Chi-Square Testing of Genetic Crosses

Real crosses never land on perfect ratios. A 3-to-1 cross with 100 offspring almost never gives exactly 75 and 25, because fertilization is random and sample sizes are small. The question isn't whether the numbers match exactly, it's whether they're close enough to chance alone. Chi-square gives a way to measure that closeness with one number instead of eyeballing it.

The test compares observed counts to the counts you'd expect if the proposed ratio were true. For each phenotype category, you find observed minus expected, square it so negatives don't cancel positives, and divide by expected to scale the error relative to the category's size. Add those values across all categories and you get the chi-square statistic. A small chi-square means the observed data sits close to the prediction. A large chi-square means the gap is big enough that chance alone is a shaky explanation.

To judge 'big enough,' you need degrees of freedom, which equals the number of phenotype categories minus one. That number points you to a critical value in a chi-square table. If your calculated chi-square exceeds the critical value, you reject the null hypothesis that the proposed ratio is correct. If it doesn't exceed the critical value, you fail to reject the null, but that's not proof the ratio is right. It just means this sample didn't give you strong enough evidence against it. This connects directly to meiosis: the ratios you're testing, like 3:1 or 9:3:3:1, come straight from independent assortment and segregation during gamete formation.