Comprehensive Guide to Random and Non-Random Sampling Methods

Fundamentals of Random Sampling

• Purpose of Random Sampling:
  • Sampling non-randomly or restrictively leads to severe sample bias. For instance, sampling exclusively from a single group (e.g., Republicans) leaves other segments of the population (e.g., Democrats and Independents) completely unrepresented.
  • Random sampling ensures that every subgroup in a population has a known, non-zero chance of inclusion, producing a sample that is representative of the entire population rather than just a specific subgroup of interest.
• Definition of Random Sampling:
  • Random sampling is the formal process of using chance to select individuals from a target population to be included in a sample.
• The Four Probability Sampling Methods:
  1. Simple Random Sampling
  2. Systematic Sampling
  3. Stratified Sampling
  4. Cluster Sampling
  • Mnemonic Note: Three methods start with the letter 'S' (Simple Random, Systematic, Stratified), and one starts with the letter 'C' (Cluster).

Simple Random Sampling

• Definition & Ideal Conditions:

  • Simple random sampling is a foundational sampling technique where every possible sample of a given size $n$ has an equal chance of being selected from a population of size $N$.
  • It is best used when a complete frame—a master list of every individual in the population—is available prior to sampling.

• Example 1: Concert Ticket Selection

  • Scenario: Sophia has $4$ tickets to a concert. Six of her friends—Yolanda, Michael, Kevin, Marissa, Annie, and Katie—express interest in attending. Sophia decides to randomly select $3$ of her $6$ friends to join her.
  • Population Size ($N$): $N = 6$
  • Sample Size ($n$): $n = 3$
  • Sampling Without Replacement: Once an individual is selected, they cannot be selected again.
  • Total Sample Combinations: Out of $6$ individuals, there are exactly $20$ distinct ways (combinations) to select $3$ people randomly.
  • Probability Calculation:
    • Each unique subset of $3$ friends has an equal likelihood of selection.
    • The probability $P$ of selecting any specific group of $3$ (such as Michael, Kevin, and Marissa) is:             $P = \frac{1}{20} = 0.05$
  • Implementation: Using a random number generator that yields an integer between $1$ and $20$, Sophia maps each integer to one of the $20$ unique combinations to decide the final group.

• Example 2: Denise and Associates Accounting Firm Survey

  • Scenario: Denise and Associates wants to evaluate client satisfaction by taking a simple random sample of $n = 5$ clients from their total population of $N = 30$ clients.
  • Population Frame ($N = 30$):
    1. Client 1
    2. Client 2
    3. …
    4. Client 14: John's Baylor
    5. Client 17: Moe's Exterminating
    6. Client 21: R and Q Realty
    7. Client 25: Thor's Robert DBS
    8. Client 30: Worldwide Wireless

• TI-84 Calculator Procedure for Simple Random Sampling:

  • Step 1: Setting the Seed
    • A random number seed is a starting whole number chosen arbitrarily (e.g., $17$) required by pseudo-random number generators to initialize the algorithm.
    • Key Sequence:
      1. Enter the chosen seed number (e.g., $17$).
      2. Press the STO> button (located directly above the ON button).
      3. Press MATH -> navigate right to the PRB (Probability) menu.
      4. Select 1: rand and press ENTER twice.
  • Step 2: Generating Unique Random Integers
    • Key Sequence (TI-84 Plus / OS with randIntNoRep):
      1. Press MATH -> navigate right to PRB.
      2. Scroll down to 8: randIntNoRep and press ENTER.
      3. Set lower: $1$
      4. Set upper: $30$
      5. Set n: $5$
      6. Select Paste and press ENTER twice.
    • Alternative Key Sequence (Older TI-84 models without randIntNoRep):
      1. Press MATH -> navigate right to PRB.
      2. Select 5: randInt(.
      3. Input syntax: randInt(1, 30).
      4. Press ENTER repeatedly $5$ times, ignoring duplicate numbers until $5$ unique values are obtained.
  • Calculator Sample Result (Seed $17$):
    • Generated integers: $21, 17, 25, 30, 14$
    • Corresponding Selected Clients:
      • $21$: R and Q Realty
      • $17$: Moe's Exterminating
      • $25$: Thor's Robert DBS
      • $30$: Worldwide Wireless
      • $14$: John's Baylor

• Alternative Digital Tool: Random.org

  • Random.org uses atmospheric noise to generate true random numbers.
  • Procedure:
    1. Navigate to Random.org and select the Integer Set Generator.
    2. Set configuration: $1$ set of $5$ unique integers ranging from $1$ to $30$
    3. Optional formatting: Sort results in ascending order, separated by commas.
    4. Click Get Sets to yield the simple random sample.

Stratified Sampling

• Definition & Process:

  • A stratified sample is obtained by separating the entire population into non-overlapping, mutually exclusive groups called strata (singular: stratum), and then obtaining a simple random sample from each stratum.
  • Individuals within each stratum must share specific characteristics (e.g., residency status, political party affiliation).
  • No single individual in the population can belong to more than one stratum.

• Example 3: DePaul University Campus Safety Survey

  • Population Size ($N$): $N = 21{,}909$ individuals across the entire university community.
  • Strata Identification & Proportions:
    1. Resident Students: $6{,}204$ individuals ($\approx 28\%$ of total population)
    2. Non-Resident Students: $13{,}304$ individuals ($\approx 61\%$ of total population)
    3. Staff Members: $2{,}401$ individuals ($\approx 11\%$ of total population)
  • Target Sample Size ($n$): $n = 100$ individuals.
  • Proportional Sample Allocation:
    • To ensure proportional representation matching the population structure:
      • Resident Students: Select $28$ individuals via simple random sampling from the $6{,}204$ residents.
      • Non-Resident Students: Select $61$ individuals via simple random sampling from the $13{,}304$ non-residents.
      • Staff Members: Select $11$ individuals via simple random sampling from the $2{,}401$ staff.
    • Total Sample Size: $28 + 61 + 11 = 100$

• Theoretical Model: National Political Parties

  • Target Population: Everyone in America.
  • Strata: Republicans, Democrats, Independents.
  • Execution: Non-overlapping groups are formed based on political affiliation. A separate simple random sample is conducted within each of the three strata to compile the full sample.

Systematic Sampling

• Definition & Use Case:

  • A systematic sample is obtained by selecting every $k$\text{-th} individual from the population.
  • Systematic sampling is particularly useful when a frame (a full list of the population) is impossible or impractical to construct (e.g., surveying all shoppers leaving a store or all $350\times 10^6$ residents of a nation).

• Procedure:

  1. Determine the desired sample size $n$.
  2. Determine or approximate the population size $N$ (if known), or set a constant sampling interval $k$
  3. Randomly select a starting integer $p$ between $1$ and $k$
  4. The sample will consist of the following sequence of individuals:         $p, \, p+k, \, p+2k, \, p+3k, \, \dots, \, p+(n-1)k$

• Example 4: Kroger Food Store Customer Satisfaction

  • Goal: Obtain a sample of $n = 40$ customers.
  • Problem: Constructing a complete frame of every customer who enters Kroger is impossible.
  • Systematic Scheme: Survey every $7$\text{-th} customer ($k = 7$) exiting the store.
  • Random Seed/Start: Randomly pick a starting number between $1$ and $7$, yielding $p = 5$
  • Selected Customer Sequence:
    • $1$\text{st} selected customer: $5$\text{th} customer
    • $2$\text{nd} selected customer: $5 + 7 = 12$\text{th} customer
    • $3$\text{rd} selected customer: $12 + 7 = 19$\text{th} customer
    • $\dots$
    • Final ($40$\text{th}) selected customer: $5 + (40-1)\times 7 = 278$\text{th} customer
  • Validity: The process remains random and free from selection bias because the starting point is chosen randomly and human selection bias is eliminated by mathematical rule.

Cluster Sampling

• Definition & Process:

  • A cluster sample is obtained by selecting all individuals within a randomly selected collection or subset of groups (clusters).
  • Step 1: Divide the target population into naturally occurring or defined groups called clusters.
  • Step 2: Obtain a simple random sample of the clusters themselves.
  • Step 3: Survey/interview every single individual contained inside the selected clusters.

• Comparison: Stratified Sampling vs. Cluster Sampling:

  • Stratified Sampling: Divide population into groups -> Sample some individuals from all groups.
  • Cluster Sampling: Divide population into groups -> Sample all individuals from some groups.

• Example 5: Boston Household Income Survey

  • Target Population: Households within the city of Boston.
  • Clusters: City blocks (Total blocks $N = 10{,}493$).
  • Target Cluster Sample Size: $20$ clusters (city blocks).
  • Procedure:
    1. Number all city blocks sequentially from $1$ to $10{,}493$
    2. Use a simple random sample generator to pick $20$ block numbers between $1$ and $10{,}493$
    3. Survey every household located on those $20$ selected city blocks.
  • Primary Advantage: Drastically reduces field travel time, logistical complexity, and research expenditure compared to simple random sampling or stratified sampling across an expansive geographical city.

• Example 6: Ball State University (BSU) Major-Based Clusters

  • Target Population: All students at Ball State University.
  • Clusters: Academic majors (e.g., Biology, Mathematics, Chemistry, Exercise Physiology, Fine Arts, etc.).
  • Procedure:
    1. Treat each academic major as a distinct cluster.
    2. Select a simple random sample of majors (e.g., randomly selecting Chemistry and Exercise Physiology).
    3. Survey every single student enrolled within those selected majors.

• Pitfalls and Limitations of Cluster Sampling:

  • Selection Bias / Cluster Homogeneity: Clusters may not be representative of the broader population structure.
  • Illustration: If only Chemistry and Exercise Physiology majors are chosen, the survey completely misses the perspectives of Fine Arts or Mathematics students (e.g., evaluations of specific faculty members, course workload experiences, or preference traits). If those two sampled clusters happen to share atypical traits, those traits will be heavily over-represented, distorting overall population estimates.

Non-Random Sampling Methods: Convenience Sampling

• Definition:
  • Convenience sampling is a non-probability sampling technique where subjects are selected based on immediate availability, physical accessibility, or voluntary participation, rather than random chance.
• Example 7: Sports Podcast Listener Survey
  • Scenario: A podcast host discussing sports requests that listeners voluntarily visit a website to complete an opinion survey.
  • Methodological Flaw:
    • Self-Selection Bias: The sample consists exclusively of individuals who already listen to that specific podcast, possess an active interest in sports, and are sufficiently motivated to fill out an unpaid voluntary survey.
    • Underrepresentation: Excludes all individuals in the general public who do not listen to the podcast or do not care about sports.
• Conclusion on Sampling Validity:
  • Convenience samples yield biased estimates and cannot be generalized to broader populations.
  • Probability/random sampling methods (Simple Random, Systematic, Stratified, Cluster) are essential for valid statistical inference.

Questions & Discussion

• Question: How do we categorize a sampling strategy where the population is split into groups like resident students, non-resident students, and staff before drawing samples?
  • Response: This is stratified sampling. The total population is partitioned into non-overlapping groups (strata), and simple random sampling is executed inside each individual group proportional to its size.
• Question: What should be done on a TI-84 calculator if the menu option randIntNoRep is missing?
  • Response: On older calculator models, use the standard command randInt(lower, upper) located under MATH -> PRB -> 5: randInt(. Execute the command repeatedly to generate individual random numbers, manually discarding and skipping any duplicate numbers until the required sample size $n$ is reached.
• Question: What is the critical risk when using academic majors as clusters in cluster sampling?
  • Response: The primary risk is that individual clusters are often internally homogeneous and may fail to reflect the diversity of unselected groups. For example, Chemistry majors might have vastly different course requirements, professor evaluations, or preferences compared to Fine Arts majors. Sampling only a few clusters risks severe over-representation of those specific majors' traits while entirely omitting other demographic groups.
• Question: Why are voluntary online podcast surveys considered invalid for making general claims?
  • Response: They rely on convenience sampling. Participation is self-selected by motivated listeners interested in that specific topic (e.g., sports), leaving non-listeners and non-sports fans completely unrepresented.