Understanding Populations, Sampling & Data Collection

Terminology & Key Definitions

Population: Entire set of observations or measurements under study; often very large or even infinite.
- Examples
- Australian Census counts $\approx 23.64\text{ million}$ residents (finite).
- Number of fish in Brisbane River (conceptually infinite).
Sample: A subset selected from the population; always smaller.
- Examples
- $100$ Australians drawn from the full population.
- Today’s catch of a single fisherman.
Census: Investigation of every element of a population.
Parameter (population characteristic)
- Symbols & meaning
- $\mu$ – population mean
- $\sigma$ – population standard deviation
- $p$ – population proportion
- Examples
- Average annual income of all Australians.
- Proportion of defective shoes produced in a factory.
Statistic (sample descriptive measure)
- Symbols & meaning
- $\bar X$ – sample mean
- $s$ – sample standard deviation
- $\hat p$ – sample proportion
- Examples
- Mean income of $100$ randomly selected Australians.
- Defect rate in $200$ sampled shoes.
Relationship
- Parameters ↔ population; Statistics ↔ sample.

Representation, Inclusion & Exclusion

Researchers must describe the sample and justify why/how it was chosen.
Core concern: Representativeness—degree to which the sample mirrors the population.
Inclusion criteria: Conditions participants must satisfy to enter the study.
Exclusion criteria: Conditions leading to participant omission.
Clear criteria sharpen focus and data quality.

Why Sample Instead of Census?

Large populations often impossible to study fully due to:
- Cost constraints
- Time constraints
- Analytical practicality / accuracy limits
- General impracticality
Well-designed samples allow inference about population parameters from sample statistics.

The Sampling Frame

Probability sampling requires that each population member has equal selection probability.
Necessitates a complete list/map/chart of population—called the sampling frame or working population.
Samples are randomly drawn from this frame.

Probability Sampling

Goal: Statistically representative, findings generalisable to entire population.
Grounded in probability theory.
Main types
1. Simple Random Sampling (SRS)
2. Stratified Random Sampling
3. Cluster Sampling
Proper use yields high precision with small sample fractions.

1. Simple Random Sampling (SRS)

Every population member equally likely to be chosen.
Procedure
- Assign unique numbers to each element.
- Select numbers using random-number table, computer generator, or “names-in-hat.”
When to use
- Only a basic list of names exists, no additional grouping info (e.g., alphabetical list of CDU students).
- No compelling reason for stratification or clustering.

2. Stratified Random Sampling

Population divided into non-overlapping strata; SRS performed within each stratum.
Enables estimates for:
- Whole population
- Each stratum
- Inter-stratum relationships
Example (tourist gender):
- Population composition – $40\%$ male, $60\%$ female.
- Desired $n=200$ ⇒ select $0.4\times200=80$ males & $0.6\times200=120$ females via SRS inside each gender.
Typical strata bases: gender, age bands, occupation categories, etc.

3. Cluster Sampling

Random sample of groups (clusters) rather than individuals.
Useful when
- Complete population list difficult/costly.
- Population geographically dispersed.
Example (Darwin household income)
1. Identify $50$ suburbs → each acts as a cluster.
2. Randomly pick, say, $3$ suburbs.
3. Survey all households within chosen suburbs.

Non-Probability Sampling

Selected to illustrate phenomenon; not statistically representative.
Employed when no full sampling frame exists (e.g., Cosmopolitan magazine readers).
Key techniques
- Judgemental/Purposive: Researcher hand-picks key informants able to contribute rich insight.
- Quota: Fill pre-set category quotas (age, gender, etc.).
- Convenience: Engage easiest accessible participants (shoppers, passers-by).
- Snowball: Each participant recommends the next; continues until sample complete, meeting inclusion criteria.

Errors in Data Collection

Two broad sources:

Sampling Error
Non-Sampling Error (Systematic)

Sampling Error

Difference between sample statistic and population parameter.
Formula: $\text{Sampling Error}=\bar X-\mu \quad\text{or}\quad \mu-\bar X$
Depends on sampling method; reducible by increasing sample size or better design.
Numerical illustration
- Population incomes ${10, 20, 30, 40, 50}$ ( $\$000$ )
- $\mu=30$
- Random sample ${10,30}$ ⇒ $\bar X=20$
- Error $=20-30=-10$ ( $\$10\,000$ under-estimate).

Non-Sampling Errors

Mistakes in design, execution, measurement.
Not mitigated by larger sample size.
Categories & examples
1. Data acquisition errors – mis-recording, misinterpretation.
2. Non-response error – sampled units unavailable/refuse.
3. Response bias – respondents slant answers consciously/unconsciously.
4. Selection error – some population members never reach sampling frame (e.g., no telephone).
Generally more serious than sampling error because persistence unaffected by sample size.

Data Collection Methods Overview

Data = backbone of statistical analysis; reliability & accuracy hinge on collection method.
Two broad approaches
- Census – whole population.
- Sample survey – subset.
Most common: Survey using a questionnaire.

Survey Administration Modes

Interviewer-Administered
- Personal interview
- Telephone interview
Self-Administered
- Postal (mail)
- Delivery & collection (drop-off/pick-up)
- Online / electronic
Direct Observation

Response Rates

Definition: Count of valid, properly completed responses.
Higher rates ⇒ more complete data, stronger generalisability.
Researchers actively pursue strategies (follow-ups, incentives) to maximise responses.

Personal Interview (Face-to-Face)

Advantages
- Highest potential response rate (often $100\%$ ).
- Interviewer clarifies questions ⇒ fewer errors; high data quality.
Disadvantages
- Expensive (travel, personnel).
- Poorly trained interviewer may bias or mis-record data.
- Hard to access certain populations (e.g., street kids).
Remedy: Rigorous interviewer training; standardised protocols.

Telephone Interview

Advantages
- Cheaper & faster than face-to-face.
- Reaches wide geography; typical response $\approx80\%$ .
Disadvantages
- Slightly lower response rate than personal.
- Coverage limited to phone owners.
- Same interviewer bias risk.
Remedies
- Train interviewers, offer incentives, call at optimal times.

Self-Administered Surveys

Types: Electronic/online, postal, delivery & collection.
Advantages
- Least expensive per respondent; feasible for large, dispersed samples.
- Rapid turnaround (especially online).
Disadvantages
- Generally low response rates.
- Higher mis-understanding risk causing incorrect answers.
- Cost escalates if multiple follow-ups needed.
Remedies
- Drop-off & pick-up strategy to boost engagement & clarify queries.
- Incentives; prepaid return envelopes; reminder contacts.

Direct Observation

Collect data by observing individuals/units directly (e.g., health conditions of Darwin residents).
No control over influencing factors; relatively low cost.

Ethical & Practical Implications

Transparency in sampling explanation safeguards study credibility.
Inclusion/exclusion criteria prevent unethical participant selection and enhance focus.
Non-probability sampling demands honesty about limited generalisability.
Minimising non-sampling errors (through training, pilot tests, robust instrument design) is ethically critical to avoid misinformation.

Mathematical & Statistical Connections

Sampling theory underpins inferential statistics (estimating $\mu, \sigma, p$ via $\bar X, s, \hat p$ ).
Stratified design improves estimator precision, analogous to weighted means where stratum weights correspond to population proportions.
Cluster sampling relates to multi-stage designs and intraclass correlation considerations in variance estimation.
Error taxonomy aligns with total survey error framework used in large-scale official statistics (e.g., ABS Census quality reports).