Simple Random Sampling

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The Purest Form of Probability Sampling

Simple Random Sampling (SRS) is the most fundamental probability sampling method, serving as the conceptual baseline against which all other sampling techniques are compared. In SRS, every member of the population has an equal and independent chance of being selected, and every possible sample of size n has the same probability of being chosen. It is the statistical equivalent of drawing names from a perfectly shuffled deck — pure chance determines who is included.

How It Works

The process is straightforward in concept:

1. Define the population: All 5,000 employees at a company, all 12,000 registered voters in a constituency, all 800 patients treated at a clinic last year.
2. Obtain a complete sampling frame: A numbered list of every population member.
3. Determine sample size: Using appropriate formulas based on desired precision and confidence level.
4. Select randomly: Using random number tables or computer-generated random numbers, select n individuals from the list.

Random Selection Methods

Random number tables: Historical method using published tables of random digits. To select from a population of 5,000 (4-digit numbers), read consecutive 4-digit numbers from the table, selecting those between 0001 and 5000.

Computer-generated random numbers: Modern standard. Software generates random numbers corresponding to population member IDs. In Excel: =RANDBETWEEN(1,5000). In Python: random.sample(range(1,5001), n). In SPSS: Data > Select Cases > Random Sample.

Lottery method: Physical method suitable for small populations. Write each member's identifier on identical pieces of paper, mix thoroughly in a container, and draw n pieces without looking. Only practical for small populations where physical mixing can be truly random.

Mathematical Properties

The beauty of SRS lies in its mathematical tractability:

Probability of selection: Each individual has probability = n/N of being included (where n = sample size, N = population size).

Expected value: The expected value of the sample mean equals the population mean (the estimator is unbiased).

Standard error of the mean: SE = σ/√n × √((N-n)/N), where σ is population standard deviation and (N-n)/N is the finite population correction factor.

For large populations (where N >> n): SE ≈ σ/√n, showing that precision depends on absolute sample size, not the proportion of the population sampled.

Practical Example

A researcher wants to study job satisfaction among 3,000 nurses registered with the state nursing council.

1. Sampling frame: Council register with nurse IDs 0001 to 3000.
2. Sample size calculation: For 95% confidence, 5% margin of error, assuming maximum variability: n = (1.96² × 0.5 × 0.5) / 0.05² = 384. With finite population correction: n_adj = 384 / (1 + 383/3000) = 384 / 1.128 = 340.
3. Selection: Computer generates 340 random numbers between 1 and 3000. These identify the nurses to be surveyed.
4. No substitution: If nurse #1847 cannot be reached, you do not replace them with nurse #1848. You either persist in contacting them or document non-response.

Advantages

Simplicity: The concept is easy to understand and explain. No complex stratification or clustering required.

Unbiased estimates: Mathematical theory guarantees that sample statistics are unbiased estimators of population parameters.

Easy calculation of sampling error: Standard formulas apply directly without complex design effect adjustments.

Foundation for theory: All sampling theory is built on SRS assumptions. Understanding SRS provides the baseline for understanding more complex designs.

No prior knowledge required: Unlike stratified sampling, you do not need to know population characteristics in advance to implement SRS.

Limitations

Requires complete sampling frame: You need a list of every population member. For many populations (all smartphone users in India, all people with diabetes, all self-employed workers), no such list exists.

May miss minority subgroups: If your population is 95% urban and 5% rural, a random sample of 200 might include only 10 rural respondents — too few for meaningful subgroup analysis.

Not the most efficient: For a given sample size, stratified sampling often produces more precise estimates than SRS because it ensures representation of key subgroups.

Geographically dispersed samples: Random selection across a large geographic area may produce a sample scattered across hundreds of locations, making data collection expensive and logistically challenging.

Large samples sometimes needed: Without the efficiency gains of stratification, SRS may require larger samples to achieve the same precision.

SRS With and Without Replacement

With replacement: After selecting a member, they are returned to the pool and could be selected again. Each draw is independent. Probability of selection remains constant (n/N for each draw). Used in theoretical derivations but rarely in practice (you would not survey the same person twice).

Without replacement: Once selected, a member cannot be chosen again. The standard approach in practice. Technically, probabilities change slightly with each draw, but for large populations the difference is negligible.

Verifying Randomness

How do you confirm your sample is actually random? Check whether sample characteristics roughly match known population characteristics. If the population is 60% female and your sample is 58% female, randomization likely worked. If your sample is 85% female, something went wrong — perhaps a non-random element entered the selection process.

Conclusion

Simple random sampling remains the theoretical ideal in probability sampling — the standard against which other methods are evaluated. While practical constraints often make more complex sampling designs necessary, understanding SRS is essential because it provides the mathematical foundation for sampling theory. When a complete sampling frame exists and the population is relatively homogeneous, SRS offers the simplest path to unbiased, generalizable research findings.