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For stricter tests use α = 0.01, which raises the bar (for df = 1, the 0.01 critical value is 6.63).","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Interpreting a Single Term","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"On its own, a per-category value isn't a significance test — that job belongs to the sum. 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Those are the categories where your observed data disagrees most with your model — and usually the ones worth writing about.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"When Per-Term Values Matter Most","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"There are three situations where calculating one chi-square term at a time isn't just convenient — it's the right tool:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Learning the formula.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Running a grid-based calculator hides the mechanics. Doing it term by term builds real intuition for why squaring matters, why division by E matters, and what makes a contribution \"big.\"","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Post-hoc analysis.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" When a full chi-square test rejects the null, per-term values tell you which categories drove the rejection. Reporting \"χ² = 24.3, p < .001, with the bulk of the deviation concentrated in categories A and D\" is far more informative than reporting just the summary.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Checking hand calculations.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Textbook problems and exam answers are graded term by term. Verifying each contribution individually catches arithmetic mistakes that a summary-only tool would bury.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"number","start":1,"tag":"ol"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Technical Notes","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Formula:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" χ² = (observed − expected)² / expected","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"This is one component of the full test statistic χ² = Σ [(Oᵢ − Eᵢ)² / Eᵢ], summed across all categories in your data.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Assumptions:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Observations are independent","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Data are counts (frequencies), not percentages or proportions","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Expected counts are generally 5 or larger per category","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Categories are mutually exclusive","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":4}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"When to use a different test:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" If expected counts are very small, use Fisher's exact test. If your data are continuous, use a t-test or ANOVA. If you're comparing two proportions directly, a z-test for proportions is often simpler.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Historical note:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" The chi-square test was introduced by Karl Pearson in 1900 and is one of the oldest tools in inferential statistics. Despite its age, it remains the default test for categorical count data in fields ranging from genetics to market research — because the logic is transparent and the math is easy to verify by hand, one term at a time.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}},"id":"69ddddaa87f33a000403bdcf"}]}]}],"$L34"]}],"$L35"]}]]}]}]}]
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34:["$","div",null,{"className":"bg-white rounded-lg shadow-sm p-8","children":[["$","h2",null,{"className":"typo-large mb-6 text-mist-950","children":"Frequently Asked Questions"}],["$","div",null,{"className":"space-y-6","children":[["$","div","0",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Does this calculator give me a p-value?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"No — it returns a single chi-square term. A p-value requires the full test statistic (summed across all categories) and your degrees of freedom, then a chi-square distribution lookup.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","1",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What are degrees of freedom?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"For a goodness-of-fit test, it's the number of categories minus 1. For a contingency table (test of independence), it's (rows − 1) × (columns − 1). Degrees of freedom tell the chi-square distribution how spread out the statistic should be by chance alone.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","2",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Why divide by the expected value?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Dividing by E keeps contributions comparable across categories of different sizes. A gap of 10 means one thing when you expected 10 and something very different when you expected 1,000. Scaling by E reflects that.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","3",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Why is the difference squared?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Squaring makes every term positive — so overestimates and underestimates both count as deviations — and it gives larger gaps disproportionately more weight, which matches how \"surprising\" data actually feels.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","4",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Can chi-square be negative?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"No. The numerator is squared and the denominator is a positive count, so every term is zero or positive.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","5",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What happens when the expected value is zero?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The formula breaks — you can't divide by zero. 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36:["$","div","6",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Do observed and expected both have to be whole numbers?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Observed counts are whole numbers because they're actual tallies. Expected values usually aren't — 65 die rolls means 10.833... expected per face. That's fine; the formula handles non-integer expected values without any adjustment.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
37:["$","div","7",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What's the difference between a goodness-of-fit test and a test of independence?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Goodness-of-fit compares one observed distribution to an expected distribution (\"is this die fair?\"). Test of independence compares observed counts in a two-way table to what you'd expect if two variables were unrelated (\"does treatment type depend on age group?\"). Both use the same per-cell formula — only the degrees-of-freedom calculation differs.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
38:["$","div","8",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"How big does chi-square need to be for significance?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Depends on your degrees of freedom and significance level. With df = 1 at α = 0.05, the critical value is 3.84. With df = 4 it's 9.49; with df = 10 it's 18.31. See the critical-values table above for common values.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
39:["$","div","9",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Why does my textbook use χ² for the statistic and the distribution?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Because they're the same thing. The test statistic is called chi-square because, under the null hypothesis, its sampling distribution follows the chi-square distribution with the appropriate degrees of freedom. That's what makes the critical-value lookup work.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
3a:["$","div","10",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What if my observed count is much larger than expected — is there a \"too significant\"?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"There isn't a ceiling, but extremely large single-term values (say, above 20–30 when others are tiny) often point to a data-entry error, a miscalculated expected value, or a category that shouldn't have been included. Worth double-checking before writing up results.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
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