From Random Assignment to a Complete Design
In Why Random Assignment Is the Key to Causation, you learned why assigning experimental units by chance helps make treatment groups comparable, on average. The next step is to describe the whole experiment clearly: what receives a treatment, what the treatments are, how the assignment happens, and what response will be measured.
A common design for making that comparison is a completely randomized design. It is called “completely” randomized because all experimental units are assigned to the treatments through a chance process, without first sorting them into groups such as age categories, greenhouse locations, or experience levels and assigning separately within those groups.
The design starts with units, not with a chart or a random number generator. As explained in Experimental Units, Factors, and Treatments, an experimental unit is the individual or object that receives an assigned treatment. Treatments are the specific conditions imposed in the experiment. In Explanatory and Response Variables in Experiments, you learned to name the explanatory variable and the response variable. A clear design identifies all of these before assignment begins.
For example, if a researcher assigns a watering schedule to each plant, the plants are the experimental units. The watering schedules are the treatments, the explanatory variable is assigned watering schedule, and the response might be plant growth after a fixed number of days. If each unit is a person, the units are often called subjects.
Plan the Assignment Before Drawing the Chart
A useful design description answers four practical questions. First, which units are eligible and available for the experiment? Second, what are the treatments, stated precisely enough that they can be applied consistently? Third, how many units will receive each treatment, and how will chance determine the assignments? Fourth, what response will be measured, when will it be measured, and in what units?
Specifying group sizes makes an assignment plan understandable. If there are 36 plants and two treatments, a researcher might randomly assign 18 plants to each treatment. Equal group sizes are often convenient for comparing treatments, but they are not required for a completely randomized design. For example, a researcher could assign 20 units to one treatment and 16 to another, as long as the chance process assigns the units to the planned groups.
A random assignment is not the same thing as letting units choose a treatment, alternating between treatments, or assigning treatments based on a unit’s characteristics. Those rules can create systematic group differences. A chance process determines which unit goes to which treatment group. The next tutorial will show ways to carry out that assignment with a number table or calculator; for now, focus on stating the plan.
Keeping other conditions consistent is part of carrying out the experiment carefully. If plants in one treatment receive more sunlight, or participants in one group are measured with a different instrument, the treatment comparison becomes harder to interpret. Consistency does not mean that all units are identical; it means that researchers plan to avoid changing additional conditions in ways that systematically favor one treatment group.
Do not confuse a control group with control of other conditions. A control group receives a comparison treatment, such as the usual practice, a placebo, or no active treatment. Controlling other conditions means holding features of the experiment steady across groups when possible. A study may include a control group and also standardize its procedures.
Reading a Flow Chart of an Experiment
A flow chart makes the sequence visible. It should show the experimental units entering the study, their random assignment to treatment groups, the treatments each group receives, and the response measurement. The final comparison should name what researchers will compare, such as group means or proportions, depending on the response.
A flow chart is not just a picture of two groups. It communicates the logic of the design. A reader should be able to tell that every unit is assigned to a treatment, that assignment is by chance, and that the response is measured after the treatments are applied. If the chart leaves out a group, a treatment, or the response, the design is incomplete.
Begin with the full set of eligible experimental units and state how many there are.
Use the planned random process to allocate units to the specified treatment groups.
Each group receives its assigned treatment under consistent procedures.
Measure the response for each unit, then compare the response results across treatment groups.
The flow chart can be read from top to bottom as one sequence, with the assignment step branching into treatment groups. The branches should rejoin conceptually at the response measurement and comparison: each group has its response recorded using the same definition and method. This common measurement plan makes the comparison meaningful.
A completely randomized design does not mean that the units are selected at random from a larger population. As discussed in Random Assignment Is Not Random Selection, random selection concerns who enters a study; random assignment concerns which treatment study units receive. A study can use random assignment without random selection. Its assignment can support cause-and-effect reasoning about the treatments, but assignment alone does not make the participants representative of a broader population.
Worked Example: Comparing Watering Schedules
Worked Example: A Completely Randomized Plant Experiment
A fictional greenhouse team has 36 similar seedlings. It wants to compare watering once each day with watering every other day. The team will measure each seedling’s height increase, in centimeters, after 21 days.
State: The experimental units are the 36 seedlings. The factor is watering schedule, with two treatment levels: daily watering and watering every other day. The response variable is height increase in centimeters over 21 days.
Plan: The team will assign 18 seedlings to each schedule by a chance process. Each seedling receives only its assigned schedule. The team will keep the soil type, pot size, light conditions, and 21-day measurement period as consistent as possible. The response will be measured the same way for every seedling.
Do—flow chart:
All eligible seedlings enter the experiment.
Assignment is by chance, not by seedling size or greenhouse location.
One group is watered daily; the other is watered every other day.
Record each increase in centimeters and compare the groups’ results.
Check the design: All 36 units are included in the assignment plan, the treatment levels are explicit, and the response and measurement period are defined. The design is completely randomized because the team assigns seedlings directly to treatment groups by chance rather than forming categories and randomizing separately within them.
Conclude about the plan: This design can provide a fair comparison of the two watering schedules for these seedlings. If the observed growth differs, random assignment supports a cause-and-effect interpretation, assuming the experiment is carried out as planned. Because the seedlings were not described as randomly selected from a broader population, the design alone does not justify generalizing the results to all seedlings.
Worked Example: Three Study-Interface Treatments
Worked Example: Designing a Comparison for Students
A fictional research team wants to compare three versions of a study website. It has 24 students who agree to participate. Each student will complete the same set of practice problems using one version. The response is the number of minutes needed to finish the set.
State: The students are the experimental units. The factor is website version, with versions A, B, and C as the three treatments. The response is completion time in minutes.
Plan: The team will randomly assign eight students to each website version. All students will receive the same instructions, practice problems, and time-measurement procedure. The team will record completion time for each student and compare the three groups.
Do—flow chart: The 24 students lead to a chance-based assignment with three branches: eight students receive version A, eight receive version B, and eight receive version C. After using the assigned website to complete the same problems, each student’s completion time is recorded in minutes. The team then compares the times across the three treatment groups.
Check the design: There are three treatments, so the diagram needs three treatment branches—not just “website” as a single condition. The equal allocation accounts for all participants: \(8+8+8=24\). Each student receives one version, and the common task and measurement plan help make the response comparison consistent.
Conclude about the plan: This is a completely randomized design because chance assigns all 24 students directly to one of the three website versions. If one group has a different typical completion time, the random assignment supports examining whether the website version caused a difference for these participants. It does not, by itself, establish that the same result would occur for all students.
Worked Example: Comparing Detergents on Fabric
Worked Example: Assigning Treatments to Swatches
A fictional testing team has 30 identical fabric swatches with the same kind of stain. It wants to compare three detergents. The response will be a stain-removal score from 0 to 10, recorded after each swatch is washed using a fixed procedure.
State: Each fabric swatch is an experimental unit. The factor is detergent, with detergents A, B, and C as the treatment levels. The response is the stain-removal score.
Plan: Assign ten swatches to each detergent by chance. Use the same washing machine settings, water temperature, wash time, and scoring procedure for all three groups. The team should not assign the most heavily stained swatches to one detergent on purpose; random assignment avoids that systematic choice.
Do—flow chart: Start with 30 swatches, randomly assign ten to detergent A, ten to detergent B, and ten to detergent C, apply each assigned detergent using the common washing procedure, and record a score for every swatch. Finally, compare the stain-removal scores across the three groups.
Check the design: The treatment group sizes account for all units because \(10+10+10=30\). The treatments and response are defined, and the procedure holds important washing and scoring conditions steady. Repeated swatches within each treatment provide more than one response for that treatment; the comparison does not rely on just one swatch per detergent.
Conclude about the plan: The design is completely randomized because all swatches are assigned to detergent groups by chance, with no preliminary grouping or separate assignment within categories. If the treatment groups have different scores, random assignment supports a causal comparison of detergents under these testing conditions. The conclusion should stay within the context of the swatches and washing procedure studied.
Common Mistakes and AP Exam Tips
- Listing treatments but not the factor. Name both: for example, the factor is watering schedule, and the treatment levels are daily and every-other-day watering.
- Calling the response a treatment. A treatment is an assigned condition. A response is what is measured afterward, such as growth in centimeters or completion time in minutes.
- Leaving the assignment vague. “The researchers divide the units into groups” does not say how. State that the units are assigned to the groups by chance and, when useful, give the number assigned to each treatment.
- Making the flow chart stop at assignment. A complete diagram also shows the treatments being applied and the response measured. It should end with a comparison of the response across treatment groups.
- Assuming equal group sizes are required. Equal sizes are often practical, but a completely randomized design can use planned unequal group sizes. The essential feature is that the allocation to treatments is by chance.
- Confusing a completely randomized design with blocking. In a completely randomized design, units are assigned directly to treatments. If researchers first make groups based on a characteristic and then randomize within each group, they have added a blocking structure rather than using a completely randomized design alone.
- Claiming random assignment guarantees identical groups. It does not. Say that chance helps make the groups comparable on average and that chance imbalances can remain in the particular experiment.
- Claiming random assignment makes the results representative. Random assignment helps with treatment comparisons; random selection is the chance process relevant to generalizing to a population. Keep those claims separate.
For a full-credit design description, make the chain explicit: identify the units, name the factor and treatment levels, describe the chance-based allocation, state how the response will be measured, and show those parts in the flow chart. Use the study context throughout, and do not claim that the design guarantees balance or supports broader generalization without the appropriate sampling plan.
Check Your Understanding
For each situation, identify the design elements and explain what a complete flow chart should show.
- A team assigns 40 seedlings by chance to two fertilizers, 20 per fertilizer, and measures height increase after four weeks. Identify the units, factor, treatment levels, and response.
- A study has 27 participants and compares three practice routines, assigning nine participants to each routine by chance. Describe the branches and final comparison in a flow chart.
- Researchers let each participant choose one of two exercise plans. Is this a completely randomized design? Explain what is missing from the assignment.
- A team first sorts plants by greenhouse shelf, then randomly assigns treatments separately within each shelf. Is the plan completely randomized without any additional structure? Explain the distinction.
- A randomized experiment uses volunteers to compare two treatments. Explain why random assignment does not, by itself, show that the volunteers represent all people who might use the treatments.