Concept Lab
Before calculating, understand the mechanics. The Chi-Square ($\chi^2$) test measures how expectations compare to actual observed data. Use the interactive graph below to see how the "Degrees of Freedom" ($df$) changes the probability distribution.
The Chi-Square Distribution
Notice how lower $df$ curves skew heavily to the left, while higher $df$ curves become more symmetrical (normal-like).
Null Hypothesis ($H_0$)
Assumes there is no relationship between the variables. Any difference is due to chance.
The Statistic
$\chi^2 = \sum \frac{(O - E)^2}{E}$
Sum of (Observed - Expected)² divided by Expected.
P-Value
The probability of seeing these results if $H_0$ were true.
P < 0.05 usually means "Significant".
The Analyst (Test of Independence)
Enter your raw count data below. This tool is perfect for analyzing Sample 1 vs Sample 2 scenarios (e.g., Treatment vs Control outcomes). It will automatically calculate the "Expected" values based on row/column totals.
| Group / Outcome | Totals |
|---|
Chi-Square Statistic
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P-Value
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Degrees of Freedom
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Conclusion
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Expected Frequencies
Values expected if groups were identical ($H_0$ is true).