Choosing a Design Starts With the Population and the Frame
A sampling design is the rule for selecting individuals from a population. Choosing among designs is not simply a matter of finding the quickest way to draw names: the design should fit the population’s structure and the information the study needs. For example, researchers may need reliable representation from several important subgroups, may have to work through naturally grouped units, or may have a list whose ordering affects selection.
The four designs considered here are simple random sampling, stratified sampling, one-stage cluster sampling, and systematic sampling. Each can be a probability sample: selection uses a chance mechanism, and each population unit has a known, positive chance of selection. The designs differ in what is randomized and how the population is organized before selection.
A useful distinction is whether the study must guarantee that particular subgroups are sampled, whether the population is already arranged in natural groups, and whether the order of the frame may affect the sample. No design can repair a frame that fails to represent the target population; the design choice assumes a suitable frame and a correctly carried-out random selection.
What Each Design Randomizes
In a simple random sample without replacement of size \(n\) from \(N\) individuals, every possible subset of \(n\) individuals has the same chance of selection. Each individual’s inclusion probability is \(n/N\). This design is direct when a complete list is available and there is no particular need to guarantee sample sizes from subgroups.
In stratified sampling, the population is divided into non-overlapping subgroups called strata, and a sample is selected at random separately within every stratum. A stratum might be a year level, region, or type of facility. The design is useful when the study needs each defined subgroup represented, or when estimates for those subgroups matter. If stratum \(h\) has \(N_h\) individuals and \(n_h\) are sampled by simple random sampling, each individual in that stratum has inclusion probability \(n_h/N_h\).
In one-stage cluster sampling, the population is divided into natural groups called clusters, a chance mechanism selects some clusters, and every individual in each selected cluster is included. Schools, clinics, and villages can serve as clusters. This can be practical when reaching scattered individuals is costly but reaching selected groups is manageable. Unlike stratified sampling, which samples individuals within every stratum, one-stage cluster sampling selects only some clusters and includes all their members.
In systematic sampling, individuals are listed in an order, a random starting position is chosen, and selection proceeds at a fixed interval through the frame. When \(N\) is a multiple of the desired sample size \(n\), the interval can be \(k=N/n\): choose one of the first \(k\) positions at random, then take every \(k\)th individual. This is convenient for an ordered list, but the order must be considered carefully. The earlier material on periodicity and ordering explains why a repeating pattern aligned with the interval can make a systematic sample unrepresentative.
A Short Justification for Inclusion Probabilities
Inclusion probabilities clarify what the randomization guarantees—and what it does not. For a simple random sample, symmetry means that each of the \(N\) individuals is equally likely to be selected. Since exactly \(n\) individuals are selected, the sum of their inclusion probabilities is \(n\), so each probability must be \(n/N\). For one-stage cluster sampling with \(C\) clusters, if \(c\) clusters are selected uniformly and every member of each selected cluster is included, each individual is included exactly when their own cluster is selected. Therefore each individual’s inclusion probability is \(c/C\), regardless of the cluster’s size.
The same reasoning also shows why a cluster sample’s total number of individuals may vary when cluster sizes differ. The identities of selected clusters vary at random, and the number of included individuals is the sum of the sizes of those selected clusters. Equal inclusion probabilities do not imply that every possible sample has the same number of individuals.
Worked Example: A Simple Random Sample
A hypothetical university has a frame listing all 1,200 enrolled students. A researcher uses a random-number procedure to choose 60 students without replacement. Assume the frame includes every student in the target population, and the random selection is implemented as specified. What is each student’s inclusion probability, and does this design guarantee that every year level is represented?
Under simple random sampling, each student has the same inclusion probability:
Thus each student has a 5% chance of inclusion. The design does not guarantee that every year level will appear; a random sample could, by chance, include few or no students from a small year level. If the aim specifically requires representation from every year level, selecting a sample separately within each year level would address that requirement more directly.
Match the Design to the Study’s Structure
When representation from specified subgroups is essential, stratified sampling is often the natural choice: define the strata, then sample within each one. This guarantees that every stratum contributes to the sample if a positive number is drawn from each. It does not automatically guarantee that the sample proportions match the population proportions; the allocation across strata determines that. The design is especially useful when results will be examined separately by subgroup.
When individuals are naturally grouped and data collection is easier within selected groups, consider one-stage cluster sampling. Its practical advantage may be reduced travel or coordination. But cluster sampling and stratified sampling answer different structural needs. Stratification samples from all strata; cluster sampling includes all individuals in only the selected clusters. Whether cluster sampling is more efficient for a particular estimate depends on the population and the costs involved; convenience alone does not establish that it will produce more precise results.
Simple random sampling is a strong straightforward option when a complete frame exists, reaching sampled individuals is feasible, and guaranteed subgroup representation is not required. Systematic sampling is a possible alternative when a usable ordered frame exists and a random start followed by a fixed interval is easy to implement. In either case, the choice should be driven by the study’s purpose and frame—not merely by which method sounds easiest.
Worked Example: Stratifying to Guarantee Representation
A hypothetical college has 200 first-year, 300 second-year, and 500 third-year students on its complete frame. A survey team wants 100 responses from a probability sample and wants each year level represented. Assume the team can identify each student’s year level and will use a random selection within each level. One possible design samples 20 first-year, 30 second-year, and 50 third-year students.
The population and sample counts are:
Within each year level, the inclusion probabilities are:
Each student therefore has a 10% inclusion probability, and the design guarantees that all three year levels are represented. The guarantee comes from sampling within each stratum, not from hoping that an unrestricted random sample happens to include all three. This example uses a proportional allocation: the same fraction is sampled from each year level. Other allocations are possible when the study has a different goal, but the selection rule and resulting inclusion probabilities should be made explicit.
Worked Example: One-Stage Cluster Sampling With Unequal Cluster Sizes
A hypothetical school district has 48 schools and a frame identifying every school. A researcher selects 8 schools uniformly at random and surveys every student in each selected school. Suppose the selected schools happen to have 180, 220, 160, 240, 200, 190, 210, and 200 students. Assume the 48 schools together contain all students in the target population, and selection of schools is genuinely uniform.
Each student is included if and only if their school is selected. Therefore, for every student, including one in a larger or smaller school:
The sample size in this particular draw is the sum of the selected schools’ student counts:
This design gives every student the same inclusion probability even though the school sizes differ. However, the sample size depends on which schools are selected: a different set of eight schools could contain more or fewer than 1,600 students. Equal-probability selection of clusters does not mean equal numbers of individuals are taken from each cluster; in one-stage cluster sampling, all individuals in selected clusters are included.
Systematic Sampling and Frame Order
Systematic sampling can be efficient when a frame is in a suitable order and the interval is easy to apply consistently. The random start matters: selecting every \(k\)th individual without randomizing the initial position is not the stated probability design. If \(N\) is a multiple of \(n\), there are \(k=N/n\) possible starting positions in the first interval, and each yields \(n\) selected positions. Each frame position appears in exactly one of those possible samples, so its inclusion probability is \(1/k=n/N\).
Worked Example: Selecting Every Twelfth Student
A hypothetical training program has a complete ordered frame of 1,200 participants. A researcher wants 100 of them and will use systematic sampling. Assume each participant appears once in the frame and the order has been checked for a repeating pattern relevant to the variable of interest.
The interval is:
The researcher randomly chooses a starting position from 1 through 12. If the random start is 7, the selected positions are 7, 19, 31, and so on, ending at 1,195. There are 100 selected positions because the sequence is \(7+12j\) for \(j=0,1,\ldots,99\); the last is \(7+12(99)=1195\), and the next would exceed 1,200. Each position has inclusion probability \(1/12=100/1200\), provided the start is chosen uniformly from those 12 positions.
The frame order still matters. For instance, if a relevant characteristic repeats every 12 positions, the selected people could repeatedly occupy the same part of that pattern. A random start alone does not eliminate the risk that the interval and a repeating order interact.
Comparison and Common Mistakes
| Design | What is selected at random? | Best reason to consider it | Key check |
|---|---|---|---|
| Simple random | Individuals from the full frame | A direct random sample is practical | Subgroups are not guaranteed to appear |
| Stratified | Individuals within every stratum | Specified subgroups must be represented | Define strata and state the allocation |
| One-stage cluster | Clusters; include all individuals in selected clusters | Natural groups make fieldwork more manageable | Sample size can vary with cluster sizes |
| Systematic | A random start, then every \(k\)th frame position | A suitable ordered frame makes regular selection practical | Check for a repeating pattern aligned with the interval |
A frequent mistake is to assume that equal-probability selection of unequal-sized clusters gives students in larger schools a lower inclusion probability. In one-stage cluster sampling, every student in a selected school is included, so a student’s inclusion chance is exactly the chance that their school is selected. If 8 of 48 schools are selected uniformly, every student has inclusion probability \(8/48\), whatever their school’s size. Unequal school sizes can make the number of sampled students vary; they do not by themselves change individual inclusion probabilities.
Another common mistake is to treat the four methods as interchangeable ways of “taking a random sample.” They randomize different units and guarantee different things. Stratification guarantees representation from each sampled stratum; cluster sampling includes all members of selected groups; systematic sampling depends on a randomized start and frame order; and simple random sampling gives every fixed-size subset the same selection chance. State what is selected, what is included, and why the design fits the study before drawing conclusions from the resulting sample.
Check Your Understanding
For each situation, identify a suitable design and explain the feature that drives your choice.
- A researcher has a complete list of 900 patients and needs a straightforward random sample of 45, with no requirement to guarantee representation by clinic. Which design could be used, and what is each patient’s inclusion probability?
- A survey must include students from each of four class years, and the registrar can identify each student’s year. Which design most directly guarantees this?
- A region has 30 community centers. A researcher randomly selects 5 centers and includes every eligible attendee at each selected center. What kind of design is this?
- A list contains 2,400 entries, and a sample of 200 is needed using a random start and a fixed interval. Calculate the interval and name one feature of the frame to check before selection.
- In one-stage cluster sampling, 6 of 36 clinics are selected uniformly and all patients in selected clinics are included. Are patients at larger clinics more likely to be included? Explain.