Stratified Random Sampling

Sampling Within Groups

In Selecting an SRS With randInt on a Calculator, we selected individuals at random from one roster. Sometimes the population contains distinct groups that matter to the question. Instead of selecting from the whole roster at once, we can divide the population into groups and select a random sample from every group. These groups are called strata.

For example, a school might want to estimate how many hours students study each week. Grade level could be useful for forming strata if study time tends to differ by grade. The aim is not to choose people who seem typical. The aim is to use a clear grouping rule, then use chance to select individuals within each group.

Definition: In stratified random sampling, divide the population into nonoverlapping groups called strata, then select a random sample from every stratum. The selected individuals from all the strata together make up the sample.

The strata must be defined so that every individual in the population belongs to exactly one stratum. The groups cannot overlap, and together they must cover the population. For example, if grade level defines a school’s strata, each student belongs to one grade, and every grade in the population must be included.

As in Population, Sampling Frame, and Sample, the sampling frame should adequately cover the population of interest. In a stratified design, it must also provide a way to identify which stratum each individual belongs to. Then, as in the earlier tutorials on simple random samples and using randInt, chance—not the sampler’s preference—determines which individuals are selected within each stratum.

How to Plan a Stratified Random Sample

Start with the question and population. Choose a characteristic for grouping that is relevant to the question and can be identified for each individual before sampling. The strata might be grade levels, regions, types of housing, or categories of a service. A useful grouping characteristic often relates to the variable being measured.

Next, decide how many individuals to select from each stratum. One straightforward choice is proportional allocation: select a sample from each stratum in proportion to that stratum’s share of the population. If a stratum contains 30% of the population, proportional allocation assigns it about 30% of the total sample.

\text{Proportional sample size for a stratum} = \text{total sample size} \times \frac{\text{size of that stratum}}{\text{population size}}

The calculated sample sizes should be whole numbers. If rounding is needed, adjust the allocations so they add up to the requested total sample size. After setting the numbers, select an SRS separately within each stratum. Combine those selected individuals to form the final sample.

Define the population and frame.
Identify who is in the population and make sure the sampling frame covers it adequately.

Set the strata.
Choose a grouping rule that puts every population member in exactly one stratum.

Set the sample size for each stratum.
Use proportional allocation for a sample that mirrors the population’s stratum sizes, unless the study has a reason to use another allocation.

Select at random within each stratum.
Use a chance method, such as an SRS procedure, separately on each stratum’s roster.

Combine and describe the sample.
Report that the final sample includes the selected individuals from every stratum.

Worked Example: Sampling Students by Grade

Worked Example: A School Study-Time Survey

A fictional high school has 800 students: 320 first-year students, 240 sophomores, 160 juniors, and 80 seniors. A counselor wants a sample of 80 students to ask about weekly study time. The goal is to include each grade in proportion to its size.

State: The population is all 800 students at the school. The variable of interest is each student’s weekly study time, measured in hours. The sample should contain 80 students.

Plan and check the design: Use grade level to define four strata: first-year students, sophomores, juniors, and seniors. These strata do not overlap, and together they include all 800 students. The school’s roster must identify each student’s grade. Select an SRS from each grade’s roster; do not select students based on who is easiest to contact.

Do: Use proportional allocation to calculate the number selected from each grade:

\begin{aligned} \text{First-year students: }&80\left(\frac{320}{800}\right)=32\\ \text{Sophomores: }&80\left(\frac{240}{800}\right)=24\\ \text{Juniors: }&80\left(\frac{160}{800}\right)=16\\ \text{Seniors: }&80\left(\frac{80}{800}\right)=8 \end{aligned}

The allocations add to \(32+24+16+8=80\). For each grade, make a separate roster, label its students, and use a chance method such as randInt to select the required number without repeats. Combine the four selections.

Conclude: The resulting stratified random sample contains 32 first-year students, 24 sophomores, 16 juniors, and 8 seniors, selected at random within their respective grades. It matches the school’s grade proportions by design. This method can support generalizing to the school’s students if the frame and selection process adequately represent that population.

When Stratification Can Reduce Variability

Sampling variability is the variation in a statistic from one sample to another when samples are selected using the same method, as discussed in Sources of Variability in Collected Data. Stratification can reduce this variability when the groups differ in the variable being measured, but individuals within each stratum tend to be relatively similar on that variable.

The reason is that the sample is guaranteed to include individuals from every stratum. In a single SRS from the whole population, the number selected from each group can vary from sample to sample. If groups have different typical values, changes in the sample’s group composition can cause the overall statistic to shift. Stratification controls how many individuals come from each group, reducing that source of variation.

For instance, suppose a fictional town has 600 households: 450 in apartment buildings and 150 in detached homes. A study measures monthly electricity use. If electricity use tends to be fairly similar among households within each housing type, but the two types tend to have different usage levels, a stratified sample can help stabilize the mix of households in the sample. A proportional sample of 60 households would include 45 apartment households and 15 detached-home households, with random selection within each group.

This benefit is not guaranteed just because a population is divided into groups. If the strata are not related to the measured variable, or if values vary greatly within each stratum, stratification may not reduce variability much. And random sampling does not remove every possible problem: an incomplete frame or substantial nonresponse can still affect the conclusions, as noted in Why Random Selection Matters and Evaluating a Study Description Critically.

Key idea: Stratification is most helpful for reducing sampling variability when individuals within a stratum tend to be similar on the measured variable, while the strata differ from one another. It controls the sample’s representation of each group; it does not guarantee a good result or eliminate bias.

Worked Example: Why Group Proportions Matter

Worked Example: Estimating Wait Time by Appointment Day

A fictional clinic has 400 appointments in a scheduling period: 300 on weekdays and 100 on weekends. A sample of 40 appointments is selected to study wait time. In the selected appointments, the mean wait is 12 minutes for weekdays and 20 minutes for weekends.

Plan: Appointment day is used to define two strata. Both are included, and appointments are selected at random within each stratum. Suppose the clinic selects 20 weekday and 20 weekend appointments. This equal allocation can be useful for comparing the two groups, but it does not match their population proportions: weekdays make up 75% of appointments and weekends 25%.

Do: If the goal is to estimate the overall mean wait time for all appointments, account for the population proportions. Weight each stratum’s sample mean by that stratum’s share of the population:

\left(\frac{300}{400}\right)(12) + \left(\frac{100}{400}\right)(20) = 0.75(12)+0.25(20) = 14\text{ minutes}

Simply averaging the two sample means would give \((12+20)/2=16\) minutes. That calculation gives equal weight to weekdays and weekends, even though the clinic has three times as many weekday appointments. For the overall population estimate, the weighted result of 14 minutes reflects the population’s group proportions.

Conclude: The weighted sample estimate of the mean wait time for all appointments is 14 minutes. The unweighted average of 16 minutes instead treats the two appointment-day groups as equally common. When the sample allocation is disproportionate, use the population group sizes when combining stratum estimates for an overall summary.

Worked Example: Choosing a Useful Grouping

Worked Example: Measuring Travel Time to School

A fictional school district wants to estimate the one-way travel time to school for its students. Its sampling frame lists each student’s school and residential zone. The district is considering strata based on school, zone, or favorite subject.

Reason through the choices: School or residential zone could be useful if travel times tend to differ between schools or zones, while students within a school or zone have more similar travel times. The district would need to define the groups so each student belongs to exactly one stratum and all students are included. It could then select a random sample within every stratum.

Favorite subject is not automatically a useful stratification choice. If it has little relationship to travel time, it may not create more internally similar groups or reduce variability. The district should choose a grouping variable based on what is known about the population and the study question, not merely because the information is available.

Conclude: Stratifying by school or residential zone is reasonable if those characteristics are related to travel time and each stratum can be clearly identified. Whether stratification actually reduces variability depends on the pattern of travel times within and between the groups.

Common Mistakes and AP Exam Tips

Using groups that overlap or leave people out. State that strata are mutually exclusive and collectively include the whole population. Every individual must belong to exactly one stratum.
Choosing a nonrandom sample within a stratum. Selecting the easiest people to reach within each group is not random selection. Describe an SRS or another chance-based method separately within every stratum.
Sampling from only some strata. A stratified sample includes a random selection from every stratum. Selecting one or two groups and leaving the others out is a different design.
Assuming stratification always reduces variability. Explain the relevant pattern: variability may be reduced when values are relatively similar within strata and differ between strata.
Confusing proportional and equal allocation. Proportional allocation makes sample counts reflect population proportions. Equal allocation gives the same number from each stratum, which can be useful for group comparisons but may not match the population composition.
Combining disproportionate samples without considering group sizes. If estimating an overall mean from unequal sampling fractions, account for each stratum’s share of the population. A plain average of stratum sample means can give the wrong overall weighting.
Claiming the design removes bias. Random selection can reduce the chance of selection bias, but it does not fix a poor frame, nonresponse, or measurement problems.

For full-credit communication, name the population and strata, explain that every individual belongs to exactly one stratum, state how many individuals are selected from each, and identify the random method used within each stratum. When asked why stratification may help, connect the reason to similarity within strata and differences between strata—not simply to the fact that the population has groups.

Key takeaway: Define nonoverlapping strata that cover the population, then select at random within every stratum. Stratification can reduce sampling variability when individuals within groups are relatively similar on the measured variable and the groups differ from one another.

Check Your Understanding

Use the ideas about strata, allocation, and variability to answer each question.

A college has 500 first-year students, 300 sophomores, and 200 juniors. Describe proportional allocation for a stratified sample of 50 students.
What two requirements should a set of strata satisfy so that every individual in the population belongs to exactly one stratum?
A researcher samples randomly within each of four regions. Explain what additional feature of the measured variable would make stratification likely to reduce sampling variability.
A population has 80% weekday appointments and 20% weekend appointments. A researcher samples 25 of each. Why might this allocation help compare the groups, and why might a plain average of the two sample means not estimate the overall population mean well?
A researcher divides a population by favorite color before estimating commute time. What should the researcher consider before claiming this stratification reduces variability?

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