Why Divide a Population Into Strata?
A single random sample can, by chance, include relatively few people from a subgroup that matters to a study. If a university wants to describe students’ experiences across different years of study, for example, it may be important to include students from every year rather than leave representation to chance. Stratified sampling addresses this by dividing the population into groups first and then sampling from every group.
The method depends on careful group definitions. Each population member must fit in exactly one group, and every member must fit somewhere. After the groups are defined, selection within each group must follow a stated random procedure if the result is to be a probability sample. A subgroup that is merely listed in a report, or a group from which no one is sampled, does not satisfy the defining idea of stratified sampling.
Define the Strata Before Drawing the Sample
Let the population contain \(N\) individuals and be divided into \(H\) strata. Write \(N_h\) for the number of population members in stratum \(h\), where \(h=1,\ldots,H\). To be a valid partition, the strata must satisfy two conditions:
- Nonoverlap: no individual belongs to more than one stratum.
- Exhaustiveness: every individual in the target population belongs to a stratum.
Thus, the stratum sizes add to the population size:
The strata should be defined using information that can be determined for each member of the target population and recorded in the sampling frame. For age strata, specify exact boundaries so there are no gaps or overlaps. For campus-year strata, specify how students with an unusual status, such as a leave of absence, will be classified if they are part of the target population. If the frame does not identify an individual’s stratum, the intended selection procedure cannot be carried out reliably for that person.
Once the strata are defined, decide how many individuals to draw from each one. Let \(n_h\) denote the number selected from stratum \(h\). A stratified design samples from every stratum, so \(n_h\) must be at least one for each stratum included in the design and cannot exceed \(N_h\). The total sample size is the sum of the stratum sample sizes:
This tutorial focuses on forming the groups and carrying out the separate random draws. The values of \(n_h\) must be specified as part of the plan; there is more than one possible choice, and the right choice depends on the study’s aims and constraints.
How to Draw a Stratified Sample
For a straightforward probability design, list the eligible members of each stratum and draw a simple random sample without replacement within that list. “Without replacement” means that a selected individual cannot be selected again in the same draw. The random selection is carried out separately for each stratum; choosing names informally or taking whoever is easiest to reach would not be a simple random draw.
Specify who is eligible and the time or setting covered. Use a frame that identifies the eligible population members and provides the information needed to assign them to strata.
Write definitions that assign every eligible individual to exactly one stratum. Check boundary cases and any categories that might otherwise be left out.
Choose \(n_h\) for each stratum, with \(1\leq n_h\leq N_h\), and calculate the total \(n\) by adding the stratum sample sizes.
Use a genuine random mechanism to select the specified number from each stratum’s list, without replacement. Combine the selected individuals into the final sample and retain their stratum labels.
Keeping the stratum labels with the selected records makes it possible to verify that every stratum contributed a sample and that the selected individuals really came from the intended groups. It also helps prevent a common operational error: drawing one overall random sample from the combined list and assuming that this is equivalent to sampling within every stratum.
Worked Example: Sampling Within Age Strata
Suppose a hypothetical city survey targets 2,400 adults on a complete frame. The study plan defines four age strata: ages 18–29, 30–44, 45–64, and 65 or older. The frame contains 900, 700, 600, and 200 adults in these groups, respectively. The plan calls for selecting 3, 2, 2, and 1 adults from the groups, respectively. Describe the design and calculate its total sample size.
Check the partition. The age bands do not overlap: for example, age 30 belongs to 30–44, not 18–29. They cover all adults on the frame because every age is either 18–29, 30–44, 45–64, or at least 65. The counts also add to the stated frame size:
Calculate the total sample size. Add the requested draws from the four strata:
Carry out the draws. Assign each adult in each age group a unique frame ID. Use a random-number generator to select 3 distinct IDs from the 900-person list, 2 from the 700-person list, 2 from the 600-person list, and 1 from the 200-person list. For example, if the random selections are IDs A014, A382, and A811 from the first list; B105 and B699 from the second; C031 and C420 from the third; and D176 from the fourth, the final sample contains eight distinct adults and includes at least one from every age stratum. These IDs are illustrative hypothetical selections, not study data.
The design is stratified because selection occurred separately in all four age groups. Drawing eight IDs at random from the combined list instead would be a simple random sample, not this stratified design, and it would not guarantee a sample from each age group.
Selection Probabilities Within Strata
If a simple random sample without replacement of size \(n_h\) is drawn from a stratum of size \(N_h\), every member of that stratum has the same chance of selection. The individual selection probability is \(n_h/N_h\). This follows because all members are treated symmetrically by the random draw and exactly \(n_h\) of the \(N_h\) members are selected. Equivalently, the sum of their individual selection probabilities must equal the fixed number selected, \(n_h\). Since their probabilities are equal, each must be \(n_h/N_h\):
This probability is for an individual in the specified stratum, under the stated random sampling design. It is not necessarily the same across different strata, because the stratum sizes and sample sizes may differ.
Worked Example: Checking Selection Chances by Campus Year
A hypothetical university has a frame of 3,200 students classified by campus year: 1,200 first-year students, 900 second-year students, 700 third-year students, and 400 fourth-year students. A study plan specifies samples of 5, 4, 3, and 2 students from those strata, respectively. Find the total sample size and the selection probability for a student in each year, assuming a simple random sample without replacement within each stratum.
Check the strata and counts. Each student is assigned to one campus year on the frame, and the group sizes sum to the stated population:
Calculate the total.
Calculate the probabilities. For a first-year student, the probability is \(5/1{,}200=0.004167\), or about 0.417%. For a second-year student, it is \(4/900=0.004444\), or about 0.444%. For a third-year student, it is \(3/700=0.004286\), or about 0.429%. For a fourth-year student, it is \(2/400=0.005000\), or 0.500%. For instance, the first-year probability is calculated as:
These probabilities describe selection before any contact or response takes place. The sample has 14 students because the four separate draws contain 5, 4, 3, and 2 students; no one is counted in more than one stratum.
Choosing Useful Strata and Checking a Plan
A stratum is a population group used to organize selection, not simply a subgroup noticed after data collection. A practical stratified plan therefore needs a usable classification for every eligible person before the random draws. It also needs enough people in each frame list to support the planned draw: if a stratum has 40 members, a request to sample 45 from it cannot be carried out without changing the plan.
The defining feature is that the design samples within every stratum. This contrasts with cluster sampling, where natural groups are used differently: a design may select some groups rather than sample individuals from every group. The distinction is about what gets selected and whether all defined groups contribute to the sample.
Stratification does not by itself guarantee a representative sample. A flawed or incomplete frame can omit eligible members, and a random selection can still be affected by nonresponse. Random selection within accurately defined strata supports a probability design, but it cannot repair missing members or ensure that selected people respond.
Worked Example: Repairing an Age-Strata Plan
A hypothetical survey plan defines two age groups as “18–40” and “40–65,” then proposes to sample from both. The target population is adults aged 18 through 65, including those who are exactly 40 or 65. Explain the problem and give a corrected partition.
The original groups overlap at age 40: a 40-year-old fits both definitions. They also omit age 65 if “40–65” is interpreted as stopping before 65, and the plan does not clearly say whether the upper endpoint is included. Consequently, the plan does not unambiguously assign every eligible adult to exactly one group.
A corrected definition is “ages 18–39” and “ages 40–65,” with both endpoints included. These groups do not overlap because no integer age can be both below 40 and at least 40. They are exhaustive for the target population: every age from 18 through 65 is either 18–39 or 40–65. Once the frame assigns each adult to one of these two lists, the study can randomly select its specified number from each list.
Common Mistakes
- Leaving a subgroup out. Calling a plan stratified is not enough if one defined stratum contributes no sample. Draw from every stratum.
- Using overlapping or incomplete definitions. If a person can fit multiple strata or none, the partition is not well defined. State boundaries and handle special cases explicitly.
- Sampling from the combined frame only. One random draw from the whole population may omit a subgroup. Separate random draws within the strata are what guarantee that all strata are represented.
- Choosing people conveniently within a stratum. Taking available or willing individuals does not implement a simple random sample. Use a random selection procedure from the full list for that stratum.
- Assuming stratification removes every source of error. The method organizes selection; it does not automatically fix an incomplete frame or ensure responses from selected individuals.
Check Your Understanding
Answer each question using the definitions and procedures in this tutorial.
- A frame contains 500 students in year one and 300 in year two. A plan selects 10 from year one and 8 from year two using separate random draws. What is the total sample size, and why is this stratified sampling?
- A survey defines age strata as 20–34 and 35–50. For a target population aged 20 through 50, do these definitions overlap or leave a gap? Explain.
- A stratum contains 250 people, and 5 are selected by simple random sampling without replacement. What is each person’s probability of selection?
- A researcher samples 20 people at random from a complete population list, then checks whether each campus year appeared. Is this the same design as drawing a separate sample from every campus year? Why or why not?
- A plan defines three strata but draws samples from only two. Which defining feature of stratified sampling has the plan failed to meet?