Understand the hypothesis test
What an omnibus test can—and cannot—say
Chi-Square Goodness-of-Fit Test. 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
χ²=Σ(O−E)²/E, df=k−1−estimated parameters.
A small p-value indicates mismatch somewhere among categories but does not identify a substantive cause.
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: Categories must be exhaustive and independent; expected counts should support the chi-square approximation and fitted-parameter count must be declared.
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 goodness-of-fit test calculator works
χ²=Σ(O−E)²/E, df=k−1−estimated parameters.
A small p-value indicates mismatch somewhere among categories but does not identify a substantive cause.
Worked example
Chi-Square Goodness-of-Fit Test example
Observed die counts can be compared with six equal expected probabilities.
χ²=Σ(O−E)²/E, df=k−1−estimated parameters.
Supported inputs
Precision and limits
Model and design
Categories must be exhaustive and independent; expected counts should support the chi-square approximation and fitted-parameter count must be declared.
Numerical scope
Inputs use double-precision numerical methods with guarded domains. Datasets accept up to 10,000 finite plain-decimal values. Extremely large parameters or probabilities deep in a numerical tail may require specialist statistical software.
Interpretation
A small p-value indicates mismatch somewhere among categories but does not identify a substantive cause.
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
Entered values and calculated results stay in this browser and are not sent to an analytics service.
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