Three independently sampled branches classify customer feedback as favorable, neutral or unfavorable. The question is whether the feedback distribution is the same across branches. Unlike sampling one population and recording two attributes, the design begins with separate groups and measures one common categorical response.
Predict
Predict whether the expected-count formula changes when the question is homogeneity rather than independence. The table arithmetic can match, but the scientific target and source of randomness differ. Also predict whether an omnibus rejection proves that every branch differs from every other branch.
Experiment
Use this original synthetic two-group example, with group sizes 40 and 60.
Group
A
B
C
Total
1 observed
12
18
10
40
2 observed
18
22
20
60
Column total
30
40
30
100
Group 1 expected
12
16
12
40
Group 2 expected
18
24
18
60
Compute contributions: 0, 4/16, 4/12, 0, 4/24, 4/18. Their sum is 35/36≈0.9722. Degrees of freedom are (2−1)(3−1)=2. Without a supplied reference tail, report the statistic and conditions rather than inventing a p-value.
Understand
The homogeneity null states that the population proportions across all response categories are the same in every group. Group sample sizes can differ; equal proportions do not imply equal raw counts. Under the null, pooled category proportions estimate the common response distribution, and expected counts multiply those proportions by each group’s sample size.
The usual statistic and degrees of freedom match the independence procedure for the same table: χ²=Σ(O−E)²/E and df=(r−1)(c−1). The interpretation differs because the design differs. A homogeneity study may independently sample several populations or randomly assign treatments and compare categorical outcomes. Generalization and causal claims follow the actual sampling and assignment, not the label on the formula.
Require independent units within and across groups, consistent mutually exclusive response categories and adequate expected counts. If the same participants are surveyed repeatedly, ordinary homogeneity calculations ignore dependence. If categories mean different things across groups, mathematical comparison may be precise but scientifically incoherent. Harmonize the measurement rule before comparing distributions.
An omnibus rejection means at least one aspect of the response distribution differs; it does not establish that every category or pair of groups differs. Follow-up comparisons should consider multiplicity and the planned analysis. Inspect conditional proportions and substantive effect sizes, not just the largest raw count or contribution.
If the test does not reject, do not claim all distributions are identical. A small study may lack power for meaningful differences, and a nonrejection is not an equivalence analysis. Report the information supplied by the data, the range of plausible effects where assessed, and the limitations of the design.
The introductory chi-square approximation may be poor with sparse expected counts. Merging categories is defensible only when it has substantive meaning and its analytical consequences are acknowledged. Adjusting the table repeatedly until a preferred p-value appears undermines the nominal error rate.
Apply
Write two study descriptions that could produce a table with the same dimensions: one independence design and one homogeneity design. State each null in words. Then explain why a significant result across four populations would motivate further planned investigation rather than a blanket statement that every population pair differs.
Continue
Next change from categorical responses to quantitative means. Estimating unknown population spread leads to a t reference rather than the usual known-spread normal reference.
Apply · explain · revise
Practice and reasoning
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Two independently sampled groups have sizes 80 and 120. A category totals 50 across both. Find its expected count in the first group under homogeneity.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Three groups are compared across five mutually exclusive outcome categories. Find chi-square degrees of freedom.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which design naturally asks a homogeneity question?
Rejecting homogeneity for observational populations means:
Explain why independence and homogeneity tests can have identical χ² calculations but different contextual hypotheses.
Structure your argument and preserve revisions
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An omnibus test across four groups is significant. Plan a responsible follow-up without declaring every group different.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
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