# Chi-square goodness of fit This test asks whether observed counts match the counts a model predicts. The statistic is the sum of (observed - expected) squared / expected, with k - 1 degrees of freedom for k categories. Statistics (university) · Part 07: Comparing categories and many groups · Card 1 of 8 · Last updated 2026-10-19 ## Analogy Checking a die by rolling it many times and comparing each face's count with the fair share. ## Example **Is the die fair?** 60 rolls, expected 10 per face Observed: 5, 8, 9, 8, 10, 20 Sum of (O-E)^2/E = (25+4+1+4+0+100)/10 = 13.4 df = 5, critical 11.07, p = 0.02 ## In practice Tests whether sales are equal across days, or whether genetics data follow a predicted ratio. ## Practice ### Flashcards - **Chi-square statistic formula?** Sum of (observed - expected) squared / expected. ## Questions people ask ### How do I read the chi-square statistic? Each term measures how far a count is from its expected value, scaled by that expected value. A large total means the data is far from the model. Compare it with the chi-square distribution for the right degrees of freedom to get a p-value. ### Can the chi-square statistic be negative? No. Every term is a squared difference divided by a positive expected count, so the sum is zero or larger. Zero means the observed counts match the expected exactly. Larger values mean worse fit, so only the upper tail matters. ## Related concepts - [Expected counts](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-7/statistics-university-part-7-concept-2) - [Chi-square test of independence](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-7/statistics-university-part-7-concept-3) Web page: https://studyvisually.com/learn/science/statistics-university/statistics-university-part-7/statistics-university-part-7-concept-1 Study this card in the app: https://studyvisually.com/app/learn?category=statistics-university&card=statistics-university-part-7-concept-1