Must group sizes be equal?

No. They can be specified differently, although allocation affects precision.

Does randomization guarantee identical groups?

No. It balances characteristics through the randomization distribution, not perfectly in every realized assignment.

Is a random sample also a randomized experiment?

Not necessarily. Sampling and treatment assignment are different operations.

AI Glossary / Statistics and probability / Intermediate

Completely randomized design

Definition

A completely randomized design is an experimental design in which treatments are assigned to experimental units using a random allocation procedure without first grouping the units into blocks. The design supports comparisons among treatment groups by making assignment independent of unit characteristics through the randomization mechanism. It does not mean haphazard allocation, equal outcomes, or guaranteed balance of every characteristic in the realized sample.

What it means

Experimental units are the entities independently assigned to treatments: people, machines, batches, or other units appropriate to the study. In a completely randomized design, the allocation is drawn from the allowed assignments according to a defined chance procedure.
The number of units assigned to each treatment may be fixed in advance. Randomization then determines which units receive each treatment. This is compatible with equal group sizes, but equal sizes are not the defining feature. The important distinction from blocking is that assignment is not restricted within prespecified groups of similar units.

How it works

Define treatments, experimental units, response measurements, and the comparison of interest. Confirm that the unit of assignment matches the unit used in analysis. Multiple measurements from one assigned unit do not automatically create additional independent experimental units.
Specify group sizes and generate the random allocation using a reproducible procedure. Protect the allocation process from discretionary changes based on anticipated outcomes. Arrange measurement and conduct to reduce unintended differences between groups.
Collect outcomes under the planned protocol. Record missing data, departures from assignment, and relevant context. Randomization protects against treatment-selection bias in the design, but it does not automatically solve measurement bias, interference between units, or selective loss of outcomes.
Analyze treatment contrasts using methods appropriate to the response and design. A one-way analysis of variance is a common model-based option for suitable continuous responses, with assumptions that need assessment. Randomization-based methods use the allocation mechanism directly. If strong nuisance structure is known beforehand, a blocked design may use resources more efficiently than complete randomization.

Three practical examples

A laboratory randomly assigns comparable experimental units to three processing methods and measures the same response for each.
A service experiment randomly allocates eligible requests to alternative workflows without creating prior blocks.
A training study assigns participants to programs according to a planned random allocation and compares outcomes using the assignment structure.

Think of it this way

Think of shuffling treatment labels and dealing them to a fixed set of units. Every allowed allocation follows the same declared chance rule. The shuffle prevents deliberate selection of favorable units, but a particular deal can still contain imbalances just by chance.

Worked example

Consider an invented experiment with six labeled units and two treatments, A and B. Exactly three units must receive A, and the remaining three receive B. There are C(6, 3) = 20 possible assignments, each selected with probability 1/20 under uniform complete randomization.
Suppose two of the six units share a particular baseline characteristic. The chance that both receive A is C(4, 1) / C(6, 3) = 4/20 = 0.20: after choosing those two, one additional A unit must be selected from the other four. By symmetry, the chance both receive B is also 0.20.
Therefore the chance the two special units are together in the same treatment group is 0.40. The chance they are split between A and B is 0.60. Randomization thus does not guarantee balance of that characteristic even though each treatment group has exactly three units.
Suppose the observed outcomes are 8, 10, and 12 in A and 5, 7, and 9 in B. Their means are 10 and 7, giving an observed difference of 3. That descriptive difference requires an appropriate uncertainty analysis; the small example alone does not establish a general treatment effect.

Illustrative case study

In this fictional case, Rowan Process Lab compares two settings on a set of experimental batches. Staff initially suggest assigning early batches to one setting and later batches to the other because it is convenient.
The team recognizes that time-related changes could then be mixed with treatment differences. It implements a documented random assignment and records the actual run conditions. When reviewing the results, it keeps repeated measurements within each batch separate from the number of independently assigned batches.
The report explains the allocation procedure and the experimental unit. It also notes whether future experiments should block on a known source of batch variation to improve precision.

Common misconceptions and mistakes

Completely randomized does not mean uncontrolled or careless. The randomization rule is deliberate and should be documented. Equal group sizes do not imply equal baseline characteristics, as the example shows.
Another mistake is confusing random assignment with random sampling. Assignment supports treatment comparisons among the studied units; sampling concerns how those units were selected from a population. Neither automatically supplies the other.
Treating repeated observations as independent units can exaggerate precision. Ignoring known clustering, interference, or changes in outcome measurement can also undermine conclusions even when the initial allocation was randomized.

Business use

The design can provide a clear and economical way to compare alternatives when units are reasonably comparable and no strong blocking factor needs to be controlled. It is useful for experiments with a manageable treatment structure and a clear unit of assignment.
Plan the allocation and analysis together. Track actual implementation, preserve outcome data regardless of whether results are convenient, and assess practical effect size alongside uncertainty. Consider blocking or another design when known variation would otherwise obscure the comparison.

Check your understanding

How many balanced assignments are possible in the example?

Answer: Twenty.

What is the chance both special units land together?

Answer: Forty percent.

Why are repeated measurements not automatically new experimental units?

Answer: They may share the same treatment assignment and underlying unit-level variation.

Related terms

Randomized blocks design
Treatment effect
Replication (statistics)
Analysis of variance
Confounding
random assignment
experimental unit
randomized block design
replication
randomization inference

Learn next

Learn next how blocking changes the allocation mechanism and analysis. Study the distinction between experimental and observational units, then compare model-based and randomization-based approaches to uncertainty.

Sources

NIST/SEMATECH e-Handbook, Completely Randomized Designs: https://www.itl.nist.gov/div898/handbook/pri/section3/pri331.htm NIST, Randomized Block Designs: https://www.itl.nist.gov/div898/handbook/pri/section3/pri332.htm NIST, Accounting for Nested Variation: https://www.itl.nist.gov/div898/handbook/pri/section5/pri55.htm

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