples: Identifying Sampling Methods

Below are examples of how to identify different sampling methods in practice:

• Stratified Sampling: A sample of 100 undergraduates is taken by organizing students by classification (freshman, sophomore, junior, senior), then selecting 25 from each group.

• Systematic Sampling: A random number generator is used to select a student from an alphabetical list, then every 50th student is included in the sample.

• Simple Random Sample: A completely random method is used to select 75 students, with each student having the same probability of being chosen at any stage.

• Convenience Sampling: An administrative assistant stands in front of the library and asks the first 100 students encountered.

Summary Table: Sampling Methods

Sampling Method	How It Works	When to Use	Example
Simple Random Sample (SRS)	Every group of n individuals has equal chance	General use, when no subgroups need special representation	Randomly select names from a list
Stratified Sampling	Divide into strata, sample from each	When subgroups differ and all must be represented	Sample by major, age, or class year
Cluster Sampling	Divide into clusters, randomly select clusters, sample all in clusters	When population is spread out geographically	Sample all students in selected classes
Systematic Sampling	Select every k-th individual after random start	When a list is available and randomization is needed	Every 10th customer on a list
Convenience Sampling	Sample those easiest to reach	Quick, but not representative	First 100 people at the library
Voluntary Response	People choose to participate	When strong opinions are likely	Online reviews