Does fixed mean there is no uncertainty anywhere in the table?

No. The cells can vary subject to the totals.

Are observed margins automatically fixed by design?

No. They may have been random before observation.

Can Fisher's exact analysis condition on random observed margins?

Yes; conditional treatment in the reference distribution is distinct from predetermining the margins during collection.

AI Glossary / Statistics and probability / Intermediate

Fixed marginal totals

Definition

Fixed marginal totals are row or column totals in a contingency table that are held constant by the sampling or allocation design, rather than varying across repetitions of that design. The source glossary emphasizes totals specified before data collection. In conditional statistical inference, analysts may also condition on observed margins, which is a different reason for treating them as fixed in the reference distribution. Fixing margins constrains the possible cell counts, but it does not fix every cell or guarantee independence between the table's variables.

What it means

A contingency table summarizes counts across categories. Its margins are the totals obtained by adding across rows or columns. If a study recruits a preset number from each group, its row totals may be fixed while its outcome totals remain random. In another design, both sets of totals may be fixed. These distinctions affect which tables could have arisen under repeated sampling and therefore which probability model is appropriate. Once observed, every total is a known number, but being observed is not the same as having been fixed by design.

How it works

Identify the unit being sampled, what the design predetermines and what remains random. A multinomial model can fix only the grand total, while product-multinomial models can fix selected group totals. In a two-by-two table with both row and column margins fixed, one cell count determines the other three. Under the appropriate independence null, its conditional distribution is hypergeometric. Conditional exact procedures evaluate tables sharing the observed margins even when those margins were not all predetermined. Report whether fixed means fixed by design or conditioned upon analytically, because those statements describe different aspects of the inference.

Three practical examples

Experimental allocation: assign a fixed number of units to each group, making group totals predetermined.
Survey design review: distinguish quotas or fixed group sample sizes from random counts of responses in outcome categories.
Exact inference: enumerate feasible contingency tables while keeping the specified marginal totals unchanged.

Think of it this way

Think of a table with totals written around its edges. Filling one cell uses up part of both a row allowance and a column allowance. When all edge totals are fixed, the remaining cells cannot be filled independently; each choice must respect the same bookkeeping constraints.

Worked example

Suppose ten hypothetical labeled items consist of seven type-A and three type-B items. A random allocation places six items in box one and four in box two. The resulting two-by-two table has fixed row totals six and four and fixed column totals seven and three. If a is the number of type-A items in box one, the cells are a, 6 - a, 7 - a and a - 3. Nonnegative counts require a to be three, four, five or six. Under uniform random allocation, their probabilities are choose(7,a) choose(3,6-a)/choose(10,6), giving 1/6, 1/2, 3/10 and 1/30 respectively. They sum to one, and the expected top-left count is 6 x 7/10 = 4.2. The margins stay fixed while the internal allocation varies. This is a design illustration, not evidence of an observed association.

Illustrative case study

Fictional case: Willow Survey Methods reviews a study that recruited fixed numbers from two customer groups. The report incorrectly states that all table margins were fixed, although the yes-or-no outcome totals were unknown until responses arrived. Analysts correct the design description and separately explain why a conditional exact analysis holds the observed outcome total fixed in its reference calculation. This invented case illustrates terminology precision without claiming that conditioning on an observed margin makes the original description of the sampling design true.

Common misconceptions and mistakes

Do not equate fixed margins with fixed individual cell counts. Do not assume all margins are fixed merely because the final table has no missing entries. A fixed total sample size does not fix each row or column total. Conditional inference and design constraints should be distinguished even when they lead to the same computational table set. The existence of fixed margins does not correct selection bias, clustered observations or measurement errors. Quota sampling can produce predetermined counts without providing the probability-sampling basis needed for broader population inference.

Business use

Businesses evaluating categorical data should document recruitment and allocation rules before choosing a test. State what could vary if the study were repeated, not only which totals appear in the finished spreadsheet. This helps analysts select suitable likelihoods and interpret exact procedures. Retain counts rather than only percentages so the constraints remain visible. Clear design documentation also prevents treating a table assembled from selected records as though it were a random sample from the entire customer population.

Check your understanding

Why can the worked example's top-left cell not equal two?

Answer: Box one would then require four type-B items to reach its total of six, but only three type-B items exist.

Related terms

Contingency table
Hypergeometric distribution
Fisher exact test
Sampling design
conditional inference
multinomial sampling
product-multinomial sampling

Learn next

compare repeated-sampling schemes that fix only the grand total, fix group totals or fix both margins, then derive the feasible cells and null distribution for a small table.

Null distribution

Sources

Sources: University of Vermont glossary, Fixed marginal totals: https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html ; SciPy, fisher_exact, conditional hypergeometric tables and observed margins: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.fisher_exact.html ; Penn State, categorical-data sampling schemes: https://online.stat.psu.edu/stat504/Lesson03 .

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