Use observed counts rather than percentages, check expected frequencies, and separate statistical evidence from practical importance.
Pair the p-value with effect size and expected-frequency checks so significance is not interpreted alone.
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Capture the current inputs, analysis settings and visible result as stable fingerprints, then flag structural method-fit issues that can be checked from the entered data. This complements the page’s existing mathematical audit; it does not prove distributional assumptions or causal validity.
Use descriptive context first, choose the method that matches the design, then keep inputs, settings and result evidence reproducible instead of copying a p-value without its analysis recipe.
Use plots and descriptive summaries before formal tests. Check the sampling/design assumptions that matter for the chosen method, report effect size and uncertainty where available, and avoid treating a threshold such as p < .05 as proof of importance, causation, or truth. Browser calculations are educational/planning utilities, not domain-specific professional advice.
Use the interactive product above first. These notes add interpretation, verification and limits below the results and product controls.
A chi-square statistic can be computed even when cells are sparse, but the usual reference distribution may be a poor approximation when expected counts are very small. Inspect the expected-count audit before interpreting the p-value.
Goodness-of-fit compares one observed category distribution with specified expected proportions or counts. A contingency-table independence test asks whether two categorical variables are associated. Use the mode that matches the sampling design.
Statistical significance can appear with a large sample even when the practical difference is modest. Pair the p-value with the effect-size evidence and the observed-versus-expected pattern rather than reporting the test statistic alone.
The total chi-square statistic is built from cell-level discrepancies. When the test is significant, inspect which observed counts differ most from expectation instead of describing every category as equally responsible.
Repeated measurements from the same subject, overlapping categories or percentages entered in place of counts can violate the model. Verify that each observation contributes to the table in the way the chosen chi-square test assumes.
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