Turning a Design Into an Actual Assignment
In Completely Randomized Design, you learned that a completely randomized design assigns all experimental units to treatment groups by chance. This tutorial makes that plan concrete: you will assign 30 subjects to two groups, 15 in each, using a number table or calculator.
The goal is a fair, reproducible procedure—not a particular pattern of group membership. Once the chance process has made the assignment, do not change it because one group seems more convenient or its members seem more alike. Random assignment helps make groups comparable on average, as discussed in Why Random Assignment Is the Key to Causation, but it does not guarantee identical groups.
A Reliable Number-Table Procedure
Suppose the subjects are listed on a roster. Give each one a unique, equal-length label. For 30 subjects, use 01 through 30. The leading zero matters: reading fixed pairs of digits is straightforward when every label has two digits. Before looking at the table, decide where to start and which direction to read. Then read the digits in pairs.
For each pair, apply the same rules in order. Accept a label from 01 through 30 if it has not already been accepted. Skip 00 and any number from 31 through 99, since those labels are not on the roster. Skip a valid label if it is a repeat. Continue until 15 different labels have been accepted. Assign those 15 subjects to Treatment A; assign the remaining 15 to Treatment B.
The stopping rule is important. The goal is 15 different subjects, not 15 pairs of digits. Invalid numbers and repeats do not add a subject to the group. Keep a record of accepted and skipped pairs so another reader can follow the procedure.
A Calculator Procedure
A calculator can generate integers from 1 through 30. Use the calculator’s integer-generating function with inclusive lower and upper bounds of 1 and 30. Process the results in the order generated: accept a label that has not appeared before, skip a repeat, and stop after 15 distinct labels have been accepted. The selected subjects receive Treatment A; everyone else receives Treatment B.
For example, a calculator may display a sequence that contains repeats. Repeats are expected because each generated value is a new chance-based result; a repeat does not mean the procedure has failed. Do not change the bounds to “make it easier,” and do not keep generating values after reaching 15 accepted labels unless you are documenting what comes next. The roster labels and calculator bounds must match.
For either device, complete the assignment before applying treatments. Decide what Treatment A and Treatment B mean in advance, or map the group labels to the treatment names according to a rule chosen before seeing the selected subjects. This avoids using the assignment to favor one treatment.
Worked Example: Selecting Subjects With a Number Table
Worked Example: Two Reminder Formats
A fictional research team will compare two formats for a weekly study reminder. It has 30 volunteer subjects, listed and labeled 01 through 30. The team plans to assign 15 subjects to Format A and 15 to Format B.
State: The 30 volunteers are the subjects. The treatment levels are reminder Format A and reminder Format B. The team will assign 15 subjects to each format by chance.
Plan: Start at a predetermined location in a random number table and read rightward in two-digit groups. Assign the first 15 distinct valid labels to Format A. Skip invalid labels and repeats. Assign every subject not selected for Format A to Format B.
Do: Suppose the successive two-digit groups read are shown below. The running count increases only when a new valid label is accepted.
| Pair read | Decision | Format A count |
|---|---|---|
| 42 | Skip: not a roster label | 0 |
| 07 | Accept | 1 |
| 18 | Accept | 2 |
| 07 | Skip: repeat | 2 |
| 30 | Accept | 3 |
| 00 | Skip: not a roster label | 3 |
| 64 | Skip: not a roster label | 3 |
| 12, 25 | Accept both | 5 |
| 31 | Skip: not a roster label | 5 |
| 04, 19, 22 | Accept all three | 8 |
| 56 | Skip: not a roster label | 8 |
| 09, 14, 28, 03, 17, 26, 11 | Accept all seven; stop at 15 | 15 |
The Format A labels, in the order selected, are 07, 18, 30, 12, 25, 04, 19, 22, 09, 14, 28, 03, 17, 26, and 11. In ordinary numerical order, the same group is 03, 04, 07, 09, 11, 12, 14, 17, 18, 19, 22, 25, 26, 28, and 30. The remaining labels—01, 02, 05, 06, 08, 10, 13, 15, 16, 20, 21, 23, 24, 27, and 29—receive Format B.
Check: Format A has 15 distinct subjects, with no repeated person. Format B contains the other 15 labels, and the groups do not overlap. Together they account for all 30 volunteers.
Conclude about the assignment: This procedure assigns every volunteer to one of the two reminder formats by chance. If the researchers later compare the response across groups, random assignment supports a cause-and-effect comparison for these subjects, provided the experiment is carried out as planned. It does not make these volunteers a random sample of all students.
Worked Example: Selecting Subjects With a Calculator
Worked Example: Comparing Two Stretching Routines
A fictional sports science team has 30 subjects who will each follow one of two stretching routines. The team plans to assign 15 subjects to Routine A and 15 to Routine B. The subjects are labeled 1 through 30.
State: The 30 subjects are the experimental units. The treatment levels are Routine A and Routine B. The planned group sizes are 15 subjects per routine.
Plan: Use a calculator to generate integers from 1 through 30. Read the output in order, accepting a label only the first time it appears. The first 15 distinct labels receive Routine A; the other labels receive Routine B.
Do: Suppose the calculator output begins 8, 24, 8, 3, 17, 29, 11, 6, 20, 14, 1, 25, 19, 4, 27, 12. The second 8 is a repeat, so skip it. The next 15 distinct labels are 8, 24, 3, 17, 29, 11, 6, 20, 14, 1, 25, 19, 4, 27, and 12. Stop when the 15th distinct label, 12, is accepted.
The 15 subjects assigned to Routine A are therefore labels 1, 3, 4, 6, 8, 11, 12, 14, 17, 19, 20, 24, 25, 27, and 29. The remaining subjects—2, 5, 7, 9, 10, 13, 15, 16, 18, 21, 22, 23, 26, 28, and 30—receive Routine B.
Check: The selected labels are all within 1 through 30 and are distinct. There are 15 labels for Routine A and 15 different remaining labels for Routine B, so every subject is assigned once.
Conclude about the assignment: The calculator’s chance process determines group membership; the team does not choose subjects based on their flexibility or athletic experience. Chance can still produce groups that differ on such characteristics in this particular assignment. The team should carry out the assignment it obtained rather than rearranging subjects to make the groups look more balanced.
Worked Example: Turning Labels Into a Group Roster
Worked Example: Two Community Workshop Methods
A fictional community team has 30 subjects participating in a workshop experiment. It will compare two ways to present the same information. The roster is labeled 01 through 30, and the team uses a calculator to generate integers from 1 through 30. The output is 2, 15, 2, 30, 9, 21, 4, 18, 7, 13, 29, 6, 24, 1, 16, 11.
State: The subjects are the 30 experimental units. The factor is presentation method, with Method A and Method B as its treatment levels. The team plans to assign 15 subjects to each method.
Plan: Accept new calculator labels in order for Method A, skip a repeated label, and stop at 15 accepted labels. Match those labels to the roster only after the assignment is complete. All unselected labels go to Method B.
Do: The second 2 is a repeat. The 15 accepted labels are 2, 15, 30, 9, 21, 4, 18, 7, 13, 29, 6, 24, 1, 16, and 11. These subjects receive Method A. The remaining labels—3, 5, 8, 10, 12, 14, 17, 19, 20, 22, 23, 25, 26, 27, and 28—receive Method B.
Check: The accepted list contains 15 unique labels, and the remaining list contains the other 15. Both lists together cover 1 through 30 without overlap. The assignment procedure is complete before the team begins presenting the workshop information.
Conclude about the assignment: The calculator assigned the subjects to methods by chance. Keeping a written record of the output, the skipped repeat, and the final rosters makes the process auditable. The roster labels themselves do not say which subjects are more likely to benefit; they simply connect each selected number to one subject.
Common Mistakes and AP Exam Tips
- Using the wrong label range. If the roster is 01 through 30, a number such as 34 is not a subject. State the label range and skip results outside it.
- Counting repeats as new subjects. A repeated label still refers to the same person. Skip it and continue until 15 distinct subjects have been selected.
- Changing the rule partway through. Do not switch reading direction, alter calculator bounds, or restart because the emerging group seems unexpected. Fix the method before using the device and follow it consistently.
- Assigning some subjects by preference. Choosing the remaining subjects based on convenience or personal characteristics would break the stated chance-based procedure. Everyone not selected for the first group goes to the second group.
- Forgetting to account for all 30 subjects. Check that the two groups each contain 15 unique subjects and together contain every roster label exactly once.
- Confusing assignment with selection. As explained in Simple Random Sample Defined and Why Random Assignment Matters, random selection concerns who enters a study, while random assignment concerns which treatment study subjects receive. Random assignment alone does not make subjects representative of a population.
- Claiming randomization guarantees balance. A chance process helps make the groups comparable on average; it does not ensure the particular groups have the same ages, experiences, or other characteristics.
For a clear AP response, give the labels and device settings, explain exactly what counts as an accepted result, say how repeats or invalid results are handled, and state the stopping rule. Then identify which group gets the selected labels and how the remaining subjects are assigned. Finish by checking that each subject appears in exactly one group.
Check Your Understanding
Use the 30-subject, two-group assignment procedures to answer each question.
- A roster is labeled 01 through 30. A table produces 00, 27, 27, 31, and 05. Which labels are accepted, and why are the others skipped?
- A calculator generates 14, 2, 14, 30, 8, 6, 19, 27, 1, 22, 5, 9, 11, 17, 23, and 10. Identify the first 15 distinct labels and state when to stop.
- After 15 subjects are selected for Treatment A, how should the other 15 be assigned? Explain why selecting them based on a personal characteristic would not follow the plan.
- Why must a student use two-digit groups when labels run from 01 through 30 in a number-table procedure?
- Explain why random assignment of 30 volunteers to two treatments does not, by itself, justify generalizing the results to all people who might use those treatments.