Count-based tests compare observed outcomes with the pattern expected under the null model.
ObservedExpected if H₀difference scaled by uncertainty
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Statistics result
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Chi-Square Test of Independence. An omnibus statistic asks whether all modeled groups, cells, conditions, or spreads follow one joint null statement.
1Partition or compare variation2Scale by null-model uncertainty3Use planned follow-up for location
Test statistic and reference model
Eij=rowᵢ total·columnⱼ total/N; χ²=Σ(O−E)²/E.
Cramér’s V describes association strength; neither statistic establishes causation.
Responsible interpretation
A small p-value says that some modeled difference exists; it does not identify which comparisons differ. Follow-up work needs multiplicity control and effect estimates.
Required assumption: Counts must be independent and mutually exclusive; sparse expected cells can require exact or consolidated analysis.
Quick guide
How to use this calculator
Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
Read the result together with its assumptions and interpretation. Statistical output summarizes uncertainty under a model; it does not repair biased data or establish causation.
Calculation method
How the chi-square test of independence calculator works
Eij=rowᵢ total·columnⱼ total/N; χ²=Σ(O−E)²/E.
Cramér’s V describes association strength; neither statistic establishes causation.
Worked example
Chi-Square Test of Independence example
Rows entered on separate lines form a contingency table with expected counts and an association test.
Eij=rowᵢ total·columnⱼ total/N; χ²=Σ(O−E)²/E.
Supported inputs
Precision and limits
Model and design
Counts must be independent and mutually exclusive; sparse expected cells can require exact or consolidated analysis.
Input scope
Category lists accept up to 10,000 nonblank observations and render up to 200 distinct labels. Matching trims outer whitespace but remains case-sensitive; no numeric ordering or distance is inferred.
Interpretation
Cramér’s V describes association strength; neither statistic establishes causation.
Decision boundary
The calculator does not validate how data were collected, diagnose dependence or bias, choose a scientifically meaningful effect, or replace review by a qualified statistician for consequential research, medical, regulatory, safety, or policy decisions.
Privacy
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