Completely Randomized Design in AP Statistics

In AP Statistics, a completely randomized design is an experimental design where treatments are assigned to experimental units completely at random, with no grouping first. Group sizes are often equal but don't have to be, and random assignment is what allows cause-and-effect conclusions.

What is Completely Randomized Design?

A completely randomized design is the simplest way to run an experiment. You take all your experimental units (people, plants, cars, whatever is being studied) and randomly assign each one to a treatment. You don't sort or group them first. Every unit goes into one big pool, and chance alone decides who gets what. The CED (1.13.B.1) puts it plainly: treatments are assigned to experimental units completely at random. Often each treatment gets the same number of units, but that isn't required. Unequal group sizes still count.

Think of it like pulling names out of a single hat. You might number 100 participants 1-100, use a random number generator to pick 50 for the new drug, and give the other 50 a placebo. That randomness is the point. It spreads out any differences between subjects (age, health, habits) roughly evenly across groups. So if the groups end up with different results, you can credit the treatment instead of some lurking variable.

Why Completely Randomized Design matters in AP® Statistics

This term lives in Topic 1.13 Experimental Design in Unit 1: Exploring One-Variable Data and Collecting Data. It hits all four learning objectives in that topic:

• 1.13.B asks you to identify experimental designs. The completely randomized design is the baseline that every other design gets compared to.
• 1.13.A lists the elements of a well-designed experiment. These are comparison of at least two treatment groups, random assignment, replication, and direct control. A completely randomized design is built around the random assignment piece.
• 1.13.C asks you to justify why one design fits better than another. That depends on the study's goals, the population, and the variables involved (1.13.C.1). Sometimes a completely randomized design is perfect. Other times blocking or matched pairs would cut down on variability.
• 1.13.D connects design to conclusions. Random assignment reduces confounding, so a well-run completely randomized experiment supports cause-and-effect claims (1.13.D.1). If the units are volunteers, though, results only generalize to units like those volunteers (1.13.D.2).

The big idea is that design decides what you're allowed to conclude. That idea comes back every time you interpret results later in the course.

How Completely Randomized Design connects across the course

Randomized Block Design (Unit 1)

A randomized block design is a completely randomized design run separately inside each block. You first group similar units by a blocking variable (like soil quality or prior knowledge), then randomize treatments within each group. Blocking removes a known source of variation that a completely randomized design would leave mixed into the results.

Matched Pairs Design (Unit 1)

Matched pairs is blocking taken to the extreme, where each block has just two units (like identical twins) or one unit gets both treatments. Compared to a completely randomized design, it controls for differences between subjects much more tightly. That's why a study using identical twins pairs them up instead of tossing everyone into one hat.

Comparing Two Groups with Inference (Units 3-4)

The design you choose shapes the analysis you run later. A completely randomized design with two treatments produces two independent groups, so you compare them with two-sample procedures for proportions (Unit 3) or means (Unit 4). A matched pairs design produces paired differences, which call for a different, paired analysis.

Statistically Significant Results and Causation (Units 3-4)

When a test finds a statistically significant difference between treatment groups, random assignment is what allows you to say the treatment caused it. Without random assignment, a significant result only shows an association. The completely randomized design you learned in Unit 1 is what justifies the causal language you use in Units 3 and 4.

Is Completely Randomized Design on the AP® Statistics exam?

Multiple choice. Expect stems that describe a study and ask you to identify the design, or to pick the best design for a situation. Common setups include a researcher with 100 participants testing a blood pressure drug and asking how a completely randomized design would assign them. Another type gives you a reason to block, like differences in prior knowledge among students or a north-to-south gradient in soil quality across a field. In those cases the right answer is usually not a completely randomized design, because blocking controls that known source of variation. Read for clues like "known differences," "gradient," or "concerned about" a particular variable.

Free response. Released FRQs use experimental contexts where you have to describe or justify a design. The 2022 FRQ Q2 used 36 pairs of identical twins in an acne drug experiment, which is the classic setup for comparing a matched pairs approach to a completely randomized one. The 2024 FRQ Q3 involved comparing mileage across four car models, A, B, C, and D. To earn credit, you typically need to:

1. Describe the random assignment process clearly. Say how units are numbered, what random process is used (random number generator, drawing slips), and how many units go to each treatment.
2. Justify the design choice. Tie it to the study's goals and to sources of variability (1.13.C).
3. State an appropriate conclusion. Random assignment supports cause and effect. Generalizing to a larger population requires random selection, not just random assignment (1.13.D).

A vague answer like "randomly assign the subjects" usually isn't enough. Be specific about the mechanism.

Completely Randomized Design vs Randomized Block Design

Both designs use random assignment, so they're easy to mix up. The difference is what happens before randomizing. In a completely randomized design, every unit goes into one pool and gets randomly assigned. In a randomized block design, you first split units into blocks of similar units based on a blocking variable, then randomly assign treatments within each block. If a problem mentions a variable you know affects the response (like soil quality or prior knowledge), that's your cue that blocking is the better choice.

Key things to remember about Completely Randomized Design

• In a completely randomized design, all experimental units are placed in one pool and treatments are assigned completely at random, with no grouping beforehand.

• Treatment groups in a completely randomized design are often the same size, but the CED says they don't have to be.

• Random assignment reduces the potential for confounding variables, which is what allows cause-and-effect conclusions between the explanatory and response variables.

• If there's a known source of variation, like a soil gradient or differences in prior knowledge, a randomized block or matched pairs design is usually more appropriate than a completely randomized design.

• Random assignment supports causation, but generalizing to a larger population requires random selection, so results from volunteers apply only to units similar to those volunteers.

• On FRQs, describe the randomization mechanism specifically (numbering units, using a random process, stating group sizes) instead of just saying "randomly assign."

Frequently asked questions about Completely Randomized Design

What is a completely randomized design in AP Stats?

It's an experimental design where treatments are assigned to experimental units completely at random, with all units drawn from one pool and no blocking. It's covered in Topic 1.13 (Experimental Design) in Unit 1, under learning objective 1.13.B.

Do the groups in a completely randomized design have to be the same size?

No. The CED (1.13.B.1) says group sizes are often equal, but they don't have to be. What makes it completely randomized is that assignment is fully random, not that the groups are balanced.

Does a completely randomized design let you generalize to the whole population?

Not by itself. Random assignment lets you conclude cause and effect, but generalizing requires that the units were randomly selected from the population. If the units are volunteers, the results apply only to units similar to those who participated (1.13.D.2).

What's the difference between a completely randomized design and a randomized block design?

A completely randomized design randomizes all units together in one group. A randomized block design first groups similar units into blocks using a blocking variable, then randomizes treatments within each block. Blocking is better when you know a variable, like a north-to-south soil gradient, affects the response.

When should you use a matched pairs design instead of a completely randomized design?

Use matched pairs when you can create very similar pairs or give each unit both treatments, like the 36 pairs of identical twins in the 2022 FRQ Q2 acne study. Pairing controls for differences between subjects that a completely randomized design would leave as random variation.