Chi-Square Goodness of Fit

Overview

The Chi-Square Goodness of Fit test determines whether observed categorical data matches an expected distribution. Whether you’re testing if a die is fair, if color preferences follow a uniform distribution, or if demographic data matches census proportions, this test quantifies how well your observations align with theoretical expectations. Watch as the chi-square statistic measures the discrepancy between what you observe and what you expect.

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Tips

• The chi-square statistic Σ(O-E)²/E sums squared standardized deviations across all categories
• Larger chi-square values indicate greater discrepancy between observed and expected frequencies
• Each category contributes to the total chi-square - examine which categories deviate most
• Degrees of freedom = number of categories - 1 (or -k if estimating k parameters)
• The test requires expected frequencies ≥ 5 in each category for validity
• Try the fair die example to see how random variation affects the test
• Compare uniform vs. custom expected distributions to understand flexibility
• P-value < 0.05 suggests the data doesn’t fit the expected distribution
• Visual comparison of observed vs expected bars makes discrepancies obvious